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Normal second parameter is variance; geometric counts trials from one. Q5.3b tight cycle lower bound corrected to max(3,c). Q8.2 prime field larger than degrees specified. Balls/bin sampling explicitly independent uniform. Markov diagrams verified on blank pages20–21. 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Composite code/transition blanks exact-match." ], "included": [ { "id": "2.1a", "source_parts": [ "2.1a" ], "repairs": [] }, { "id": "2.1b", "source_parts": [ "2.1b" ], "repairs": [] }, { "id": "2.1c", "source_parts": [ "2.1c" ], "repairs": [] }, { "id": "2.2a", "source_parts": [ "2.2a" ], "repairs": [] }, { "id": "2.2b", "source_parts": [ "2.2b" ], "repairs": [] }, { "id": "2.2c", "source_parts": [ "2.2c" ], "repairs": [] }, { "id": "2.3", "source_parts": [ "2.3" ], "repairs": [] }, { "id": "3.2", "source_parts": [ "3.2" ], "repairs": [] }, { "id": "3.3a", "source_parts": [ "3.3a" ], "repairs": [] }, { "id": "3.3b", "source_parts": [ "3.3b" ], "repairs": [] }, { "id": "4.1a", "source_parts": [ "4.1a" ], "repairs": [] }, { "id": "4.1b", "source_parts": [ "4.1b" ], "repairs": [] }, { "id": "4.2a", "source_parts": [ "4.2a" ], "repairs": [] }, { "id": "4.2b", "source_parts": [ "4.2b" ], "repairs": [] }, { "id": "5.1a", "source_parts": [ "5.1a" ], "repairs": [] }, { "id": "5.1b", "source_parts": [ "5.1b" ], "repairs": [] }, { "id": "5.1c", "source_parts": [ "5.1c" ], "repairs": [] }, { "id": "5.2a", "source_parts": [ "5.2a" ], "repairs": [] }, { "id": "5.2b", "source_parts": [ "5.2b" ], "repairs": [] }, { "id": "5.3", "source_parts": [ "5.3" ], "repairs": [] }, { "id": "6.1", "source_parts": [ "6.1" ], "repairs": [] }, { "id": "6.2", "source_parts": [ "6.2" ], "repairs": [] }, { "id": "6.3", "source_parts": [ "6.3" ], "repairs": [] }, { "id": "6.4a", "source_parts": [ "6.4a" ], "repairs": [] }, { "id": "6.4b", "source_parts": [ "6.4b" ], "repairs": [] }, { "id": "6.4c", "source_parts": [ "6.4c" ], "repairs": [] }, { "id": "6.4d", "source_parts": [ "6.4d" ], "repairs": [] }, { "id": "6.4e", "source_parts": [ "6.4e" ], "repairs": [] }, { "id": "6.4f", "source_parts": [ "6.4f" ], "repairs": [] }, { "id": "6.5", "source_parts": [ "6.5" ], "repairs": [ "Exclude N=1 and strengthen to iff: the original one-way implication also permits vacuously true distractors." ] }, { "id": "6.6", "source_parts": [ "6.6" ], "repairs": [] }, { "id": "7.1", "source_parts": [ "7.1" ], "repairs": [] }, { "id": "7.2a", "source_parts": [ "7.2a" ], "repairs": [] }, { "id": "7.2b", "source_parts": [ "7.2b" ], "repairs": [] }, { "id": "7.3a", "source_parts": [ "7.3a" ], "repairs": [] }, { "id": "7.3b", "source_parts": [ "7.3b" ], "repairs": [] }, { "id": "7.3c", "source_parts": [ "7.3c" ], "repairs": [] }, { "id": "8.1", "source_parts": [ "8.1" ], "repairs": [] }, { "id": "8.2", "source_parts": [ "8.2" ], "repairs": [] }, { "id": "8.3a", "source_parts": [ "8.3a" ], "repairs": [] }, { "id": "8.3b", "source_parts": [ "8.3b" ], "repairs": [] }, { "id": "8.4", "source_parts": [ "8.4" ], "repairs": [] }, { "id": "9.1", "source_parts": [ "9.1" ], "repairs": [] }, { "id": "9.2", "source_parts": [ "9.2" ], "repairs": [] }, { "id": "9.3", "source_parts": [ "9.3" ], "repairs": [] }, { "id": "10.1", "source_parts": [ "10.1" ], "repairs": [] }, { "id": "10.2", "source_parts": [ "10.2" ], "repairs": [] }, { "id": "10.3", "source_parts": [ "10.3" ], "repairs": [] }, { "id": "11.1", "source_parts": [ "11.1", "11.2", "11.3", "11.4" ], "repairs": [] }, { "id": "12.1a", "source_parts": [ "12.1a" ], "repairs": [] }, { "id": "12.1b", "source_parts": [ "12.1b" ], "repairs": [] }, { "id": "12.1c", "source_parts": [ "12.1c" ], "repairs": [] }, { "id": "12.1d", "source_parts": [ "12.1d" ], "repairs": [] }, { "id": "12.2a", "source_parts": [ "12.2a" ], "repairs": [] }, { "id": "12.2bi", "source_parts": [ "12.2bi" ], "repairs": [] }, { "id": "12.2bii", "source_parts": [ "12.2bii" ], "repairs": [] }, { "id": "12.2biii", "source_parts": [ "12.2biii" ], "repairs": [] }, { "id": "12.3a", "source_parts": [ "12.3a" ], "repairs": [] }, { "id": "12.3b", "source_parts": [ "12.3b" ], "repairs": [] }, { "id": "12.3c", "source_parts": [ "12.3c" ], "repairs": [] }, { "id": "12.4a", "source_parts": [ "12.4a" ], "repairs": [] }, { "id": "12.4b", "source_parts": [ "12.4b" ], "repairs": [] }, { "id": "13.1a", "source_parts": [ "13.1a" ], "repairs": [] }, { "id": "13.1b", "source_parts": [ "13.1b" ], "repairs": [] }, { "id": "13.2", "source_parts": [ "13.2" ], "repairs": [ "Correct official 1/p key: >= retains a possible zero residual; 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Q6 uses n>=8 to retain intended eight-color result and corrects expectation index to n−1. Q10 path/cycle orientations explicit. Q12 CLT is an approximation, not a rigorous upper bound; Chebyshev question specifies direct two-sided application. Q14 count independent of lifetimes explicit. Q21 independent uniform allocations, n>=4. 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Native alternatives retained. Q12 grid is fully defined as a cycle; illustrative drawing adds no data. Q14 directed diagram visually checked on blank page17. Q13 makes Poisson arrival-process assumption explicit: mean spacing and informal independent arrivals alone do not specify a Poisson process. Q9 statements preserve available moment information. 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Course-context birthday warmup retained with its original cross-exam clue, separately tagged. Source drawing of the pepper chain visually checked on blank page 21. Tree drawings represented by edge sets. Geometric support uses trials 1,2,...; normal variance convention standard. Q13.1 clips the union bound at 1; Q13.5b/c correct an incomplete printed sharp lower bound by considering both B and its complement. Finite second moments and positive regression variance explicit. 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Original page 11 complements and page 15 lattice diagram visually checked. Q8.1 fixes degree ambiguity to at most 3 (officially 100 and 125 both accepted). Q8.3a asks maximum supported message length. Q9.6 fixes starting checkpoint and distinguishes traversal directions. Q15 composite moment boxes scored jointly. Q16.2c no continuity correction, explicitly an approximation. Q18 requires independent arrivals. Finite second moments assumed in variance proof." ], "included": [ { "id": "3.1", "source_parts": [ "3.1" ], "repairs": [] }, { "id": "3.2", "source_parts": [ "3.2" ], "repairs": [] }, { "id": "3.3", "source_parts": [ "3.3" ], "repairs": [] }, { "id": "3.4", "source_parts": [ "3.4" ], "repairs": [] }, { "id": "3.5", "source_parts": [ "3.5" ], "repairs": [] }, { "id": "3.6", "source_parts": [ "3.6" ], "repairs": [] }, { "id": "4.1", "source_parts": [ "4.1" ], "repairs": [] }, { "id": "4.2", "source_parts": [ "4.2" ], "repairs": [] }, { "id": "5.1", "source_parts": [ "5.1" ], "repairs": [] }, { "id": "5.2", "source_parts": [ "5.2" ], "repairs": [] }, { "id": "5.3", "source_parts": [ "5.3" ], "repairs": [] }, { "id": "5.4a", "source_parts": [ "5.4a" ], "repairs": [ "Require positive dimension to exclude a single vertex." ] }, { "id": "5.4b", "source_parts": [ "5.4b" ], "repairs": [] }, { "id": "5.4c", "source_parts": [ "5.4c" ], "repairs": [] }, { "id": "6.1", "source_parts": [ "6.1" ], "repairs": [] }, { "id": "6.2", "source_parts": [ "6.2" ], "repairs": [] }, { "id": "6.3", "source_parts": [ "6.3" ], "repairs": [] }, { "id": "6.4", "source_parts": [ "6.4" ], "repairs": [] }, { "id": "7.1a", "source_parts": [ "7.1a" ], "repairs": [] }, { "id": "7.1b", "source_parts": [ "7.1b" ], "repairs": [] }, { "id": "7.2a", "source_parts": [ "7.2a" ], "repairs": [] }, { "id": "7.2b", "source_parts": [ "7.2b" ], "repairs": [] }, { "id": "7.3a", "source_parts": [ "7.3a" ], "repairs": [] }, { "id": "7.3b", "source_parts": [ "7.3b" ], "repairs": [] }, { "id": "8.1", "source_parts": [ "8.1" ], "repairs": [ "Resolve source degree ambiguity as at most three; original grading accepted both strict and non-strict interpretations." ] }, { "id": "8.2a", "source_parts": [ "8.2a" ], "repairs": [] }, { "id": "8.2b", "source_parts": [ "8.2b" ], "repairs": [] }, { "id": "8.3a", "source_parts": [ "8.3a" ], "repairs": [ "Ask maximum length; shorter messages also meet the stated guarantee." ] }, { "id": "8.3b", "source_parts": [ "8.3b" ], "repairs": [ "Specify minimal locator to exclude padding roots." ] }, { "id": "9.1", "source_parts": [ "9.1" ], "repairs": [] }, { "id": "9.2", "source_parts": [ "9.2" ], "repairs": [] }, { "id": "9.3", "source_parts": [ "9.3" ], "repairs": [] }, { "id": "9.4", "source_parts": [ "9.4" ], "repairs": [] }, { "id": "9.5", "source_parts": [ "9.5" ], "repairs": [] }, { "id": "9.6", "source_parts": [ "9.6" ], "repairs": [ "Fix the starting checkpoint and distinguish reverse traversal, matching the source’s intended count." ] }, { "id": "9.7", "source_parts": [ "9.7" ], "repairs": [] }, { "id": "10.1", "source_parts": [ "10.1" ], "repairs": [] }, { "id": "10.2", "source_parts": [ "10.2" ], "repairs": [] }, { "id": "10.3", "source_parts": [ "10.3" ], "repairs": [] }, { "id": "10.4", "source_parts": [ "10.4" ], "repairs": [] }, { "id": "10.5", "source_parts": [ "10.5" ], "repairs": [] }, { "id": "10.6", "source_parts": [ "10.6" ], "repairs": [] }, { "id": "11.1", "source_parts": [ "11.1" ], "repairs": [] }, { "id": "11.2", "source_parts": [ "11.2" ], "repairs": [] }, { "id": "11.3", "source_parts": [ "11.3" ], "repairs": [] }, { "id": "11.4", "source_parts": [ "11.4" ], "repairs": [] }, { "id": "11.5", "source_parts": [ "11.5" ], "repairs": [] }, { "id": "11.6", "source_parts": [ "11.6" ], "repairs": [] }, { "id": "11.7", "source_parts": [ "11.7" ], "repairs": [] }, { "id": "12.1", "source_parts": [ "12.1" ], "repairs": [] }, { "id": "12.2a", "source_parts": [ "12.2a" ], "repairs": [] }, { "id": "12.2b", "source_parts": [ "12.2b" ], "repairs": [] }, { "id": "12.3", "source_parts": [ "12.3" ], "repairs": [] }, { "id": "13.1", "source_parts": [ "13.1" ], "repairs": [] }, { "id": "13.2", "source_parts": [ "13.2" ], "repairs": [] }, { "id": "13.3", "source_parts": [ "13.3" ], "repairs": [] }, { "id": "13.4a", "source_parts": [ "13.4a" ], "repairs": [] }, { "id": "13.4b", "source_parts": [ "13.4b" ], "repairs": [] }, { "id": "13.4c", "source_parts": [ "13.4c" ], "repairs": [] }, { "id": "13.4d", "source_parts": [ "13.4d" ], "repairs": [] }, { "id": "14.1", "source_parts": [ "14.1" ], "repairs": [] }, { "id": "14.2", "source_parts": [ "14.2" ], "repairs": [] }, { "id": "15.1a", "source_parts": [ "15.1a.EX", "15.1a.EY", "15.1a.VarY", "15.1a.Cov" ], "repairs": [] }, { "id": "15.1b", "source_parts": [ "15.1b" ], "repairs": [] }, { "id": "15.1c", "source_parts": [ "15.1c" ], "repairs": [] }, { "id": "15.1d", "source_parts": [ "15.1d.EX", "15.1d.EY", "15.1d.VarY", "15.1d.Cov" ], "repairs": [] }, { "id": "15.1e", "source_parts": [ "15.1e.EX", "15.1e.EY", "15.1e.VarY", "15.1e.Cov" ], "repairs": [] }, { "id": "16.1a", "source_parts": [ "16.1a" ], "repairs": [] }, { "id": "16.1b", "source_parts": [ "16.1b" ], "repairs": [] }, { "id": "16.1c", "source_parts": [ "16.1c" ], "repairs": [] }, { "id": "16.2a", "source_parts": [ "16.2a" ], "repairs": [] }, { "id": "16.2b", "source_parts": [ "16.2b" ], "repairs": [] }, { "id": "16.2c", "source_parts": [ "16.2c" ], "repairs": [ "Explicitly request no continuity correction; this is a tail approximation, not an exact probability." ] }, { "id": "17.1", "source_parts": [ "17.1" ], "repairs": [] }, { "id": "17.2", "source_parts": [ "17.2" ], "repairs": [] }, { "id": "17.3", "source_parts": [ "17.3.c", "17.3.expected" ], "repairs": [] }, { "id": "17.4", "source_parts": [ "17.4.c", "17.4.expected" ], "repairs": [] }, { "id": "17.5", "source_parts": [ "17.5" ], "repairs": [] }, { "id": "18.1a", "source_parts": [ "18.1a" ], "repairs": [ "State independence, required for the source formulas." ] }, { "id": "18.1b", "source_parts": [ "18.1b" ], "repairs": [] }, { "id": "18.1c", "source_parts": [ "18.1c" ], "repairs": [] }, { "id": "18.1d", "source_parts": [ "18.1d" ], "repairs": [] }, { "id": "18.2a", "source_parts": [ "18.2a" ], "repairs": [ "State independence explicitly." ] }, { "id": "18.2b", "source_parts": [ "18.2b" ], "repairs": [] }, { "id": "18.2c", "source_parts": [ "18.2c" ], "repairs": [] }, { "id": "19.1", "source_parts": [ "19.1" ], "repairs": [] }, { "id": "20.1", "source_parts": [ "20.1" ], "repairs": [] }, { "id": "20.2", "source_parts": [ "20.2" ], "repairs": [] }, { "id": "20.3", "source_parts": [ "20.3" ], "repairs": [] }, { "id": "20.4", "source_parts": [ "20.4" ], "repairs": [] }, { "id": "20.5", "source_parts": [ "20.5" ], "repairs": [] } ], "not_scored": [ { "source_part": "1", "status": "excluded", "reason": "Administrative pledge." }, { "source_part": "2", "status": "withheld", "reason": "Self-reference to unavailable classroom responses." } ] }, "midterm_fa23": { "items": 62, "notes": [ "Exclude pledge Q1. Withhold self-referential classroom-answer gcd warmup Q2 from mathematical benchmark. Q3 ratios require nonzero denominator. Q9.2c uses unit group of size q(q−1), not the whole residue set mistakenly referenced in solution prose. Q10.2 means for each fixed i a possibly different a exists; no single a≠1 can satisfy every distinct prime exponent. Q10.3 uses the published corrected question, ungraded on original exam. Q12 division assumes dividend degree at least divisor degree. Grouped Q13 blanks use exact tuple scoring." ], "included": [ { "id": "3.1a", "source_parts": [ "3.1a" ], "repairs": [] }, { "id": "3.1b", "source_parts": [ "3.1b" ], "repairs": [] }, { "id": "3.2", "source_parts": [ "3.2" ], "repairs": [] }, { "id": "3.3", "source_parts": [ "3.3" ], "repairs": [] }, { "id": "3.4", "source_parts": [ "3.4" ], "repairs": [] }, { "id": "3.5", "source_parts": [ "3.5" ], "repairs": [] }, { "id": "4.1", "source_parts": [ "4.1" ], "repairs": [] }, { "id": "4.2", "source_parts": [ "4.2" ], "repairs": [] }, { "id": "4.3", "source_parts": [ "4.3" ], "repairs": [] }, { "id": "5.1", "source_parts": [ "5.1" ], "repairs": [] }, { "id": "5.2", "source_parts": [ "5.2" ], "repairs": [] }, { "id": "6.1", "source_parts": [ "6.1" ], "repairs": [] }, { "id": "6.2", "source_parts": [ "6.2" ], "repairs": [] }, { "id": "6.3", "source_parts": [ "6.3" ], "repairs": [] }, { "id": "6.4", "source_parts": [ "6.4" ], "repairs": [] }, { "id": "7.1", "source_parts": [ "7.1" ], "repairs": [] }, { "id": "7.2", "source_parts": [ "7.2" ], "repairs": [] }, { "id": "7.3a", "source_parts": [ "7.3a" ], "repairs": [] }, { "id": "7.3b", "source_parts": [ "7.3b" ], "repairs": [] }, { "id": "7.3c", "source_parts": [ "7.3c" ], "repairs": [] }, { "id": "7.4", "source_parts": [ "7.4" ], "repairs": [] }, { "id": "7.5", "source_parts": [ "7.5" ], "repairs": [] }, { "id": "7.6", "source_parts": [ "7.6" ], "repairs": [] }, { "id": "8.1a", "source_parts": [ "8.1a" ], "repairs": [] }, { "id": "8.1b", "source_parts": [ "8.1b" ], "repairs": [] }, { "id": "8.1c", "source_parts": [ "8.1c" ], "repairs": [] }, { "id": "8.1di", "source_parts": [ "8.1di" ], "repairs": [] }, { "id": "8.1dii", "source_parts": [ "8.1dii" ], "repairs": [] }, { "id": "8.1diii", "source_parts": [ "8.1diii" ], "repairs": [] }, { "id": "8.2", "source_parts": [ "8.2" ], "repairs": [] }, { "id": "9.1", "source_parts": [ "9.1" ], "repairs": [] }, { "id": "9.2a", "source_parts": [ "9.2a" ], "repairs": [] }, { "id": "9.2b", "source_parts": [ "9.2b" ], "repairs": [] }, { "id": "9.2c", "source_parts": [ "9.2c" ], "repairs": [ "Use an explicit exponent >1 and the correct unit-group cardinality." ] }, { "id": "9.3", "source_parts": [ "9.3" ], "repairs": [] }, { "id": "9.4a", "source_parts": [ "9.4a" ], "repairs": [] }, { "id": "9.4b", "source_parts": [ "9.4b" ], "repairs": [] }, { "id": "9.4c", "source_parts": [ "9.4c" ], "repairs": [] }, { "id": "9.5", "source_parts": [ "9.5" ], "repairs": [] }, { "id": "9.6a", "source_parts": [ "9.6a" ], "repairs": [ "Remove irrelevant x≠0 restriction; original arbitrary a could otherwise exclude the purported first k." ] }, { "id": "9.6b", "source_parts": [ "9.6b" ], "repairs": [] }, { "id": "10.1", "source_parts": [ "10.1" ], "repairs": [ "Require n>=1; n=0 makes the claim false." ] }, { "id": "10.2", "source_parts": [ "10.2" ], "repairs": [ "Clarify quantifiers: a may depend on i." ] }, { "id": "10.3", "source_parts": [ "10.3" ], "repairs": [ "Use corrected published modulus p; original exam typo caused this part to be ungraded." ] }, { "id": "11.1", "source_parts": [ "11.1" ], "repairs": [] }, { "id": "12.1", "source_parts": [ "12.1" ], "repairs": [] }, { "id": "12.2a", "source_parts": [ "12.2a" ], "repairs": [] }, { "id": "12.2bi", "source_parts": [ "12.2bi" ], "repairs": [ "Specify dP>=dQ, required for the nonzero quotient degree formula." ] }, { "id": "12.2bii", "source_parts": [ "12.2bii" ], "repairs": [] }, { "id": "12.2biii", "source_parts": [ "12.2biii" ], "repairs": [] }, { "id": "12.2biv", "source_parts": [ "12.2biv" ], "repairs": [] }, { "id": "12.3a", "source_parts": [ "12.3a" ], "repairs": [] }, { "id": "12.3b", "source_parts": [ "12.3b" ], "repairs": [] }, { "id": "13.1", "source_parts": [ "13.1a", "13.1b", "13.1c", "13.1d", "13.1e", "13.1f", "13.1g", "13.1h" ], "repairs": [] }, { "id": "13.2", "source_parts": [ "13.2a", "13.2b", "13.2c", "13.2d", "13.2e", "13.2f" ], "repairs": [] }, { "id": "14.1", "source_parts": [ "14.1" ], "repairs": [] }, { "id": "14.2", "source_parts": [ "14.2" ], "repairs": [] }, { "id": "14.3", "source_parts": [ "14.3" ], "repairs": [] }, { "id": "14.4", "source_parts": [ "14.4" ], "repairs": [] }, { "id": "14.5", "source_parts": [ "14.5" ], "repairs": [] }, { "id": "14.6", "source_parts": [ "14.6" ], "repairs": [] }, { "id": "15.1", "source_parts": [ "15.1" ], "repairs": [] } ], "not_scored": [ { "source_part": "1", "status": "excluded", "reason": "Administrative pledge." }, { "source_part": "2", "status": "withheld", "reason": "Self-referential gcd of unavailable classroom answers." } ] }, "midterm_fa24": { "items": 72, "notes": [ "Q1 pledge excluded. Q2 self-referential student-answer conjunction is unresolved without actual class responses, so withheld rather than assuming a student answered False. Q7.1g assumes at least two vertices; Q7.5 makes n-vertex premise explicit. Q12.3 distinct abscissas and Q14.7 distinct roots explicit. All polynomials treated formally." ], "included": [ { "id": "3.1a", "source_parts": [ "3.1a" ], "repairs": [] }, { "id": "3.1b", "source_parts": [ "3.1b" ], "repairs": [] }, { "id": "3.1c", "source_parts": [ "3.1c" ], "repairs": [] }, { "id": "3.2a", "source_parts": [ "3.2a" ], "repairs": [] }, { "id": "3.2b", "source_parts": [ "3.2b" ], "repairs": [] }, { "id": "4.1", "source_parts": [ "4.1" ], "repairs": [] }, { "id": "4.2", "source_parts": [ "4.2" ], "repairs": [] }, { "id": "5.1", "source_parts": [ "5.1" ], "repairs": [] }, { "id": "6.1a", "source_parts": [ "6.1a" ], "repairs": [] }, { "id": "6.1b", "source_parts": [ "6.1b" ], "repairs": [] }, { "id": "6.1c", "source_parts": [ "6.1c" ], "repairs": [] }, { "id": "6.2", "source_parts": [ "6.2" ], "repairs": [] }, { "id": "6.3", "source_parts": [ "6.3" ], "repairs": [] }, { "id": "6.4", "source_parts": [ "6.4" ], "repairs": [] }, { "id": "6.5", "source_parts": [ "6.5" ], "repairs": [] }, { "id": "7.1a", "source_parts": [ "7.1a" ], "repairs": [] }, { "id": "7.1b", "source_parts": [ "7.1b" ], "repairs": [] }, { "id": "7.1c", "source_parts": [ "7.1c" ], "repairs": [] }, { "id": "7.1d", "source_parts": [ "7.1d" ], "repairs": [] }, { "id": "7.1e", "source_parts": [ "7.1e" ], "repairs": [] }, { "id": "7.1f", "source_parts": [ "7.1f" ], "repairs": [] }, { "id": "7.1g", "source_parts": [ "7.1g" ], "repairs": [ "Require at least two vertices to exclude the one-vertex tree." ] }, { "id": "7.1h", "source_parts": [ "7.1h" ], "repairs": [] }, { "id": "7.2", "source_parts": [ "7.2" ], "repairs": [] }, { "id": "7.3", "source_parts": [ "7.3" ], "repairs": [] }, { "id": "7.4", "source_parts": [ "7.4" ], "repairs": [] }, { "id": "7.5", "source_parts": [ "7.5" ], "repairs": [ "Make n-vertex premise explicit; source omitted it in the local sentence." ] }, { "id": "8.1", "source_parts": [ "8.1" ], "repairs": [] }, { "id": "8.2", "source_parts": [ "8.2" ], "repairs": [] }, { "id": "9.1a", "source_parts": [ "9.1a" ], "repairs": [] }, { "id": "9.1b", "source_parts": [ "9.1b" ], "repairs": [] }, { "id": "9.2a", "source_parts": [ "9.2a" ], "repairs": [] }, { "id": "9.2b", "source_parts": [ "9.2b" ], "repairs": [] }, { "id": "9.2c", "source_parts": [ "9.2c" ], "repairs": [] }, { "id": "10.1", "source_parts": [ "10.1" ], "repairs": [] }, { "id": "10.2", "source_parts": [ "10.2" ], "repairs": [] }, { "id": "10.3", "source_parts": [ "10.3" ], "repairs": [] }, { "id": "10.4", "source_parts": [ "10.4" ], "repairs": [] }, { "id": "11.1", "source_parts": [ "11.1" ], "repairs": [] }, { "id": "11.2", "source_parts": [ "11.2" ], "repairs": [] }, { "id": "11.3", "source_parts": [ "11.3" ], "repairs": [] }, { "id": "11.4", "source_parts": [ "11.4" ], "repairs": [] }, { "id": "11.5a", "source_parts": [ "11.5a" ], "repairs": [] }, { "id": "11.5b", "source_parts": [ "11.5b" ], "repairs": [] }, { "id": "11.5c", "source_parts": [ "11.5c" ], "repairs": [] }, { "id": "12.1", "source_parts": [ "12.1" ], "repairs": [] }, { "id": "12.2", "source_parts": [ "12.2a", "12.2b", "12.2c" ], "repairs": [] }, { "id": "12.3", "source_parts": [ "12.3" ], "repairs": [ "Specify distinct x-coordinates." ] }, { "id": "12.4a", "source_parts": [ "12.4a" ], "repairs": [] }, { "id": "12.4b", "source_parts": [ "12.4b" ], "repairs": [] }, { "id": "13.1a", "source_parts": [ "13.1a" ], "repairs": [] }, { "id": "13.1b", "source_parts": [ "13.1b" ], "repairs": [] }, { "id": "13.1c", "source_parts": [ "13.1c" ], "repairs": [] }, { "id": "13.2a", "source_parts": [ "13.2a" ], "repairs": [] }, { "id": "13.2bi", "source_parts": [ "13.2bi" ], "repairs": [] }, { "id": "13.2bii", "source_parts": [ "13.2bii" ], "repairs": [] }, { "id": "13.2biii", "source_parts": [ "13.2biii" ], "repairs": [] }, { "id": "14.1", "source_parts": [ "14.1" ], "repairs": [] }, { "id": "14.2", "source_parts": [ "14.2" ], "repairs": [] }, { "id": "14.3", "source_parts": [ "14.3" ], "repairs": [] }, { "id": "14.4", "source_parts": [ "14.4" ], "repairs": [] }, { "id": "14.5", "source_parts": [ "14.5" ], "repairs": [] }, { "id": "14.6", "source_parts": [ "14.6" ], "repairs": [] }, { "id": "14.7", "source_parts": [ "14.7" ], "repairs": [ "Require distinct roots so different exponent tuples give distinct formal polynomials." ] }, { "id": "15.1a", "source_parts": [ "15.1a" ], "repairs": [] }, { "id": "15.1b", "source_parts": [ "15.1b" ], "repairs": [] }, { "id": "15.2", "source_parts": [ "15.2" ], "repairs": [] }, { "id": "15.3a", "source_parts": [ "15.3a" ], "repairs": [] }, { "id": "15.3b", "source_parts": [ "15.3b" ], "repairs": [] }, { "id": "15.3c", "source_parts": [ "15.3c" ], "repairs": [] }, { "id": "15.4", "source_parts": [ "15.4" ], "repairs": [] }, { "id": "15.5", "source_parts": [ "15.5" ], "repairs": [] } ], "not_scored": [ { "source_part": "1", "status": "excluded", "reason": "Administrative pledge." }, { "source_part": "2", "status": "withheld", "reason": "Self-referential conjunction of unavailable classroom answers." } ] }, "midterm_fa25": { "items": 43, "notes": [ "Exclude Q13 ungraded drawing and administrative identification. Q6 numbering restarts: ids 6.4–6.6 denote the three later Euler subparts. Diagram transcribed from original page 8 and visually checked. Composite answers preserve the requested witness/proof." ], "included": [ { "id": "1.1", "source_parts": [ "1.1" ], "repairs": [] }, { "id": "1.2", "source_parts": [ "1.2" ], "repairs": [] }, { "id": "1.3", "source_parts": [ "1.3" ], "repairs": [] }, { "id": "2.1", "source_parts": [ "2.1" ], "repairs": [] }, { "id": "2.2", "source_parts": [ "2.2" ], "repairs": [] }, { "id": "2.3", "source_parts": [ "2.3" ], "repairs": [] }, { "id": "3.1", "source_parts": [ "3.1" ], "repairs": [] }, { "id": "3.2", "source_parts": [ "3.2" ], "repairs": [] }, { "id": "4.1a", "source_parts": [ "4.1a" ], "repairs": [] }, { "id": "4.1b", "source_parts": [ "4.1b" ], "repairs": [] }, { "id": "4.2", "source_parts": [ "4.2" ], "repairs": [] }, { "id": "4.3", "source_parts": [ "4.3" ], "repairs": [] }, { "id": "4.4", "source_parts": [ "4.4" ], "repairs": [] }, { "id": "5.1", "source_parts": [ "5.1" ], "repairs": [] }, { "id": "5.2", "source_parts": [ "5.2" ], "repairs": [] }, { "id": "5.3", "source_parts": [ "5.3" ], "repairs": [] }, { "id": "5.4", "source_parts": [ "5.4" ], "repairs": [] }, { "id": "6.1", "source_parts": [ "6.1" ], "repairs": [] }, { "id": "6.2", "source_parts": [ "6.2" ], "repairs": [] }, { "id": "6.3", "source_parts": [ "6.3" ], "repairs": [] }, { "id": "6.4", "source_parts": [ "6.Euler.1" ], "repairs": [] }, { "id": "6.5", "source_parts": [ "6.Euler.2" ], "repairs": [] }, { "id": "6.6", "source_parts": [ "6.Euler.3" ], "repairs": [] }, { "id": "7.1", "source_parts": [ "7.1" ], "repairs": [] }, { "id": "8.1", "source_parts": [ "8.1" ], "repairs": [] }, { "id": "8.2", "source_parts": [ "8.2" ], "repairs": [] }, { "id": "8.3", "source_parts": [ "8.3" ], "repairs": [] }, { "id": "8.4", "source_parts": [ "8.4" ], "repairs": [] }, { "id": "8.5", "source_parts": [ "8.5" ], "repairs": [] }, { "id": "8.6", "source_parts": [ "8.6" ], "repairs": [] }, { "id": "8.7", "source_parts": [ "8.7" ], "repairs": [] }, { "id": "9.1", "source_parts": [ "9.1" ], "repairs": [] }, { "id": "10.1a", "source_parts": [ "10.1a" ], "repairs": [] }, { "id": "10.1b", "source_parts": [ "10.1b" ], "repairs": [] }, { "id": "10.2", "source_parts": [ "10.2" ], "repairs": [] }, { "id": "10.3", "source_parts": [ "10.3" ], "repairs": [] }, { "id": "10.4", "source_parts": [ "10.4" ], "repairs": [ "Specify distinct x-coordinates, needed for independent constraints." ] }, { "id": "11.1", "source_parts": [ "11.1a", "11.1b", "11.1c", "11.1d", "11.1e" ], "repairs": [] }, { "id": "11.2", "source_parts": [ "11.2" ], "repairs": [] }, { "id": "12.1", "source_parts": [ "12.1" ], "repairs": [] }, { "id": "12.2", "source_parts": [ "12.2" ], "repairs": [] }, { "id": "12.3", "source_parts": [ "12.3" ], "repairs": [] }, { "id": "12.4", "source_parts": [ "12.4" ], "repairs": [] } ], "not_scored": [ { "source_part": "13", "status": "excluded", "reason": "Ungraded drawing; not a mathematical task." } ] }, "midterm_sp23": { "items": 68, "notes": [ "Exclude Q1 pledge and Q2 self-referential classroom product. Clarify nonempty logical domain, nonempty subset in pigeonhole proof, distinct primes/coordinates, minimal error locator. Q11.4 asks for a guaranteed divisor, not the invalid universal claim that pq is the largest possible. All repairs precede inference." ], "included": [ { "id": "3.1", "source_parts": [ "3.1" ], "repairs": [] }, { "id": "3.2", "source_parts": [ "3.2" ], "repairs": [] }, { "id": "3.3", "source_parts": [ "3.3" ], "repairs": [] }, { "id": "3.4a", "source_parts": [ "3.4a" ], "repairs": [] }, { "id": "3.4b", "source_parts": [ "3.4b" ], "repairs": [] }, { "id": "4.1", "source_parts": [ "4.1" ], "repairs": [] }, { "id": "4.2", "source_parts": [ "4.2" ], "repairs": [ "Require nonempty subset explicitly, as intended by the supplied proof." ] }, { "id": "5.1", "source_parts": [ "5.1" ], "repairs": [] }, { "id": "5.2", "source_parts": [ "5.2" ], "repairs": [] }, { "id": "6.1", "source_parts": [ "6.1" ], "repairs": [] }, { "id": "7.1", "source_parts": [ "7.1" ], "repairs": [] }, { "id": "7.2", "source_parts": [ "7.2" ], "repairs": [] }, { "id": "7.3", "source_parts": [ "7.3" ], "repairs": [] }, { "id": "7.4", "source_parts": [ "7.4" ], "repairs": [] }, { "id": "7.5", "source_parts": [ "7.5" ], "repairs": [] }, { "id": "8.1", "source_parts": [ "8.1" ], "repairs": [] }, { "id": "8.2", "source_parts": [ "8.2" ], "repairs": [] }, { "id": "8.3", "source_parts": [ "8.3" ], "repairs": [] }, { "id": "8.4", "source_parts": [ "8.4" ], "repairs": [] }, { "id": "8.5a", "source_parts": [ "8.5a" ], "repairs": [] }, { "id": "8.5b", "source_parts": [ "8.5b" ], "repairs": [] }, { "id": "8.5c", "source_parts": [ "8.5c" ], "repairs": [] }, { "id": "8.5d", "source_parts": [ "8.5d" ], "repairs": [] }, { "id": "8.6", "source_parts": [ "8.6" ], "repairs": [] }, { "id": "8.7", "source_parts": [ "8.7" ], "repairs": [] }, { "id": "8.8", "source_parts": [ "8.8" ], "repairs": [] }, { "id": "8.9", "source_parts": [ "8.9" ], "repairs": [] }, { "id": "9.1a", "source_parts": [ "9.1a" ], "repairs": [] }, { "id": "9.1b", "source_parts": [ "9.1b" ], "repairs": [] }, { "id": "9.2a", "source_parts": [ "9.2a" ], "repairs": [] }, { "id": "9.2b", "source_parts": [ "9.2b" ], "repairs": [] }, { "id": "9.3", "source_parts": [ "9.3" ], "repairs": [] }, { "id": "10.1", "source_parts": [ "10.1" ], "repairs": [] }, { "id": "10.2", "source_parts": [ "10.2" ], "repairs": [] }, { "id": "10.3a", "source_parts": [ "10.3a" ], "repairs": [] }, { "id": "10.3b", "source_parts": [ "10.3b" ], "repairs": [] }, { "id": "10.4", "source_parts": [ "10.4" ], "repairs": [] }, { "id": "10.5", "source_parts": [ "10.5" ], "repairs": [ "Specify finite residue domain and distinct primes." ] }, { "id": "11.1", "source_parts": [ "11.1" ], "repairs": [] }, { "id": "11.2a", "source_parts": [ "11.2a" ], "repairs": [] }, { "id": "11.2b", "source_parts": [ "11.2b" ], "repairs": [] }, { "id": "11.3", "source_parts": [ "11.3" ], "repairs": [] }, { "id": "11.4", "source_parts": [ "11.4" ], "repairs": [ "Ask guaranteed divisor among alternatives. Original largest-divisor wording can fail: e.g. p=3,q=5 gives an extra universal factor 2." ] }, { "id": "11.5a", "source_parts": [ "11.5a" ], "repairs": [] }, { "id": "11.5b", "source_parts": [ "11.5b" ], "repairs": [] }, { "id": "12.1", "source_parts": [ "12.1" ], "repairs": [] }, { "id": "12.2", "source_parts": [ "12.2" ], "repairs": [] }, { "id": "12.3", "source_parts": [ "12.3" ], "repairs": [] }, { "id": "13.1", "source_parts": [ "13.1" ], "repairs": [] }, { "id": "13.2", "source_parts": [ "13.2" ], "repairs": [ "Distinct x-coordinates explicitly required." ] }, { "id": "13.3a", "source_parts": [ "13.3a" ], "repairs": [] }, { "id": "13.3bi", "source_parts": [ "13.3bi" ], "repairs": [] }, { "id": "13.3bii", "source_parts": [ "13.3bii" ], "repairs": [] }, { "id": "13.3c", "source_parts": [ "13.3c" ], "repairs": [] }, { "id": "14.1", "source_parts": [ "14.1" ], "repairs": [] }, { "id": "14.2", "source_parts": [ "14.2" ], "repairs": [ "Specify minimal error locator, excluding arbitrary extra roots from padding." ] }, { "id": "14.3", "source_parts": [ "14.3" ], "repairs": [] }, { "id": "14.4", "source_parts": [ "14.4" ], "repairs": [] }, { "id": "15.1", "source_parts": [ "15.1" ], "repairs": [] }, { "id": "15.2", "source_parts": [ "15.2" ], "repairs": [] }, { "id": "15.3a", "source_parts": [ "15.3a" ], "repairs": [] }, { "id": "15.3b", "source_parts": [ "15.3b" ], "repairs": [] }, { "id": "16.1", "source_parts": [ "16.1" ], "repairs": [] }, { "id": "16.2", "source_parts": [ "16.2" ], "repairs": [] }, { "id": "16.3", "source_parts": [ "16.3" ], "repairs": [] }, { "id": "16.4", "source_parts": [ "16.4" ], "repairs": [] }, { "id": "16.5a", "source_parts": [ "16.5a" ], "repairs": [] }, { "id": "16.5b", "source_parts": [ "16.5b" ], "repairs": [] } ], "not_scored": [ { "source_part": "1", "status": "excluded", "reason": "Administrative pledge." }, { "source_part": "2", "status": "withheld", "reason": "Self-referential product of unavailable classroom answers." } ] }, "midterm_sp24": { "items": 42, "notes": [ "Graphs on blank exam page 7 visually transcribed. Native 5-option Q1 and 6-option Q4c preserved. Q5 combines selected proof line with requested explanation. Q9 uses degree at most two, allows hint coordinates 0 and 1 as later subparts do, and excludes physical treasure-location queries in error detection. Q9e repeats original hints, never the solved polynomial." ], "included": [ { "id": "1.1", "source_parts": [ "1.1" ], "repairs": [] }, { "id": "2.1", "source_parts": [ "2.1" ], "repairs": [] }, { "id": "2.2", "source_parts": [ "2.2" ], "repairs": [] }, { "id": "2.3", "source_parts": [ "2.3" ], "repairs": [] }, { "id": "2.4", "source_parts": [ "2.4" ], "repairs": [] }, { "id": "2.5", "source_parts": [ "2.5" ], "repairs": [] }, { "id": "2.6", "source_parts": [ "2.6" ], "repairs": [] }, { "id": "3.1", "source_parts": [ "3.1" ], "repairs": [] }, { "id": "3.2", "source_parts": [ "3.2" ], "repairs": [] }, { "id": "3.3", "source_parts": [ "3.3" ], "repairs": [] }, { "id": "3.4", "source_parts": [ "3.4" ], "repairs": [] }, { "id": "3.5", "source_parts": [ "3.5" ], "repairs": [] }, { "id": "4.a", "source_parts": [ "4.a" ], "repairs": [] }, { "id": "4.b", "source_parts": [ "4.b" ], "repairs": [] }, { "id": "4.c", "source_parts": [ "4.c" ], "repairs": [] }, { "id": "4.d", "source_parts": [ "4.d" ], "repairs": [] }, { "id": "5.a", "source_parts": [ "5.a" ], "repairs": [] }, { "id": "5.b", "source_parts": [ "5.b" ], "repairs": [] }, { "id": "6.a", "source_parts": [ "6.a" ], "repairs": [] }, { "id": "6.b", "source_parts": [ "6.b" ], "repairs": [] }, { "id": "7.a", "source_parts": [ "7.a" ], "repairs": [] }, { "id": "7.b", "source_parts": [ "7.b" ], "repairs": [] }, { "id": "7.c", "source_parts": [ "7.c" ], "repairs": [] }, { "id": "7.d", "source_parts": [ "7.d" ], "repairs": [] }, { "id": "8.ai", "source_parts": [ "8.ai" ], "repairs": [] }, { "id": "8.aii", "source_parts": [ "8.aii" ], "repairs": [] }, { "id": "8.aiii", "source_parts": [ "8.aiii" ], "repairs": [] }, { "id": "8.aiv", "source_parts": [ "8.aiv" ], "repairs": [] }, { "id": "8.av", "source_parts": [ "8.av" ], "repairs": [] }, { "id": "8.avi", "source_parts": [ "8.avi" ], "repairs": [] }, { "id": "8.b", "source_parts": [ "8.b" ], "repairs": [] }, { "id": "9.a", "source_parts": [ "9.a" ], "repairs": [ "Use degree at most 2 to match the source’s interpolation/counting convention." ] }, { "id": "9.b", "source_parts": [ "9.b" ], "repairs": [] }, { "id": "9.c", "source_parts": [ "9.c" ], "repairs": [] }, { "id": "9.d", "source_parts": [ "9.d" ], "repairs": [ "Allow coordinates 0 and 1, used by source despite its earlier k>=2 restriction." ] }, { "id": "9.e", "source_parts": [ "9.e" ], "repairs": [] }, { "id": "9.f", "source_parts": [ "9.f" ], "repairs": [ "Disallow physical location checks; source accepted an alternative three-hint strategy using extra observations." ] }, { "id": "9.g", "source_parts": [ "9.g" ], "repairs": [] }, { "id": "10.a", "source_parts": [ "10.a" ], "repairs": [] }, { "id": "10.b", "source_parts": [ "10.b" ], "repairs": [] }, { "id": "10.c", "source_parts": [ "10.c" ], "repairs": [] }, { "id": "10.d", "source_parts": [ "10.d" ], "repairs": [] } ], "not_scored": [] }, "midterm_sp25": { "items": 62, "notes": [ "Exclude Q1 pledge. Retain the non-mathematical TA warmup, with the original cross-question clue supplied; analyze separately from mathematics. Girth bound assumes a cycle exists. Q9.6c uses residue classes, avoiding the printed range excluding zero. Necklace equivalence is rotation only. Diagram-free exam." ], "included": [ { "id": "2.1", "source_parts": [ "2.1" ], "repairs": [ "Supply original Q9.3 cross-question clue to avoid testing unavailable course attendance context." ] }, { "id": "3.1a", "source_parts": [ "3.1a" ], "repairs": [] }, { "id": "3.1b", "source_parts": [ "3.1b" ], "repairs": [] }, { "id": "3.1c", "source_parts": [ "3.1c" ], "repairs": [] }, { "id": "3.1d", "source_parts": [ "3.1d" ], "repairs": [] }, { "id": "3.2a", "source_parts": [ "3.2a" ], "repairs": [] }, { "id": "3.2b", "source_parts": [ "3.2b" ], "repairs": [] }, { "id": "3.2c", "source_parts": [ "3.2c" ], "repairs": [] }, { "id": "3.2d", "source_parts": [ "3.2d" ], "repairs": [] }, { "id": "4.1", "source_parts": [ "4.1" ], "repairs": [] }, { "id": "4.2", "source_parts": [ "4.2" ], "repairs": [ "Use the intended common-gcd definition explicitly, consistent with the official grading note." ] }, { "id": "5.1", "source_parts": [ "5.1" ], "repairs": [] }, { "id": "6.1", "source_parts": [ "6.1" ], "repairs": [] }, { "id": "6.2", "source_parts": [ "6.2" ], "repairs": [] }, { "id": "6.3", "source_parts": [ "6.3" ], "repairs": [] }, { "id": "6.4", "source_parts": [ "6.4" ], "repairs": [] }, { "id": "6.5a", "source_parts": [ "6.5a" ], "repairs": [] }, { "id": "6.5b", "source_parts": [ "6.5b" ], "repairs": [] }, { "id": "6.5c", "source_parts": [ "6.5c" ], "repairs": [] }, { "id": "7.1a", "source_parts": [ "7.1a" ], "repairs": [] }, { "id": "7.1b", "source_parts": [ "7.1b" ], "repairs": [ "Require a cycle; without it a tiny tree can violate the stated girth bound for arbitrarily large g." ] }, { "id": "7.1c", "source_parts": [ "7.1c" ], "repairs": [] }, { "id": "7.2a", "source_parts": [ "7.2a" ], "repairs": [] }, { "id": "7.2b", "source_parts": [ "7.2b" ], "repairs": [] }, { "id": "7.3a", "source_parts": [ "7.3a" ], "repairs": [] }, { "id": "7.3b", "source_parts": [ "7.3b" ], "repairs": [] }, { "id": "7.3c", "source_parts": [ "7.3c" ], "repairs": [] }, { "id": "7.3d", "source_parts": [ "7.3d" ], "repairs": [] }, { "id": "7.3e", "source_parts": [ "7.3e" ], "repairs": [] }, { "id": "7.3f", "source_parts": [ "7.3f" ], "repairs": [] }, { "id": "7.3g", "source_parts": [ "7.3g" ], "repairs": [] }, { "id": "8.1", "source_parts": [ "8.1" ], "repairs": [] }, { "id": "9.1", "source_parts": [ "9.1" ], "repairs": [] }, { "id": "9.2", "source_parts": [ "9.2" ], "repairs": [] }, { "id": "9.3", "source_parts": [ "9.3" ], "repairs": [] }, { "id": "9.4", "source_parts": [ "9.4" ], "repairs": [] }, { "id": "9.5a", "source_parts": [ "9.5a" ], "repairs": [] }, { "id": "9.5b", "source_parts": [ "9.5b" ], "repairs": [] }, { "id": "9.5c", "source_parts": [ "9.5c" ], "repairs": [] }, { "id": "9.6a", "source_parts": [ "9.6a" ], "repairs": [] }, { "id": "9.6b", "source_parts": [ "9.6b" ], "repairs": [ "State a nonzero explicitly, needed for Fermat exponent reduction." ] }, { "id": "9.6c", "source_parts": [ "9.6c" ], "repairs": [] }, { "id": "9.6d", "source_parts": [ "9.6d" ], "repairs": [] }, { "id": "9.6e", "source_parts": [ "9.6e" ], "repairs": [] }, { "id": "10.1", "source_parts": [ "10.1a", "10.1b", "10.1c" ], "repairs": [] }, { "id": "10.2a", "source_parts": [ "10.2a" ], "repairs": [] }, { "id": "10.2b", "source_parts": [ "10.2b" ], "repairs": [] }, { "id": "10.3", "source_parts": [ "10.3" ], "repairs": [] }, { "id": "10.4a", "source_parts": [ "10.4a" ], "repairs": [ "Quantify sufficiently large prime as p>2d to permit disjoint root sets." ] }, { "id": "10.4b", "source_parts": [ "10.4b" ], "repairs": [] }, { "id": "10.4c", "source_parts": [ "10.4c" ], "repairs": [] }, { "id": "10.4d", "source_parts": [ "10.4d" ], "repairs": [] }, { "id": "10.5", "source_parts": [ "10.5" ], "repairs": [] }, { "id": "10.6a", "source_parts": [ "10.6a" ], "repairs": [] }, { "id": "10.6b", "source_parts": [ "10.6b" ], "repairs": [] }, { "id": "11.1", "source_parts": [ "11.1" ], "repairs": [] }, { "id": "12.1", "source_parts": [ "12.1" ], "repairs": [] }, { "id": "12.2", "source_parts": [ "12.2" ], "repairs": [] }, { "id": "12.3", "source_parts": [ "12.3" ], "repairs": [] }, { "id": "12.4", "source_parts": [ "12.4" ], "repairs": [] }, { "id": "12.5", "source_parts": [ "12.5" ], "repairs": [] }, { "id": "12.6", "source_parts": [ "12.6" ], "repairs": [ "Explicitly identify rotations only, consistent with the source answer." ] } ], "not_scored": [ { "source_part": "1", "status": "excluded", "reason": "Administrative pledge." } ] }, "midterm_su25": { "items": 71, "notes": [ "Exclude Q1 pledge. Correct official recurrence arithmetic: S3=9,S4=14,S5=25,S6=57; requested bound c=1 is unchanged. RSA smallest exponent explicitly requires e>1. Polynomial minimum questions explicitly assume equal degrees to repair the printed answer. Error locator uses the minimal locator; packet rate ignores integrality as a rate bound. These repairs precede inference." ], "included": [ { "id": "2.1a", "source_parts": [ "2.1a" ], "repairs": [] }, { "id": "2.1b", "source_parts": [ "2.1b" ], "repairs": [] }, { "id": "2.1c", "source_parts": [ "2.1c" ], "repairs": [] }, { "id": "2.1d", "source_parts": [ "2.1d" ], "repairs": [] }, { "id": "2.2a", "source_parts": [ "2.2a" ], "repairs": [] }, { "id": "2.2b", "source_parts": [ "2.2b" ], "repairs": [] }, { "id": "2.3", "source_parts": [ "2.3" ], "repairs": [] }, { "id": "3.1", "source_parts": [ "3.1" ], "repairs": [] }, { "id": "3.2a", "source_parts": [ "3.2a" ], "repairs": [] }, { "id": "3.2b", "source_parts": [ "3.2b" ], "repairs": [] }, { "id": "4.1a", "source_parts": [ "4.1a" ], "repairs": [] }, { "id": "4.1b", "source_parts": [ "4.1b" ], "repairs": [] }, { "id": "4.1c", "source_parts": [ "4.1c" ], "repairs": [ "Official intermediate terms miscomputed; reference bound remains 1." ] }, { "id": "4.2", "source_parts": [ "4.2" ], "repairs": [] }, { "id": "5.1a", "source_parts": [ "5.1a" ], "repairs": [] }, { "id": "5.1b", "source_parts": [ "5.1b" ], "repairs": [] }, { "id": "5.1c", "source_parts": [ "5.1c" ], "repairs": [] }, { "id": "5.2", "source_parts": [ "5.2" ], "repairs": [] }, { "id": "5.3", "source_parts": [ "5.3" ], "repairs": [] }, { "id": "5.4", "source_parts": [ "5.4" ], "repairs": [] }, { "id": "5.5", "source_parts": [ "5.5" ], "repairs": [] }, { "id": "5.6", "source_parts": [ "5.6" ], "repairs": [] }, { "id": "5.7", "source_parts": [ "5.7" ], "repairs": [] }, { "id": "6.1a", "source_parts": [ "6.1a" ], "repairs": [] }, { "id": "6.1b", "source_parts": [ "6.1b" ], "repairs": [] }, { "id": "6.1c", "source_parts": [ "6.1c" ], "repairs": [] }, { "id": "6.1d", "source_parts": [ "6.1d" ], "repairs": [] }, { "id": "6.1e", "source_parts": [ "6.1e" ], "repairs": [] }, { "id": "6.1f", "source_parts": [ "6.1f" ], "repairs": [] }, { "id": "6.2", "source_parts": [ "6.2" ], "repairs": [] }, { "id": "6.3a", "source_parts": [ "6.3a" ], "repairs": [] }, { "id": "6.3b", "source_parts": [ "6.3b" ], "repairs": [] }, { "id": "6.4a", "source_parts": [ "6.4a" ], "repairs": [] }, { "id": "6.4b", "source_parts": [ "6.4b" ], "repairs": [] }, { "id": "6.5", "source_parts": [ "6.5" ], "repairs": [] }, { "id": "7.1", "source_parts": [ "7.1" ], "repairs": [] }, { "id": "7.2", "source_parts": [ "7.2" ], "repairs": [] }, { "id": "8.1", "source_parts": [ "8.1" ], "repairs": [] }, { "id": "8.2", "source_parts": [ "8.2" ], "repairs": [] }, { "id": "8.3", "source_parts": [ "8.3" ], "repairs": [] }, { "id": "8.4", "source_parts": [ "8.4" ], "repairs": [] }, { "id": "8.5", "source_parts": [ "8.5" ], "repairs": [] }, { "id": "8.6", "source_parts": [ "8.6" ], "repairs": [] }, { "id": "8.7", "source_parts": [ "8.7" ], "repairs": [] }, { "id": "8.8a", "source_parts": [ "8.8a" ], "repairs": [ "Require e>1 explicitly; without it, exponent 1 is mathematically invertible." ] }, { "id": "8.8b", "source_parts": [ "8.8b" ], "repairs": [] }, { "id": "9.1", "source_parts": [ "9.1" ], "repairs": [] }, { "id": "9.2", "source_parts": [ "9.2" ], "repairs": [] }, { "id": "9.3a", "source_parts": [ "9.3a" ], "repairs": [] }, { "id": "9.3b", "source_parts": [ "9.3b" ], "repairs": [] }, { "id": "9.3c", "source_parts": [ "9.3c" ], "repairs": [] }, { "id": "9.3d", "source_parts": [ "9.3d" ], "repairs": [] }, { "id": "10.1a", "source_parts": [ "10.1a" ], "repairs": [] }, { "id": "10.1b", "source_parts": [ "10.1b" ], "repairs": [ "Equal degrees added: with unequal degrees the printed reference 0 is false." ] }, { "id": "10.1c", "source_parts": [ "10.1c" ], "repairs": [] }, { "id": "10.1d", "source_parts": [ "10.1d" ], "repairs": [ "Equal degrees added: degree-one versus constant polynomials necessarily agree once." ] }, { "id": "10.2a", "source_parts": [ "10.2a" ], "repairs": [] }, { "id": "10.2b", "source_parts": [ "10.2b" ], "repairs": [] }, { "id": "10.2c", "source_parts": [ "10.2c" ], "repairs": [] }, { "id": "10.3", "source_parts": [ "10.3" ], "repairs": [] }, { "id": "11.1a", "source_parts": [ "11.1a" ], "repairs": [] }, { "id": "11.1b", "source_parts": [ "11.1b" ], "repairs": [] }, { "id": "11.1c", "source_parts": [ "11.1c" ], "repairs": [ "Specify minimal locator; padded Berlekamp–Welch locators need not have exactly the actual error count as degree." ] }, { "id": "11.1d", "source_parts": [ "11.1d" ], "repairs": [ "Specify minimal locator rather than a padded degree-k locator." ] }, { "id": "11.2", "source_parts": [ "11.2" ], "repairs": [ "Continuous rate bound; floor-dependent finite block rounding is not being tested." ] }, { "id": "12.1", "source_parts": [ "12.1" ], "repairs": [] }, { "id": "12.2", "source_parts": [ "12.2" ], "repairs": [] }, { "id": "12.3", "source_parts": [ "12.3" ], "repairs": [] }, { "id": "12.4", "source_parts": [ "12.4" ], "repairs": [] }, { "id": "12.5", "source_parts": [ "12.5" ], "repairs": [] }, { "id": "12.6", "source_parts": [ "12.6" ], "repairs": [] } ], "not_scored": [ { "source_part": "1", "status": "excluded", "reason": "Administrative pledge." } ] } }, "follow_up_run": { "started_at": "2026-09-18T08:11:58.318223+00:00", "freeze_sha256": "b2ac3ee0d79407ddbaa1e3d711c49a9b392927b6e97279207f5db72350c8c96f", "sections": [ { "exam": "final_fa23", "section": 2, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "2.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: P AND NOT P is equivalent to True.", "criteria": { "A": "False", "B": "True" } }, "2.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Simplify (P implies R) AND (P implies NOT R).", "criteria": { "A": "P", "B": "NOT R", "C": "NOT P", "D": "True" } }, "2.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: (P AND Q implies NOT R) is equivalent to NOT R OR NOT P OR NOT Q.", "criteria": { "A": "True", "B": "False" } }, "2.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: NOT forall x exists y Q(x,y) is equivalent to exists x forall y NOT Q(x,y).", "criteria": { "A": "False", "B": "True" } }, "2.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Suppose [forall x exists y Q(x,y)] implies [forall x forall y P(x,y)], and some P(a,b) is false. There exists x for which every Q(x,y) is __.", "criteria": { "A": "True", "B": "Undetermined", "C": "False" } } } }, "attempts": [ { "status": 200, "full_response_ms": 518.7205000001995, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"2.1a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"A\":1.0}},\"2.1b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"C\":1.0,\"B\":0.0,\"A\":0.0}},\"2.1c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"A\":1.0}},\"2.2a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.18,\"probabilities\":{\"B\":0.41,\"A\":0.59}},\"2.2b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.84,\"probabilities\":{\"C\":0.89,\"B\":0.07,\"A\":0.04}}},\"usage\":{\"input_tokens\":785,\"output_tokens\":179}}" } ], "response": { "model": "jev-1.13.0", "answers": { "2.1a": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "B": 0.0, "A": 1.0 } }, "2.1b": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "D": 0.0, "C": 1.0, "B": 0.0, "A": 0.0 } }, "2.1c": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "B": 0.0, "A": 1.0 } }, "2.2a": { "type": "choice", "choice": "A", "confidence": 0.18, "probabilities": { "B": 0.41, "A": 0.59 } }, "2.2b": { "type": "choice", "choice": "C", "confidence": 0.84, "probabilities": { "C": 0.89, "B": 0.07, "A": 0.04 } } }, "usage": { "input_tokens": 785, "output_tokens": 179 } } }, { "exam": "final_fa23", "section": 3, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "3.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For natural a,b,c, prove a|b and b|c imply a|c.", "criteria": { "A": "The product abc is divisible by a, which implies c is divisible by a.", "B": "Divisibility is symmetric, so c|a implies a|c.", "C": "Write b=ka and c=lb for integers k,l; then c=(lk)a.", "D": "Since a<=b<=c, a must divide c." } }, "3.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prove an integer whose decimal digit sum is divisible by nine is divisible by nine.", "criteria": { "A": "Since n and its digit sum have the same parity, they have the same residue modulo nine.", "B": "A decimal expansion has nine possible digits, so the pigeonhole principle proves divisibility.", "C": "Each decimal digit is separately divisible by nine whenever their sum is.", "D": "Write n=sum ak*10^k. Since 10=1 mod9, n=sum ak mod9; zero residue of the digit sum therefore gives zero residue of n." } } } }, "attempts": [ { "status": 200, "full_response_ms": 401.8495829986932, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"3.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"C\":1.0,\"B\":0.0}},\"3.2\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"A\":0.0,\"C\":0.0,\"B\":0.0}}},\"usage\":{\"input_tokens\":681,\"output_tokens\":91}}" } ], "response": { "model": "jev-1.13.0", "answers": { "3.1": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 1.0, "B": 0.0 } }, "3.2": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "A": 0.0, "C": 0.0, "B": 0.0 } } }, "usage": { "input_tokens": 681, "output_tokens": 91 } } }, { "exam": "final_fa23", "section": 4, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "4.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A job receiving its first-choice candidate in some stable matching must receive that candidate in every stable matching.", "criteria": { "A": "False", "B": "True" } }, "4.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A job ranked last by every candidate must be rejected n−1 times in job-proposing deferred acceptance.", "criteria": { "A": "True", "B": "False" } }, "4.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If some job proposes to its final-ranked candidate during synchronous job-proposing deferred acceptance, the algorithm terminates that day.", "criteria": { "A": "False", "B": "True" } } } }, "attempts": [ { "status": 200, "full_response_ms": 374.150916999497, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"4.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.13,\"probabilities\":{\"A\":0.56,\"B\":0.44}},\"4.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.5,\"probabilities\":{\"A\":0.75,\"B\":0.25}},\"4.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.62,\"probabilities\":{\"A\":0.81,\"B\":0.19}}},\"usage\":{\"input_tokens\":582,\"output_tokens\":93}}" } ], "response": { "model": "jev-1.13.0", "answers": { "4.1": { "type": "choice", "choice": "A", "confidence": 0.13, "probabilities": { "A": 0.56, "B": 0.44 } }, "4.2": { "type": "choice", "choice": "A", "confidence": 0.5, "probabilities": { "A": 0.75, "B": 0.25 } }, "4.3": { "type": "choice", "choice": "A", "confidence": 0.62, "probabilities": { "A": 0.81, "B": 0.19 } } }, "usage": { "input_tokens": 582, "output_tokens": 93 } } }, { "exam": "final_fa23", "section": 5, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "5.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Adding a new edge either creates a cycle containing that edge or reduces the component count.", "criteria": { "A": "False", "B": "True" } }, "5.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Edges in a tree on n>0 vertices?", "criteria": { "A": "2n−2", "B": "n+1", "C": "n", "D": "n−1" } }, "5.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Delete k edges from a tree, keeping all vertices. Component count?", "criteria": { "A": "k", "B": "k−1", "C": "2k", "D": "k+1" } }, "5.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A tree has exactly two degree-five vertices. Sharp minimum number of leaves?", "criteria": { "A": "10", "B": "6", "C": "5", "D": "8" } }, "5.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: After removing a simple cycle’s edges from a connected Eulerian graph, every component is Eulerian, allowing the trivial zero-edge tour at an isolated vertex.", "criteria": { "A": "True", "B": "False" } }, "5.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: After removing a simple cycle’s edges from a connected Eulerian simple graph, c components remain. Sharp universal lower bound on removed cycle length?", "criteria": { "A": "c", "B": "c+1", "C": "max(3,c)", "D": "2c" } }, "5.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: The four-dimensional hypercube has an Eulerian tour.", "criteria": { "A": "False", "B": "True" } }, "5.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: G is k-colorable and H is bipartite on the same vertices. Sharp universal color bound for their edge union, in terms of k alone?", "criteria": { "A": "k", "B": "2k", "C": "k+1", "D": "k²" } }, "5.6a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Connected planar embedding: n vertices,2n edges. Faces?", "criteria": { "A": "n", "B": "2n", "C": "n−2", "D": "n+2" } }, "5.6b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Connected planar embedding: n vertices, average face boundary length five. Edges?", "criteria": { "A": "2n−4", "B": "3n−6", "C": "5(n−2)/3", "D": "5n/2" } } } }, "attempts": [ { "status": 200, "full_response_ms": 337.21541599879856, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"5.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.94,\"probabilities\":{\"A\":0.03,\"B\":0.97}},\"5.2a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"A\":0.0,\"B\":0.0,\"C\":0.0}},\"5.2b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"A\":0.0,\"B\":0.0,\"C\":0.0}},\"5.2c\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.69,\"probabilities\":{\"A\":0.07,\"D\":0.77,\"B\":0.14,\"C\":0.02}},\"5.3a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.82,\"probabilities\":{\"A\":0.91,\"B\":0.09}},\"5.3b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.33,\"probabilities\":{\"D\":0.07,\"A\":0.18,\"B\":0.25,\"C\":0.5}},\"5.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.66,\"probabilities\":{\"A\":0.17,\"B\":0.83}},\"5.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.56,\"probabilities\":{\"D\":0.16,\"A\":0.05,\"B\":0.68,\"C\":0.11}},\"5.6a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.25,\"probabilities\":{\"D\":0.18,\"A\":0.37,\"B\":0.01,\"C\":0.44}},\"5.6b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.68,\"probabilities\":{\"D\":0.06,\"A\":0.16,\"B\":0.02,\"C\":0.76}}},\"usage\":{\"input_tokens\":1354,\"output_tokens\":408}}" } ], "response": { "model": "jev-1.13.0", "answers": { "5.1": { "type": "choice", "choice": "B", "confidence": 0.94, "probabilities": { "A": 0.03, "B": 0.97 } }, "5.2a": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "A": 0.0, "B": 0.0, "C": 0.0 } }, "5.2b": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "A": 0.0, "B": 0.0, "C": 0.0 } }, "5.2c": { "type": "choice", "choice": "D", "confidence": 0.69, "probabilities": { "A": 0.07, "D": 0.77, "B": 0.14, "C": 0.02 } }, "5.3a": { "type": "choice", "choice": "A", "confidence": 0.82, "probabilities": { "A": 0.91, "B": 0.09 } }, "5.3b": { "type": "choice", "choice": "C", "confidence": 0.33, "probabilities": { "D": 0.07, "A": 0.18, "B": 0.25, "C": 0.5 } }, "5.4": { "type": "choice", "choice": "B", "confidence": 0.66, "probabilities": { "A": 0.17, "B": 0.83 } }, "5.5": { "type": "choice", "choice": "B", "confidence": 0.56, "probabilities": { "D": 0.16, "A": 0.05, "B": 0.68, "C": 0.11 } }, "5.6a": { "type": "choice", "choice": "C", "confidence": 0.25, "probabilities": { "D": 0.18, "A": 0.37, "B": 0.01, "C": 0.44 } }, "5.6b": { "type": "choice", "choice": "C", "confidence": 0.68, "probabilities": { "D": 0.06, "A": 0.16, "B": 0.02, "C": 0.76 } } }, "usage": { "input_tokens": 1354, "output_tokens": 408 } } }, { "exam": "final_fa23", "section": 6, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "6.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Choose integers (a,b) solving 47a+49b=1.", "criteria": { "A": "(−24,23)", "B": "(24,−23)", "C": "(23,−22)", "D": "(49,−47)" } }, "6.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: 2^30 modulo 35?", "criteria": { "A": "8", "B": "29", "C": "1", "D": "16" } }, "6.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct primes p,q. Simplify q^p mod pq.", "criteria": { "A": "p", "B": "1", "C": "q", "D": "0" } }, "6.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: There are pq−__ invertible residues modulo pq for distinct primes p,q. Fill the blank.", "criteria": { "A": "p+q−1", "B": "p+q", "C": "p+q+1", "D": "pq−1" } } } }, "attempts": [ { "status": 200, "full_response_ms": 419.9863750000077, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"6.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.38,\"probabilities\":{\"B\":0.53,\"D\":0.02,\"A\":0.34,\"C\":0.11}},\"6.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.21,\"probabilities\":{\"B\":0.41,\"D\":0.34,\"A\":0.17,\"C\":0.08}},\"6.3a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.34,\"probabilities\":{\"C\":0.36,\"B\":0.04,\"A\":0.09,\"D\":0.51}},\"6.3b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.93,\"probabilities\":{\"B\":0.03,\"D\":0.0,\"A\":0.95,\"C\":0.02}}},\"usage\":{\"input_tokens\":778,\"output_tokens\":181}}" } ], "response": { "model": "jev-1.13.0", "answers": { "6.1": { "type": "choice", "choice": "B", "confidence": 0.38, "probabilities": { "B": 0.53, "D": 0.02, "A": 0.34, "C": 0.11 } }, "6.2": { "type": "choice", "choice": "B", "confidence": 0.21, "probabilities": { "B": 0.41, "D": 0.34, "A": 0.17, "C": 0.08 } }, "6.3a": { "type": "choice", "choice": "D", "confidence": 0.34, "probabilities": { "C": 0.36, "B": 0.04, "A": 0.09, "D": 0.51 } }, "6.3b": { "type": "choice", "choice": "A", "confidence": 0.93, "probabilities": { "B": 0.03, "D": 0.0, "A": 0.95, "C": 0.02 } } }, "usage": { "input_tokens": 778, "output_tokens": 181 } } }, { "exam": "final_fa23", "section": 7, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "7.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prime p: why is no a in 2..p−2 its own inverse?", "criteria": { "A": "The inverse operation has no fixed points in any field.", "B": "a=a^(-1) would imply (a−1)(a+1)=0 modp. A field has no zero divisors, forcing a=1 or−1, outside the range.", "C": "All nonzero residues have inverse one.", "D": "Every inverse is larger than its original residue." } }, "7.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Product of all residues 1,...,p−1 modulo prime p?", "criteria": { "A": "0", "B": "p−2", "C": "p−1", "D": "1" } }, "7.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct odd primes p,q: construct a nontrivial self-inverse unit modulo pq.", "criteria": { "A": "Set a=p, since any prime is its own inverse.", "B": "Set a=0, whose inverse equals itself.", "C": "CRT choose a=1 modp and a=−1 modq. Then a²=1 modulo both primes, but a is neither 1 nor−1 modulo pq.", "D": "Use a=pq−1, which lies in the required nontrivial range 2..pq−2." } } } }, "attempts": [ { "status": 200, "full_response_ms": 427.8359579984681, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"7.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"B\":1.0,\"C\":0.0}},\"7.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.98,\"probabilities\":{\"C\":0.99,\"A\":0.0,\"B\":0.0,\"D\":0.01}},\"7.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"B\":0.0,\"C\":1.0}}},\"usage\":{\"input_tokens\":785,\"output_tokens\":135}}" } ], "response": { "model": "jev-1.13.0", "answers": { "7.1": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 1.0, "C": 0.0 } }, "7.2": { "type": "choice", "choice": "C", "confidence": 0.98, "probabilities": { "C": 0.99, "A": 0.0, "B": 0.0, "D": 0.01 } }, "7.3": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 0.0, "C": 1.0 } } }, "usage": { "input_tokens": 785, "output_tokens": 135 } } }, { "exam": "final_fa23", "section": 8, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "8.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Give a degree-two polynomial over GF(5) with roots 0 and1.", "criteria": { "A": "x(x−1)", "B": "(x−1)²", "C": "x²+1", "D": "x(x+1)" } }, "8.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct formal P,Q have degrees dP,dQ over GF(p),p>max(dP,dQ). General tight upper bound on coordinates where P=Q?", "criteria": { "A": "min(dP,dQ)", "B": "dP+dQ", "C": "max(dP,dQ)", "D": "p" } }, "8.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P(x)=mx+b over GF(p), prime p>2. One child receives P(1), another P(2), a third m; b is secret.\nProblem: Any two children can reconstruct b.", "criteria": { "A": "True", "B": "False" } }, "8.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P(x)=mx+b over GF(p), prime p>2. One child receives P(1), another P(2), a third m; b is secret.\nProblem: Recover b given m and P(1).", "criteria": { "A": "P(1)+m modp", "B": "P(1)−m modp", "C": "P(1)/m modp", "D": "m−P(1) modp" } } } }, "attempts": [ { "status": 200, "full_response_ms": 386.2307499985036, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"8.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":1.0,\"C\":0.0}},\"8.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.33,\"probabilities\":{\"B\":0.21,\"D\":0.01,\"A\":0.49,\"C\":0.29}},\"8.3a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.43,\"probabilities\":{\"B\":0.28,\"A\":0.72}},\"8.3b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"B\":1.0,\"D\":0.0,\"A\":0.0,\"C\":0.0}}},\"usage\":{\"input_tokens\":835,\"output_tokens\":167}}" } ], "response": { "model": "jev-1.13.0", "answers": { "8.1": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 1.0, "C": 0.0 } }, "8.2": { "type": "choice", "choice": "A", "confidence": 0.33, "probabilities": { "B": 0.21, "D": 0.01, "A": 0.49, "C": 0.29 } }, "8.3a": { "type": "choice", "choice": "A", "confidence": 0.43, "probabilities": { "B": 0.28, "A": 0.72 } }, "8.3b": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "B": 1.0, "D": 0.0, "A": 0.0, "C": 0.0 } } }, "usage": { "input_tokens": 835, "output_tokens": 167 } } }, { "exam": "final_fa23", "section": 9, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "9.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: The set of all subsets of the primes is countable.", "criteria": { "A": "True", "B": "False" } }, "9.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For sets A1,...,An, let D={i in 1..n: i is not in Ai}. Then D differs from every Ai.", "criteria": { "A": "True", "B": "False" } }, "9.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Programs that halt on input1 form a countable set.", "criteria": { "A": "False", "B": "True" } }, "9.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A total algorithm can decide for any P,x and natural n whether P(x) runs for more than n steps.", "criteria": { "A": "True", "B": "False" } } } }, "attempts": [ { "status": 200, "full_response_ms": 328.3510840010422, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"9.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.93,\"probabilities\":{\"B\":0.97,\"A\":0.03}},\"9.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.93,\"probabilities\":{\"B\":0.03,\"A\":0.97}},\"9.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.9,\"probabilities\":{\"B\":0.95,\"A\":0.05}},\"9.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.85,\"probabilities\":{\"B\":0.93,\"A\":0.07}}},\"usage\":{\"input_tokens\":656,\"output_tokens\":123}}" } ], "response": { "model": "jev-1.13.0", "answers": { "9.1": { "type": "choice", "choice": "B", "confidence": 0.93, "probabilities": { "B": 0.97, "A": 0.03 } }, "9.2": { "type": "choice", "choice": "A", "confidence": 0.93, "probabilities": { "B": 0.03, "A": 0.97 } }, "9.3": { "type": "choice", "choice": "B", "confidence": 0.9, "probabilities": { "B": 0.95, "A": 0.05 } }, "9.4": { "type": "choice", "choice": "B", "confidence": 0.85, "probabilities": { "B": 0.93, "A": 0.07 } } }, "usage": { "input_tokens": 656, "output_tokens": 123 } } }, { "exam": "final_fa23", "section": 10, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "10.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct arrangements of CAT?", "criteria": { "A": "6", "B": "1", "C": "9", "D": "3" } }, "10.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Histograms with k named bins and n observations: nonnegative counts sum to n. Count?", "criteria": { "A": "k^n", "B": "C(n+k−1,k−1)", "C": "C(n−1,k−1)", "D": "C(n,k)" } }, "10.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A staircase has k steps of heights one or two and total height n,k<=n<=2k. Number of height-two steps?", "criteria": { "A": "n−k", "B": "2k−n", "C": "k", "D": "n−2k" } }, "10.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of ordered staircases of k steps, each height one or two, total n?", "criteria": { "A": "C(k,n−k)", "B": "C(n−k,k)", "C": "C(n,k)", "D": "2^k" } } } }, "attempts": [ { "status": 200, "full_response_ms": 344.4564169985824, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"10.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"A\":1.0,\"D\":0.0,\"B\":0.0}},\"10.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"C\":0.0,\"A\":0.0,\"D\":0.0,\"B\":1.0}},\"10.3a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"C\":0.0,\"A\":0.99,\"D\":0.0,\"B\":0.01}},\"10.3b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.53,\"probabilities\":{\"C\":0.01,\"A\":0.64,\"D\":0.0,\"B\":0.35}}},\"usage\":{\"input_tokens\":787,\"output_tokens\":185}}" } ], "response": { "model": "jev-1.13.0", "answers": { "10.1": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "C": 0.0, "A": 1.0, "D": 0.0, "B": 0.0 } }, "10.2": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "C": 0.0, "A": 0.0, "D": 0.0, "B": 1.0 } }, "10.3a": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "C": 0.0, "A": 0.99, "D": 0.0, "B": 0.01 } }, "10.3b": { "type": "choice", "choice": "A", "confidence": 0.53, "probabilities": { "C": 0.01, "A": 0.64, "D": 0.0, "B": 0.35 } } }, "usage": { "input_tokens": 787, "output_tokens": 185 } } }, { "exam": "final_fa23", "section": 11, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "11.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Uniformly choose bag A:4 red,4 blue or bag B:3 red,5 blue. Draw two marbles without replacement. E:both blue.\nProblem: P(E given A)?", "criteria": { "A": "1/2", "B": "5/14", "C": "3/14", "D": "1/4" } }, "11.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Uniformly choose bag A:4 red,4 blue or bag B:3 red,5 blue. Draw two marbles without replacement. E:both blue.\nProblem: P(A given E)?", "criteria": { "A": "3/8", "B": "5/8", "C": "1/2", "D": "3/14" } }, "11.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Roll two independent fair six-sided dice. X=sum modulo3,Y=sum modulo4.\nProblem: P(X=1)?", "criteria": { "A": "1/4", "B": "1/3", "C": "5/18", "D": "1/6" } }, "11.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Roll two independent fair six-sided dice. X=sum modulo3,Y=sum modulo4.\nProblem: P(Y=3)?", "criteria": { "A": "1/4", "B": "1/6", "C": "5/18", "D": "1/3" } }, "11.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Roll two independent fair six-sided dice. X=sum modulo3,Y=sum modulo4.\nProblem: P(X=1 and Y=3)?", "criteria": { "A": "1/3", "B": "1/6", "C": "5/54", "D": "1/12" } }, "11.2d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Roll two independent fair six-sided dice. X=sum modulo3,Y=sum modulo4.\nProblem: P(X=1 or Y=3)?", "criteria": { "A": "11/18", "B": "4/9", "C": "5/54", "D": "1/2" } }, "11.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Prize uniformly behind one of three doors. You choose door1. Host, unaware of prize, uniformly opens door2 or3; if prize revealed you lose. A:host reveals no prize; B:prize is behind door1.\nProblem: P(A)?", "criteria": { "A": "2/3", "B": "1/2", "C": "1", "D": "1/3" } }, "11.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Prize uniformly behind one of three doors. You choose door1. Host, unaware of prize, uniformly opens door2 or3; if prize revealed you lose. A:host reveals no prize; B:prize is behind door1.\nProblem: P(B given A)?", "criteria": { "A": "1/3", "B": "2/3", "C": "1", "D": "1/2" } } } }, "attempts": [ { "status": 200, "full_response_ms": 374.01329199929023, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"11.1a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.77,\"probabilities\":{\"D\":0.02,\"B\":0.15,\"A\":0.01,\"C\":0.82}},\"11.1b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.3,\"probabilities\":{\"B\":0.37,\"C\":0.03,\"A\":0.48,\"D\":0.12}},\"11.2a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.63,\"probabilities\":{\"B\":0.72,\"C\":0.24,\"A\":0.02,\"D\":0.02}},\"11.2b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.23,\"probabilities\":{\"C\":0.43,\"D\":0.04,\"A\":0.35,\"B\":0.18}},\"11.2c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.36,\"probabilities\":{\"B\":0.12,\"C\":0.51,\"A\":0.01,\"D\":0.36}},\"11.2d\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.43,\"probabilities\":{\"B\":0.23,\"C\":0.01,\"A\":0.57,\"D\":0.19}},\"11.3a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.89,\"probabilities\":{\"B\":0.04,\"C\":0.03,\"A\":0.91,\"D\":0.02}},\"11.3b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.26,\"probabilities\":{\"B\":0.24,\"C\":0.01,\"A\":0.3,\"D\":0.45}}},\"usage\":{\"input_tokens\":1358,\"output_tokens\":371}}" } ], "response": { "model": "jev-1.13.0", "answers": { "11.1a": { "type": "choice", "choice": "C", "confidence": 0.77, "probabilities": { "D": 0.02, "B": 0.15, "A": 0.01, "C": 0.82 } }, "11.1b": { "type": "choice", "choice": "A", "confidence": 0.3, "probabilities": { "B": 0.37, "C": 0.03, "A": 0.48, "D": 0.12 } }, "11.2a": { "type": "choice", "choice": "B", "confidence": 0.63, "probabilities": { "B": 0.72, "C": 0.24, "A": 0.02, "D": 0.02 } }, "11.2b": { "type": "choice", "choice": "C", "confidence": 0.23, "probabilities": { "C": 0.43, "D": 0.04, "A": 0.35, "B": 0.18 } }, "11.2c": { "type": "choice", "choice": "C", "confidence": 0.36, "probabilities": { "B": 0.12, "C": 0.51, "A": 0.01, "D": 0.36 } }, "11.2d": { "type": "choice", "choice": "A", "confidence": 0.43, "probabilities": { "B": 0.23, "C": 0.01, "A": 0.57, "D": 0.19 } }, "11.3a": { "type": "choice", "choice": "A", "confidence": 0.89, "probabilities": { "B": 0.04, "C": 0.03, "A": 0.91, "D": 0.02 } }, "11.3b": { "type": "choice", "choice": "D", "confidence": 0.26, "probabilities": { "B": 0.24, "C": 0.01, "A": 0.3, "D": 0.45 } } }, "usage": { "input_tokens": 1358, "output_tokens": 371 } } }, { "exam": "final_fa23", "section": 12, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "12.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: n distinct balls independently choose uniform bins among n named bins. X counts balls in bin1.\nProblem: P(X=0)?", "criteria": { "A": "exp(−n)", "B": "(1−1/n)^n", "C": "(1/n)^n", "D": "1−1/n" } }, "12.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: n distinct balls independently choose uniform bins among n named bins. X counts balls in bin1.\nProblem: Distribution of X?", "criteria": { "A": "Poisson(n)", "B": "Uniform{0,...,n}", "C": "Geom(1/n)", "D": "Bin(n,1/n)" } }, "12.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: n distinct balls independently choose uniform bins among n named bins. X counts balls in bin1.\nProblem: Limit of P(X=10) as n grows?", "criteria": { "A": "1/10", "B": "0", "C": "exp(−1)/10!", "D": "exp(−10)" } }, "12.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: 100 independent fair flips. X=number of heads minus number of tails; Y=X².\nProblem: P(X=0)?", "criteria": { "A": "1/2", "B": "C(100,0)/2^100", "C": "1/101", "D": "C(100,50)/2^100" } }, "12.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: 100 independent fair flips. X=number of heads minus number of tails; Y=X².\nProblem: E[X]?", "criteria": { "A": "50", "B": "0", "C": "1/2", "D": "100" } }, "12.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: 100 independent fair flips. X=number of heads minus number of tails; Y=X².\nProblem: Are X,Y independent? Justify.", "criteria": { "A": "Yes; their covariance is zero.", "B": "No; E[Y] is negative.", "C": "Yes; independent flips imply any functions of them are independent.", "D": "No; Y=0 forces X=0, whereas P(X=0)<1." } }, "12.2d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: 100 independent fair flips. X=number of heads minus number of tails; Y=X².\nProblem: Cov(X,Y)?", "criteria": { "A": "0", "B": "100", "C": "−100", "D": "10000" } } } }, "attempts": [ { "status": 200, "full_response_ms": 330.14270900093834, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"12.1a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":1.0,\"A\":0.0,\"C\":0.0}},\"12.1b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":1.0,\"A\":0.0,\"C\":0.0}},\"12.1c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.52,\"probabilities\":{\"D\":0.0,\"B\":0.36,\"A\":0.0,\"C\":0.64}},\"12.2a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"B\":0.0,\"A\":0.0,\"C\":0.0}},\"12.2b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"B\":1.0,\"D\":0.0,\"A\":0.0,\"C\":0.0}},\"12.2c\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"B\":0.0,\"A\":0.0,\"C\":0.0}},\"12.2d\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.83,\"probabilities\":{\"D\":0.01,\"B\":0.11,\"A\":0.87,\"C\":0.01}}},\"usage\":{\"input_tokens\":1251,\"output_tokens\":325}}" } ], "response": { "model": "jev-1.13.0", "answers": { "12.1a": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 1.0, "A": 0.0, "C": 0.0 } }, "12.1b": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 1.0, "A": 0.0, "C": 0.0 } }, "12.1c": { "type": "choice", "choice": "C", "confidence": 0.52, "probabilities": { "D": 0.0, "B": 0.36, "A": 0.0, "C": 0.64 } }, "12.2a": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "A": 0.0, "C": 0.0 } }, "12.2b": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "B": 1.0, "D": 0.0, "A": 0.0, "C": 0.0 } }, "12.2c": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "A": 0.0, "C": 0.0 } }, "12.2d": { "type": "choice", "choice": "A", "confidence": 0.83, "probabilities": { "D": 0.01, "B": 0.11, "A": 0.87, "C": 0.01 } } }, "usage": { "input_tokens": 1251, "output_tokens": 325 } } }, { "exam": "final_fa23", "section": 13, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "13.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Uniform random linear permutation of 2n distinct socks, n>=2 labeled left-right matching pairs. X counts pairs whose two socks are adjacent.\nProblem: Probability pair1 is adjacent?", "criteria": { "A": "1/n", "B": "1/(2n−1)", "C": "2/n", "D": "2/(2n−1)" } }, "13.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Uniform random linear permutation of 2n distinct socks, n>=2 labeled left-right matching pairs. X counts pairs whose two socks are adjacent.\nProblem: E[X]?", "criteria": { "A": "1", "B": "n", "C": "n/2", "D": "2" } }, "13.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Uniform random linear permutation of 2n distinct socks, n>=2 labeled left-right matching pairs. X counts pairs whose two socks are adjacent.\nProblem: Probability both pairs1 and2 are adjacent?", "criteria": { "A": "2/((2n)(2n−1))", "B": "4/((2n−1)(2n−2))", "C": "1/n²", "D": "4/((2n)(2n−1))" } }, "13.1d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Uniform random linear permutation of 2n distinct socks, n>=2 labeled left-right matching pairs. X counts pairs whose two socks are adjacent.\nProblem: Let p=P(pair1 adjacent),q=P(pairs1,2 adjacent). Var(X)?", "criteria": { "A": "nq−n²p²", "B": "np(1−p)", "C": "np+n(n−1)q−n²p²", "D": "np+n(n−1)q" } }, "13.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: k students have independent uniform birthdays on 365 days.\nProblem: Expected unordered pairs sharing a birthday?", "criteria": { "A": "C(k,2)/365", "B": "C(k,2)/365²", "C": "365/C(k,2)", "D": "k/365" } }, "13.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: k students have independent uniform birthdays on 365 days.\nProblem: Expected unordered triples sharing a birthday?", "criteria": { "A": "C(k,3)/365³", "B": "C(k,3)/365²", "C": "k/365²", "D": "C(k,3)/365" } }, "13.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: k students have independent uniform birthdays on 365 days.\nProblem: Variance of number of matching-birthday pairs?", "criteria": { "A": "C(k,2)*(1/365)*(364/365)", "B": "C(k,2)/365²", "C": "C(k,2)/365", "D": "C(k,2)²*(1/365)*(364/365)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 418.6213749999297, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"13.1a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.53,\"probabilities\":{\"D\":0.64,\"A\":0.03,\"B\":0.32,\"C\":0.01}},\"13.1b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.5,\"probabilities\":{\"D\":0.05,\"A\":0.62,\"B\":0.11,\"C\":0.22}},\"13.1c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.34,\"probabilities\":{\"D\":0.25,\"A\":0.23,\"B\":0.51,\"C\":0.01}},\"13.1d\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.98,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"B\":0.01,\"C\":0.99}},\"13.2a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":1.0,\"B\":0.0,\"C\":0.0}},\"13.2b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.85,\"probabilities\":{\"D\":0.04,\"A\":0.07,\"B\":0.89,\"C\":0.0}},\"13.2c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.66,\"probabilities\":{\"D\":0.11,\"A\":0.75,\"B\":0.02,\"C\":0.12}}},\"usage\":{\"input_tokens\":1325,\"output_tokens\":325}}" } ], "response": { "model": "jev-1.13.0", "answers": { "13.1a": { "type": "choice", "choice": "D", "confidence": 0.53, "probabilities": { "D": 0.64, "A": 0.03, "B": 0.32, "C": 0.01 } }, "13.1b": { "type": "choice", "choice": "A", "confidence": 0.5, "probabilities": { "D": 0.05, "A": 0.62, "B": 0.11, "C": 0.22 } }, "13.1c": { "type": "choice", "choice": "B", "confidence": 0.34, "probabilities": { "D": 0.25, "A": 0.23, "B": 0.51, "C": 0.01 } }, "13.1d": { "type": "choice", "choice": "C", "confidence": 0.98, "probabilities": { "D": 0.0, "A": 0.0, "B": 0.01, "C": 0.99 } }, "13.2a": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 1.0, "B": 0.0, "C": 0.0 } }, "13.2b": { "type": "choice", "choice": "B", "confidence": 0.85, "probabilities": { "D": 0.04, "A": 0.07, "B": 0.89, "C": 0.0 } }, "13.2c": { "type": "choice", "choice": "A", "confidence": 0.66, "probabilities": { "D": 0.11, "A": 0.75, "B": 0.02, "C": 0.12 } } }, "usage": { "input_tokens": 1325, "output_tokens": 325 } } }, { "exam": "final_fa23", "section": 14, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "14.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent biased coin flips with heads probability p>0. X counts flips through the kth head, k>=1; Y=X/k.\nProblem: Why is Y unbiased for 1/p?", "criteria": { "A": "The inverse of any unbiased estimator of p is unbiased for1/p.", "B": "X is geometric with parameter kp, so dividing by k yields1/p.", "C": "X always equals k/p, so Y has no sampling variability.", "D": "Split X into k independent geometric waiting times, each mean1/p; E[X]=k/p and E[Y]=1/p." } }, "14.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent biased coin flips with heads probability p>0. X counts flips through the kth head, k>=1; Y=X/k.\nProblem: Var(Y)?", "criteria": { "A": "k*(1−p)/p²", "B": "(1−p)/(k²*p²)", "C": "(1−p)/p²", "D": "(1−p)/(k*p²)" } }, "14.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent biased coin flips with heads probability p>0. X counts flips through the kth head, k>=1; Y=X/k.\nProblem: Chebyshev bound for P(|Y−1/p|>=epsilon/p), clipped at1?", "criteria": { "A": "min(1,(1−p)/(epsilon²*k*p²))", "B": "min(1,(1−p)/(epsilon*k))", "C": "min(1,p/(epsilon²*k))", "D": "min(1,(1−p)/(epsilon²*k))" } }, "14.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent biased coin flips with heads probability p>0. X counts flips through the kth head, k>=1; Y=X/k.\nProblem: Only p>=1/2 is known. Smallest k certified by that Chebyshev bound for relative error strictly below10% with probability at least95%?", "criteria": { "A": "100", "B": "2000", "C": "500", "D": "1000" } } } }, "attempts": [ { "status": 200, "full_response_ms": 309.46979200234637, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"14.1\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"A\":0.0,\"D\":1.0,\"B\":0.0}},\"14.2\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.95,\"probabilities\":{\"C\":0.0,\"B\":0.03,\"A\":0.0,\"D\":0.97}},\"14.3\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.92,\"probabilities\":{\"D\":0.95,\"B\":0.0,\"C\":0.0,\"A\":0.05}},\"14.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.1,\"probabilities\":{\"C\":0.23,\"B\":0.33,\"A\":0.15,\"D\":0.29}}},\"usage\":{\"input_tokens\":990,\"output_tokens\":183}}" } ], "response": { "model": "jev-1.13.0", "answers": { "14.1": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "C": 0.0, "A": 0.0, "D": 1.0, "B": 0.0 } }, "14.2": { "type": "choice", "choice": "D", "confidence": 0.95, "probabilities": { "C": 0.0, "B": 0.03, "A": 0.0, "D": 0.97 } }, "14.3": { "type": "choice", "choice": "D", "confidence": 0.92, "probabilities": { "D": 0.95, "B": 0.0, "C": 0.0, "A": 0.05 } }, "14.4": { "type": "choice", "choice": "B", "confidence": 0.1, "probabilities": { "C": 0.23, "B": 0.33, "A": 0.15, "D": 0.29 } } }, "usage": { "input_tokens": 990, "output_tokens": 183 } } }, { "exam": "final_fa23", "section": 15, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "15.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Drive 360 miles at one speed S chosen uniformly in [40,60] mph. Expected hours? Give a valid integral and value.", "criteria": { "A": "E[S]/360=50/360", "B": "(1/20)integral(40..60)360/s ds=18ln(3/2)", "C": "360/E[S]=360/50", "D": "integral(40..60)360/s ds=360ln(3/2)" } }, "15.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X,Y,Z~Uniform[0,1].\nProblem: CDF of max(X,Y,Z) on [0,1]?", "criteria": { "A": "t³", "B": "1−(1−t)³", "C": "3t²", "D": "t" } }, "15.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X,Y,Z~Uniform[0,1].\nProblem: PDF of max(X,Y,Z) on [0,1]?", "criteria": { "A": "3(1−t)²", "B": "3t²", "C": "1", "D": "t³" } }, "15.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X,Y,Z~Uniform[0,1].\nProblem: E[max(X,Y,Z)]?", "criteria": { "A": "3/4", "B": "1/2", "C": "1/4", "D": "2/3" } }, "15.2d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X,Y,Z~Uniform[0,1].\nProblem: PDF of min(X,Y,Z) on [0,1]?", "criteria": { "A": "3t²", "B": "1−t³", "C": "(1−t)³", "D": "3(1−t)²" } }, "15.2e": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X,Y,Z~Uniform[0,1].\nProblem: Expected median of X,Y,Z?", "criteria": { "A": "1/4", "B": "1/2", "C": "2/3", "D": "3/4" } }, "15.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent TS~N(mean4,variance0.2),TJ~N(mean4,variance0.3).\nProblem: Distribution of TS+TJ, second parameter variance?", "criteria": { "A": "N(8,0.13)", "B": "N(4,0.5)", "C": "N(8,0.1)", "D": "N(8,0.5)" } }, "15.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent TS~N(mean4,variance0.2),TJ~N(mean4,variance0.3).\nProblem: Distribution of TS−TJ, second parameter variance?", "criteria": { "A": "N(0,0.5)", "B": "N(0,0.1)", "C": "N(8,0.5)", "D": "N(0,−0.1)" } }, "15.3c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent TS~N(mean4,variance0.2),TJ~N(mean4,variance0.3).\nProblem: P(TS>TJ)?", "criteria": { "A": "2/5", "B": "0", "C": "1/2", "D": "3/5" } } } }, "attempts": [ { "status": 200, "full_response_ms": 312.2406250004133, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"15.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"C\":0.0,\"B\":0.99,\"A\":0.0,\"D\":0.01}},\"15.2a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.89,\"probabilities\":{\"D\":0.0,\"B\":0.08,\"C\":0.0,\"A\":0.92}},\"15.2b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"C\":0.0,\"B\":1.0}},\"15.2c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"D\":0.0,\"A\":1.0,\"C\":0.0,\"B\":0.0}},\"15.2d\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.99,\"probabilities\":{\"C\":0.0,\"B\":0.0,\"A\":0.01,\"D\":0.99}},\"15.2e\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"D\":0.0,\"B\":1.0,\"C\":0.0,\"A\":0.0}},\"15.3a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"B\":0.0,\"A\":0.0,\"C\":0.0}},\"15.3b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.91,\"probabilities\":{\"D\":0.0,\"A\":0.93,\"C\":0.0,\"B\":0.07}},\"15.3c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.76,\"probabilities\":{\"C\":0.82,\"B\":0.01,\"A\":0.11,\"D\":0.06}}},\"usage\":{\"input_tokens\":1537,\"output_tokens\":416}}" } ], "response": { "model": "jev-1.13.0", "answers": { "15.1": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "C": 0.0, "B": 0.99, "A": 0.0, "D": 0.01 } }, "15.2a": { "type": "choice", "choice": "A", "confidence": 0.89, "probabilities": { "D": 0.0, "B": 0.08, "C": 0.0, "A": 0.92 } }, "15.2b": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.0, "B": 1.0 } }, "15.2c": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "D": 0.0, "A": 1.0, "C": 0.0, "B": 0.0 } }, "15.2d": { "type": "choice", "choice": "D", "confidence": 0.99, "probabilities": { "C": 0.0, "B": 0.0, "A": 0.01, "D": 0.99 } }, "15.2e": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "D": 0.0, "B": 1.0, "C": 0.0, "A": 0.0 } }, "15.3a": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "A": 0.0, "C": 0.0 } }, "15.3b": { "type": "choice", "choice": "A", "confidence": 0.91, "probabilities": { "D": 0.0, "A": 0.93, "C": 0.0, "B": 0.07 } }, "15.3c": { "type": "choice", "choice": "C", "confidence": 0.76, "probabilities": { "C": 0.82, "B": 0.01, "A": 0.11, "D": 0.06 } } }, "usage": { "input_tokens": 1537, "output_tokens": 416 } } }, { "exam": "final_fa23", "section": 16, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "16.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Uniformly choose one of three circular boards with positive radii a,b,c, then throw a dart uniformly over the chosen board’s area.\nProblem: Expected distance to center?", "criteria": { "A": "(a+b+c)/6", "B": "(a+b+c)/3", "C": "2(a²+b²+c²)/9", "D": "2(a+b+c)/9" } }, "16.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Uniformly choose one of three circular boards with positive radii a,b,c, then throw a dart uniformly over the chosen board’s area.\nProblem: Justify the expected distance.", "criteria": { "A": "The mean distance equals the radius by rotational symmetry.", "B": "Boards must be weighted proportional to their area, despite uniform board selection.", "C": "Distance is uniform on[0,r], so each conditional mean is r/2.", "D": "At fixed radius r, radial density2x/r² yields conditional mean2r/3. Average these means with weight1/3 over the three boards." } }, "16.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Y has mean7,variance2. Independent Z~N(mean3,variance5). X=Y+Z.\nProblem: Given Y=y, give the minimum-MSE estimate of X and its conditional MSE.", "criteria": { "A": "(y+3,7)", "B": "(y+7,2)", "C": "(y+3,5)", "D": "(10,7)" } }, "16.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Y has mean7,variance2. Independent Z~N(mean3,variance5). X=Y+Z.\nProblem: Best affine estimator of X from Y?", "criteria": { "A": "(2/7)Y+8", "B": "Y+7", "C": "10", "D": "Y+3" } }, "16.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: X~N(69,3),Y1,Y2~N(0,1) mutually independent, second parameters variances. Twins have Z1=X+Y1,Z2=X+Y2. Best affine estimator of Z2 given Z1?", "criteria": { "A": "Z1", "B": "69+(4/3)(Z1−69)", "C": "69+(3/4)(Z1−69)", "D": "69+(1/4)(Z1−69)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 354.43083299833233, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"16.1a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.97,\"probabilities\":{\"B\":0.01,\"D\":0.98,\"A\":0.01,\"C\":0.0}},\"16.1b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"D\":1.0,\"C\":0.0,\"A\":0.0}},\"16.2a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.85,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"C\":0.89,\"A\":0.11}},\"16.2b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.96,\"probabilities\":{\"B\":0.01,\"D\":0.97,\"C\":0.0,\"A\":0.02}},\"16.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.48,\"probabilities\":{\"B\":0.37,\"D\":0.01,\"C\":0.6,\"A\":0.02}}},\"usage\":{\"input_tokens\":1117,\"output_tokens\":232}}" } ], "response": { "model": "jev-1.13.0", "answers": { "16.1a": { "type": "choice", "choice": "D", "confidence": 0.97, "probabilities": { "B": 0.01, "D": 0.98, "A": 0.01, "C": 0.0 } }, "16.1b": { "type": "choice", "choice": "D", "confidence": 0.99, "probabilities": { "B": 0.0, "D": 1.0, "C": 0.0, "A": 0.0 } }, "16.2a": { "type": "choice", "choice": "C", "confidence": 0.85, "probabilities": { "B": 0.0, "D": 0.0, "C": 0.89, "A": 0.11 } }, "16.2b": { "type": "choice", "choice": "D", "confidence": 0.96, "probabilities": { "B": 0.01, "D": 0.97, "C": 0.0, "A": 0.02 } }, "16.3": { "type": "choice", "choice": "C", "confidence": 0.48, "probabilities": { "B": 0.37, "D": 0.01, "C": 0.6, "A": 0.02 } } }, "usage": { "input_tokens": 1117, "output_tokens": 232 } } }, { "exam": "final_fa23", "section": 17, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "17.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Chain states0,1,2. Row0:[1−p,p,0]; row1:[1−p,0,p]; row2:[0,1−p,p].\nProblem: p=1/3. Choose stationary equations plus normalization.", "criteria": { "A": "pi0=pi1=pi2=1/3", "B": "pi0=2pi0/3; pi1=pi0/3; pi2=pi1/3+pi2; sum pi=1", "C": "pi0=2(pi0+pi1)/3; pi1=pi0/3+2pi2/3; pi2=(pi1+pi2)/3; sum pi=1", "D": "pi0=2pi0/3+pi1/3; pi1=2pi0/3+pi2/3; pi2=2pi1/3+pi2/3; sum pi=1" } }, "17.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Chain states0,1,2. Row0:[1−p,p,0]; row1:[1−p,0,p]; row2:[0,1−p,p]. Now choose p such that stationary masses are (1,3,9)/13. X has that stationary law and Y=0 for X=0 or1, Y=1 for X=2.\nProblem: E[X given Y]?", "criteria": { "A": "3/4 if Y=0;2 if Y=1", "B": "1/2 if Y=0;2 if Y=1", "C": "3/13 if Y=0;18/13 if Y=1", "D": "0 if Y=0;1 if Y=1" } }, "17.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Chain states0,1,2. Row0:[1−p,p,0]; row1:[1−p,0,p]; row2:[0,1−p,p]. Now choose p such that stationary masses are (1,3,9)/13. X has that stationary law and Y=0 for X=0 or1, Y=1 for X=2.\nProblem: Best affine estimate of X as aY+b?", "criteria": { "A": "(5/4)Y+3/4", "B": "(3/4)Y+5/4", "C": "(18/13)Y+3/13", "D": "2Y" } }, "17.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Roll a fair six-sided die until the product of the last two rolls is15. StateA=start or previous roll in{1,2,4,6}; B=previous in{3,5} but target not hit; C=target hit, then stop and remain atC.\nProblem: Give (PAA,PAB,PBA,PBB,PBC,PCC).", "criteria": { "A": "(1/3,2/3,1/3,1/3,1/3,1)", "B": "(2/3,1/3,2/3,0,1/3,1)", "C": "(2/3,1/3,1/6,2/3,1/6,1)", "D": "(2/3,1/3,2/3,1/6,1/6,1)" } }, "17.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Roll a fair six-sided die until the product of the last two rolls is15. StateA=start or previous roll in{1,2,4,6}; B=previous in{3,5} but target not hit; C=target hit, then stop and remain atC.\nProblem: This stopped chain is irreducible.", "criteria": { "A": "True", "B": "False" } }, "17.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Roll a fair six-sided die until the product of the last two rolls is15. StateA=start or previous roll in{1,2,4,6}; B=previous in{3,5} but target not hit; C=target hit, then stop and remain atC.\nProblem: Every state of this chain has period one.", "criteria": { "A": "False", "B": "True" } }, "17.2d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Roll a fair six-sided die until the product of the last two rolls is15. StateA=start or previous roll in{1,2,4,6}; B=previous in{3,5} but target not hit; C=target hit, then stop and remain atC.\nProblem: Let a0=PAA,a1=PAB,b1=PBA,b0=PBB,b2=PBC,c0=PCC. Equations for expected remaining rolls beta to hit C?", "criteria": { "A": "betaA=a0*betaA+a1*betaB; betaB=b1*betaA+b0*betaB+b2*betaC; betaC=0", "B": "betaA=1+a0*betaA+b1*betaB; betaB=1+a1*betaA+b0*betaB; betaC=0", "C": "betaA=1+a0*betaA+a1*betaB; betaB=1+b1*betaA+b0*betaB+b2*betaC; betaC=1+c0*betaC", "D": "betaA=1+a0*betaA+a1*betaB; betaB=1+b1*betaA+b0*betaB+b2*betaC; betaC=0" } } } }, "attempts": [ { "status": 200, "full_response_ms": 368.06929200247396, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"17.1a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.21,\"probabilities\":{\"C\":0.4,\"B\":0.02,\"D\":0.37,\"A\":0.21}},\"17.1b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.66,\"probabilities\":{\"C\":0.01,\"D\":0.01,\"B\":0.23,\"A\":0.75}},\"17.1c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.23,\"probabilities\":{\"C\":0.32,\"B\":0.16,\"D\":0.09,\"A\":0.43}},\"17.2a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.54,\"probabilities\":{\"C\":0.14,\"B\":0.08,\"D\":0.66,\"A\":0.12}},\"17.2b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.98,\"probabilities\":{\"B\":0.99,\"A\":0.01}},\"17.2c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.19,\"probabilities\":{\"B\":0.6,\"A\":0.4}},\"17.2d\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.92,\"probabilities\":{\"C\":0.01,\"B\":0.04,\"D\":0.94,\"A\":0.01}}},\"usage\":{\"input_tokens\":1860,\"output_tokens\":297}}" } ], "response": { "model": "jev-1.13.0", "answers": { "17.1a": { "type": "choice", "choice": "C", "confidence": 0.21, "probabilities": { "C": 0.4, "B": 0.02, "D": 0.37, "A": 0.21 } }, "17.1b": { "type": "choice", "choice": "A", "confidence": 0.66, "probabilities": { "C": 0.01, "D": 0.01, "B": 0.23, "A": 0.75 } }, "17.1c": { "type": "choice", "choice": "A", "confidence": 0.23, "probabilities": { "C": 0.32, "B": 0.16, "D": 0.09, "A": 0.43 } }, "17.2a": { "type": "choice", "choice": "D", "confidence": 0.54, "probabilities": { "C": 0.14, "B": 0.08, "D": 0.66, "A": 0.12 } }, "17.2b": { "type": "choice", "choice": "B", "confidence": 0.98, "probabilities": { "B": 0.99, "A": 0.01 } }, "17.2c": { "type": "choice", "choice": "B", "confidence": 0.19, "probabilities": { "B": 0.6, "A": 0.4 } }, "17.2d": { "type": "choice", "choice": "D", "confidence": 0.92, "probabilities": { "C": 0.01, "B": 0.04, "D": 0.94, "A": 0.01 } } }, "usage": { "input_tokens": 1860, "output_tokens": 297 } } }, { "exam": "final_fa24", "section": 2, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "2.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Truth status of P OR Q OR R OR NOT R?", "criteria": { "A": "False", "B": "Could be either", "C": "True" } }, "2.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Truth status of (NOT P implies (R AND NOT R)) implies P?", "criteria": { "A": "True", "B": "False", "C": "Could be either" } }, "2.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Truth status of (R AND NOT R) OR (Q AND P)?", "criteria": { "A": "Could be either", "B": "False", "C": "True" } }, "2.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: For all x in nonempty U, P(x) implies [Q(x) OR exists y R(x,y)].\nProblem: P(x) is true and all R(x,y) false. Truth status of Q(x)?", "criteria": { "A": "Could be either", "B": "True", "C": "False" } }, "2.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: For all x in nonempty U, P(x) implies [Q(x) OR exists y R(x,y)].\nProblem: Q(x) is false and all R(x,y) false. Truth status of P(x)?", "criteria": { "A": "True", "B": "Could be either", "C": "False" } }, "2.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: For all x in nonempty U, P(x) implies [Q(x) OR exists y R(x,y)].\nProblem: P(x) is true and some R(x,y) true. Truth status of Q(x)?", "criteria": { "A": "False", "B": "True", "C": "Could be either" } }, "2.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Express |a−b| using only min(a,b) and max(a,b).", "criteria": { "A": "max(a,b)−min(a,b)", "B": "max(a,b)*min(a,b)", "C": "min(a,b)−max(a,b)", "D": "max(a,b)+min(a,b)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 350.9275839969632, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"2.1a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"B\":0.01,\"A\":0.0,\"C\":0.99}},\"2.1b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.92,\"probabilities\":{\"B\":0.02,\"A\":0.95,\"C\":0.03}},\"2.1c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.88,\"probabilities\":{\"B\":0.07,\"A\":0.93,\"C\":0.0}},\"2.2a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"B\":0.99,\"A\":0.01,\"C\":0.0}},\"2.2b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.98,\"probabilities\":{\"B\":0.01,\"A\":0.0,\"C\":0.99}},\"2.2c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"A\":0.0,\"C\":1.0}},\"2.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":1.0,\"C\":0.0}}},\"usage\":{\"input_tokens\":1089,\"output_tokens\":275}}" } ], "response": { "model": "jev-1.13.0", "answers": { "2.1a": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "B": 0.01, "A": 0.0, "C": 0.99 } }, "2.1b": { "type": "choice", "choice": "A", "confidence": 0.92, "probabilities": { "B": 0.02, "A": 0.95, "C": 0.03 } }, "2.1c": { "type": "choice", "choice": "A", "confidence": 0.88, "probabilities": { "B": 0.07, "A": 0.93, "C": 0.0 } }, "2.2a": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "B": 0.99, "A": 0.01, "C": 0.0 } }, "2.2b": { "type": "choice", "choice": "C", "confidence": 0.98, "probabilities": { "B": 0.01, "A": 0.0, "C": 0.99 } }, "2.2c": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "A": 0.0, "C": 1.0 } }, "2.3": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 1.0, "C": 0.0 } } }, "usage": { "input_tokens": 1089, "output_tokens": 275 } } }, { "exam": "final_fa24", "section": 3, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "3.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Complete: forall n [P(n)=>P(n+1)] is equivalent to NOT exists n [P(n) AND __].", "criteria": { "A": "NOT P(n+1)", "B": "P(n+1)", "C": "P(n)", "D": "NOT P(n)" } }, "3.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: F0=F1=1 and Fi=F(i−1)+F(i−2).\nProblem: Largest real c>=1 such that Fi>=c^(i−1) for every i>=1?", "criteria": { "A": "3/2", "B": "2", "C": "(1+sqrt(5))/2", "D": "sqrt(2)" } }, "3.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: F0=F1=1 and Fi=F(i−1)+F(i−2).\nProblem: Which proves the optimal exponential lower bound?", "criteria": { "A": "Let phi=(1+sqrt(5))/2, so phi²=phi+1. Bases i=1,2 hold, and adding inductive bounds gives Fi>=phi^(i−1). Similarly Fi<=phi^i; any c>phi eventually has c^(i−1)>phi^i, so cannot work for all i.", "B": "A lower bound that works for one i must work for every later i.", "C": "Since F2=2, c=2 works for every i without checking the recurrence.", "D": "The recurrence implies Fi=phi^(i−1) exactly for all i." } } } }, "attempts": [ { "status": 200, "full_response_ms": 382.2543750029581, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"3.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"C\":0.0,\"A\":0.99,\"B\":0.0,\"D\":0.01}},\"3.3a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.75,\"probabilities\":{\"C\":0.8099999999999999,\"B\":0.0,\"A\":0.04,\"D\":0.15}},\"3.3b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":0.0,\"A\":1.0,\"D\":0.0}}},\"usage\":{\"input_tokens\":822,\"output_tokens\":137}}" } ], "response": { "model": "jev-1.13.0", "answers": { "3.2": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "C": 0.0, "A": 0.99, "B": 0.0, "D": 0.01 } }, "3.3a": { "type": "choice", "choice": "C", "confidence": 0.75, "probabilities": { "C": 0.8099999999999999, "B": 0.0, "A": 0.04, "D": 0.15 } }, "3.3b": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 1.0, "D": 0.0 } } }, "usage": { "input_tokens": 822, "output_tokens": 137 } } }, { "exam": "final_fa24", "section": 4, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "4.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Jobs A:1>2,B:2>1; candidates 1:B>A,2:A>B.\nProblem: Job-optimal matching?", "criteria": { "A": "A–2,B–1", "B": "A–1,B–2" } }, "4.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Jobs A:1>2,B:2>1; candidates 1:B>A,2:A>B.\nProblem: Candidate-optimal matching?", "criteria": { "A": "A–1,B–2", "B": "A–2,B–1" } }, "4.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Jobs A:1>2>3>4; B:2>1>4>3; C:3>4>1>2; D:4>3>2>1. Candidates 1:B>A>C>D; 2:A>B>D>C; 3:D>C>A>B; 4:C>D>B>A.\nProblem: Number of stable matchings?", "criteria": { "A": "1", "B": "2", "C": "8", "D": "4" } }, "4.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Jobs A:1>2>3>4; B:2>1>4>3; C:3>4>1>2; D:4>3>2>1. Candidates 1:B>A>C>D; 2:A>B>D>C; 3:D>C>A>B; 4:C>D>B>A.\nProblem: Choose a stable matching optimal for neither entire side.", "criteria": { "A": "A–2,B–1,C–4,D–3", "B": "A–1,B–2,C–3,D–4", "C": "A–1,B–2,C–4,D–3", "D": "A–3,B–4,C–1,D–2" } } } }, "attempts": [ { "status": 200, "full_response_ms": 321.4624169995659, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"4.1a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.63,\"probabilities\":{\"A\":0.81,\"B\":0.19}},\"4.1b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.35,\"probabilities\":{\"A\":0.32,\"B\":0.68}},\"4.2a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.17,\"probabilities\":{\"D\":0.38,\"B\":0.18,\"A\":0.21,\"C\":0.23}},\"4.2b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.33,\"probabilities\":{\"D\":0.49,\"A\":0.24,\"B\":0.16,\"C\":0.11}}},\"usage\":{\"input_tokens\":895,\"output_tokens\":155}}" } ], "response": { "model": "jev-1.13.0", "answers": { "4.1a": { "type": "choice", "choice": "A", "confidence": 0.63, "probabilities": { "A": 0.81, "B": 0.19 } }, "4.1b": { "type": "choice", "choice": "B", "confidence": 0.35, "probabilities": { "A": 0.32, "B": 0.68 } }, "4.2a": { "type": "choice", "choice": "D", "confidence": 0.17, "probabilities": { "D": 0.38, "B": 0.18, "A": 0.21, "C": 0.23 } }, "4.2b": { "type": "choice", "choice": "D", "confidence": 0.33, "probabilities": { "D": 0.49, "A": 0.24, "B": 0.16, "C": 0.11 } } }, "usage": { "input_tokens": 895, "output_tokens": 155 } } }, { "exam": "final_fa24", "section": 5, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "5.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Graph G has k connected components. Sum of vertex degrees?", "criteria": { "A": "2|V|", "B": "|E|", "C": "2|E|", "D": "|V|−k" } }, "5.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Maximum edges in a simple graph with v vertices and k components,1<=k<=v?", "criteria": { "A": "C(v,2)−k", "B": "v−k", "C": "C(v−k+1,2)", "D": "k*C(v/k,2)" } }, "5.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Minimum edges with v vertices and k components?", "criteria": { "A": "v+k", "B": "k−1", "C": "v−1", "D": "v−k" } }, "5.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Minimum vertex colors for a positive-dimensional hypercube?", "criteria": { "A": "log2(|V|)", "B": "log2(|V|)+1", "C": "2", "D": "1" } }, "5.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Minimum edge colors for a hypercube on |V| vertices?", "criteria": { "A": "2", "B": "log2(|V|)+1", "C": "log2(|V|)", "D": "|V|/2" } }, "5.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Delete all edges of a simple k-cycle from a connected graph, retaining vertices. Sharp upper bound on component count?", "criteria": { "A": "k", "B": "2k", "C": "k−1", "D": "k+1" } } } }, "attempts": [ { "status": 200, "full_response_ms": 364.28287500166334, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"5.1a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":0.0,\"C\":1.0}},\"5.1b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":0.0,\"C\":1.0}},\"5.1c\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"B\":0.0,\"A\":0.0,\"C\":0.0}},\"5.2a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.96,\"probabilities\":{\"B\":0.01,\"D\":0.0,\"A\":0.01,\"C\":0.98}},\"5.2b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.94,\"probabilities\":{\"B\":0.03,\"D\":0.0,\"A\":0.01,\"C\":0.96}},\"5.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.48,\"probabilities\":{\"B\":0.04,\"D\":0.29,\"A\":0.62,\"C\":0.05}}},\"usage\":{\"input_tokens\":1004,\"output_tokens\":272}}" } ], "response": { "model": "jev-1.13.0", "answers": { "5.1a": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 } }, "5.1b": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 } }, "5.1c": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "A": 0.0, "C": 0.0 } }, "5.2a": { "type": "choice", "choice": "C", "confidence": 0.96, "probabilities": { "B": 0.01, "D": 0.0, "A": 0.01, "C": 0.98 } }, "5.2b": { "type": "choice", "choice": "C", "confidence": 0.94, "probabilities": { "B": 0.03, "D": 0.0, "A": 0.01, "C": 0.96 } }, "5.3": { "type": "choice", "choice": "A", "confidence": 0.48, "probabilities": { "B": 0.04, "D": 0.29, "A": 0.62, "C": 0.05 } } }, "usage": { "input_tokens": 1004, "output_tokens": 272 } } }, { "exam": "final_fa24", "section": 6, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "6.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: 2^63 modulo 77?", "criteria": { "A": "8", "B": "2", "C": "16", "D": "1" } }, "6.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: 1*2*3*4*5*6 modulo 7?", "criteria": { "A": "0", "B": "5", "C": "1", "D": "6" } }, "6.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Inverse of 63 modulo 71 in 0..70?", "criteria": { "A": "62", "B": "8", "C": "63", "D": "9" } }, "6.4a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: p,q are distinct primes; a,b range independently over residues 0..pq−1.\nProblem: Number of a coprime to p?", "criteria": { "A": "(p−1)q", "B": "pq−p", "C": "pq−1", "D": "(p−1)(q−1)" } }, "6.4b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: p,q are distinct primes; a,b range independently over residues 0..pq−1.\nProblem: Number of a coprime to both p and q?", "criteria": { "A": "(p−1)(q−1)", "B": "pq−p−q", "C": "pq−1", "D": "p+q−2" } }, "6.4c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: p,q are distinct primes; a,b range independently over residues 0..pq−1.\nProblem: Number of pairs (a,b) for which ax=b mod pq has exactly one solution?", "criteria": { "A": "pq−1", "B": "(pq)²", "C": "pq(p−1)(q−1)", "D": "(p−1)(q−1)" } }, "6.4d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: p,q are distinct primes; a,b range independently over residues 0..pq−1.\nProblem: gcd(a,pq)=p and x is a solution. Smallest positive shift t for which x+t is another solution?", "criteria": { "A": "pq", "B": "1", "C": "q", "D": "p" } }, "6.4e": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: p,q are distinct primes; a,b range independently over residues 0..pq−1.\nProblem: Number of pairs (a,b) with exactly p solutions?", "criteria": { "A": "q(q−1)", "B": "pq", "C": "q(p−1)", "D": "p(p−1)" } }, "6.4f": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: p,q are distinct primes; a,b range independently over residues 0..pq−1.\nProblem: Number of pairs (a,b) with no solutions?", "criteria": { "A": "(q−1)(pq−q)+(p−1)(pq−p)+(pq−1)", "B": "(pq−1)²", "C": "(p−1)(q−1)pq", "D": "(q−1)(pq−q)+(p−1)(pq−p)" } }, "6.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For every N>=2, N is prime if and only if no a,b in {1,...,N−1} have ab equal to __ mod N. Choose the residue that makes this equivalence valid.", "criteria": { "A": "1", "B": "2", "C": "0", "D": "N−1" } }, "6.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Use CRT to count units modulo pq for distinct primes p,q.", "criteria": { "A": "CRT bijects residues mod pq with pairs mod p and q; a unit corresponds exactly to a pair with both coordinates nonzero, giving (p−1)(q−1).", "B": "All nonzero residues modulo pq are units, giving pq−1.", "C": "CRT requires both coordinates to match as integers, giving min(p,q)−1.", "D": "Exactly one coordinate must be nonzero, giving p+q−2." } } } }, "attempts": [ { "status": 200, "full_response_ms": 399.7862910000549, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"6.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.17,\"probabilities\":{\"D\":0.13,\"C\":0.35,\"B\":0.14,\"A\":0.38}},\"6.2\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.89,\"probabilities\":{\"D\":0.91,\"C\":0.03,\"B\":0.03,\"A\":0.03}},\"6.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.22,\"probabilities\":{\"D\":0.18,\"C\":0.02,\"B\":0.38,\"A\":0.42}},\"6.4a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.45,\"probabilities\":{\"D\":0.0,\"C\":0.0,\"B\":0.41,\"A\":0.59}},\"6.4b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"D\":0.0,\"C\":0.0,\"B\":0.01,\"A\":0.99}},\"6.4c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.8,\"probabilities\":{\"D\":0.05,\"C\":0.85,\"B\":0.05,\"A\":0.05}},\"6.4d\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.26,\"probabilities\":{\"D\":0.17,\"C\":0.38,\"B\":0.01,\"A\":0.44}},\"6.4e\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.3,\"probabilities\":{\"D\":0.07,\"C\":0.36,\"B\":0.1,\"A\":0.47}},\"6.4f\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.33,\"probabilities\":{\"D\":0.18,\"C\":0.27,\"B\":0.05,\"A\":0.5}},\"6.5\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.9,\"probabilities\":{\"D\":0.01,\"C\":0.93,\"B\":0.0,\"A\":0.06}},\"6.6\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"C\":0.0,\"B\":0.0,\"A\":1.0}}},\"usage\":{\"input_tokens\":1764,\"output_tokens\":493}}" } ], "response": { "model": "jev-1.13.0", "answers": { "6.1": { "type": "choice", "choice": "A", "confidence": 0.17, "probabilities": { "D": 0.13, "C": 0.35, "B": 0.14, "A": 0.38 } }, "6.2": { "type": "choice", "choice": "D", "confidence": 0.89, "probabilities": { "D": 0.91, "C": 0.03, "B": 0.03, "A": 0.03 } }, "6.3": { "type": "choice", "choice": "A", "confidence": 0.22, "probabilities": { "D": 0.18, "C": 0.02, "B": 0.38, "A": 0.42 } }, "6.4a": { "type": "choice", "choice": "A", "confidence": 0.45, "probabilities": { "D": 0.0, "C": 0.0, "B": 0.41, "A": 0.59 } }, "6.4b": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "D": 0.0, "C": 0.0, "B": 0.01, "A": 0.99 } }, "6.4c": { "type": "choice", "choice": "C", "confidence": 0.8, "probabilities": { "D": 0.05, "C": 0.85, "B": 0.05, "A": 0.05 } }, "6.4d": { "type": "choice", "choice": "A", "confidence": 0.26, "probabilities": { "D": 0.17, "C": 0.38, "B": 0.01, "A": 0.44 } }, "6.4e": { "type": "choice", "choice": "A", "confidence": 0.3, "probabilities": { "D": 0.07, "C": 0.36, "B": 0.1, "A": 0.47 } }, "6.4f": { "type": "choice", "choice": "A", "confidence": 0.33, "probabilities": { "D": 0.18, "C": 0.27, "B": 0.05, "A": 0.5 } }, "6.5": { "type": "choice", "choice": "C", "confidence": 0.9, "probabilities": { "D": 0.01, "C": 0.93, "B": 0.0, "A": 0.06 } }, "6.6": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "C": 0.0, "B": 0.0, "A": 1.0 } } }, "usage": { "input_tokens": 1764, "output_tokens": 493 } } }, { "exam": "final_fa24", "section": 7, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "7.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: GF(5): normalized Lagrange Delta0 for x-coordinates 0,1,4?", "criteria": { "A": "−x²+4", "B": "−x²−4", "C": "x²−1", "D": "x²+4" } }, "7.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: GF(5), a constant message polynomial is encoded; at most one error. Received points (0,1),(2,1),(3,4).\nProblem: Original polynomial?", "criteria": { "A": "3", "B": "1", "C": "x+1", "D": "4" } }, "7.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: GF(5), a constant message polynomial is encoded; at most one error. Received points (0,1),(2,1),(3,4).\nProblem: With Q(x)=q1*x+q0 and E(x)=x+b0, write the BW equation for received (3,4).", "criteria": { "A": "q1+q0=4b0 mod 5", "B": "3q1+q0=4(3+b0) mod 5", "C": "3q1+q0=3+b0 mod 5", "D": "4q1+q0=3(4+b0) mod 5" } }, "7.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Normalized Lagrange basis on distinct field coordinates: Delta_i(i)?", "criteria": { "A": "0", "B": "i", "C": "1", "D": "yi" } }, "7.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Normalized Lagrange basis: Delta_i(j) for a different interpolation coordinate j?", "criteria": { "A": "i−j", "B": "yj", "C": "1", "D": "0" } }, "7.3c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Interpolate (0,y0),(1,y1),(2,y2) using normalized basis D0,D1,D2.", "criteria": { "A": "D0+D1+D2", "B": "y0*D1+y1*D2+y2*D0", "C": "y0+y1*x+y2*x²", "D": "y0*D0+y1*D1+y2*D2" } } } }, "attempts": [ { "status": 200, "full_response_ms": 315.1168339973083, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"7.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.5,\"probabilities\":{\"C\":0.05,\"A\":0.63,\"B\":0.13,\"D\":0.19}},\"7.2a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.59,\"probabilities\":{\"C\":0.01,\"A\":0.09,\"B\":0.69,\"D\":0.21}},\"7.2b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.93,\"probabilities\":{\"C\":0.01,\"B\":0.94,\"A\":0.02,\"D\":0.03}},\"7.3a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"C\":0.99,\"A\":0.01,\"B\":0.0,\"D\":0.0}},\"7.3b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.29,\"probabilities\":{\"C\":0.46,\"A\":0.06,\"B\":0.02,\"D\":0.45999999999999996}},\"7.3c\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.99,\"probabilities\":{\"C\":0.0,\"A\":0.01,\"B\":0.0,\"D\":0.99}}},\"usage\":{\"input_tokens\":1133,\"output_tokens\":272}}" } ], "response": { "model": "jev-1.13.0", "answers": { "7.1": { "type": "choice", "choice": "A", "confidence": 0.5, "probabilities": { "C": 0.05, "A": 0.63, "B": 0.13, "D": 0.19 } }, "7.2a": { "type": "choice", "choice": "B", "confidence": 0.59, "probabilities": { "C": 0.01, "A": 0.09, "B": 0.69, "D": 0.21 } }, "7.2b": { "type": "choice", "choice": "B", "confidence": 0.93, "probabilities": { "C": 0.01, "B": 0.94, "A": 0.02, "D": 0.03 } }, "7.3a": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "C": 0.99, "A": 0.01, "B": 0.0, "D": 0.0 } }, "7.3b": { "type": "choice", "choice": "D", "confidence": 0.29, "probabilities": { "C": 0.46, "A": 0.06, "B": 0.02, "D": 0.45999999999999996 } }, "7.3c": { "type": "choice", "choice": "D", "confidence": 0.99, "probabilities": { "C": 0.0, "A": 0.01, "B": 0.0, "D": 0.99 } } }, "usage": { "input_tokens": 1133, "output_tokens": 272 } } }, { "exam": "final_fa24", "section": 8, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "8.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct anagrams of BERKELEY?", "criteria": { "A": "7!/3!", "B": "8!/(2!2!)", "C": "8!", "D": "8!/3!" } }, "8.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Assign n labeled balls to m labeled bins without restrictions. Count?", "criteria": { "A": "n^m", "B": "m^n", "C": "C(n+m−1,m−1)", "D": "n!" } }, "8.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of unordered five-card hands all of one suit in a standard deck, including straight flushes?", "criteria": { "A": "4*13^5", "B": "C(52,5)", "C": "4*C(13,5)", "D": "C(13,5)" } }, "8.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: You hold two hearts. Choose five unordered cards from the remaining 50. Count selections with at least three additional hearts.", "criteria": { "A": "C(13,3)C(39,2)", "B": "C(11,3)C(47,2)", "C": "C(11,3)C(39,2)", "D": "C(11,3)C(39,2)+C(11,4)C(39,1)+C(11,5)" } }, "8.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Count integer solutions sum(xi,i=1..k)=n, each xi>=−1, n>=0.", "criteria": { "A": "k^n", "B": "C(n+2k−1,k−1)", "C": "C(n+k−1,k−1)", "D": "C(n−1,k−1)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 322.78100000257837, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"8.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.32,\"probabilities\":{\"B\":0.48,\"C\":0.01,\"D\":0.43,\"A\":0.08}},\"8.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"B\":0.99,\"A\":0.01,\"D\":0.0,\"C\":0.0}},\"8.3a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"A\":0.0,\"D\":0.0,\"C\":1.0}},\"8.3b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.96,\"probabilities\":{\"B\":0.01,\"A\":0.0,\"D\":0.97,\"C\":0.02}},\"8.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.96,\"probabilities\":{\"B\":0.97,\"C\":0.03,\"D\":0.0,\"A\":0.0}}},\"usage\":{\"input_tokens\":1000,\"output_tokens\":225}}" } ], "response": { "model": "jev-1.13.0", "answers": { "8.1": { "type": "choice", "choice": "B", "confidence": 0.32, "probabilities": { "B": 0.48, "C": 0.01, "D": 0.43, "A": 0.08 } }, "8.2": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "B": 0.99, "A": 0.01, "D": 0.0, "C": 0.0 } }, "8.3a": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "A": 0.0, "D": 0.0, "C": 1.0 } }, "8.3b": { "type": "choice", "choice": "D", "confidence": 0.96, "probabilities": { "B": 0.01, "A": 0.0, "D": 0.97, "C": 0.02 } }, "8.4": { "type": "choice", "choice": "B", "confidence": 0.96, "probabilities": { "B": 0.97, "C": 0.03, "D": 0.0, "A": 0.0 } } }, "usage": { "input_tokens": 1000, "output_tokens": 225 } } }, { "exam": "final_fa24", "section": 9, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "9.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Unit square grid from A=(0,0) to B=(7,5); only right and up steps. P=(3,2).\nProblem: Number of paths A to B?", "criteria": { "A": "2^12", "B": "C(35,5)", "C": "C(12,5)", "D": "7!*5!" } }, "9.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Unit square grid from A=(0,0) to B=(7,5); only right and up steps. P=(3,2).\nProblem: Number of paths A to B through P?", "criteria": { "A": "C(12,5)", "B": "C(7,3)C(5,3)/2", "C": "C(5,3)+C(7,4)", "D": "C(5,2)C(7,3)" } }, "9.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Unit square grid from A=(0,0) to B=(7,5); only right and up steps. P=(3,2).\nProblem: Combinatorially prove C(12,7)=sum(i=0..5)C(3+i,i)C(8−i,5−i).", "criteria": { "A": "Choose any i vertical steps twice; this counts each path once.", "B": "Classify by every vertex visited in column 3; each path visits exactly one such vertex.", "C": "Each of the six possible heights gives equally many paths, so the answer is six times any summand.", "D": "Classify each path by the unique right step from column 3 to 4 at height i. There are C(3+i,i) prefixes to (3,i), and C(8−i,5−i) suffixes from (4,i) to B." } } } }, "attempts": [ { "status": 200, "full_response_ms": 408.7955830000283, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"9.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.98,\"probabilities\":{\"B\":0.01,\"A\":0.0,\"C\":0.99,\"D\":0.0}},\"9.2\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.89,\"probabilities\":{\"D\":0.91,\"B\":0.05,\"C\":0.01,\"A\":0.03}},\"9.3\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.89,\"probabilities\":{\"C\":0.0,\"A\":0.0,\"D\":0.92,\"B\":0.08}}},\"usage\":{\"input_tokens\":886,\"output_tokens\":135}}" } ], "response": { "model": "jev-1.13.0", "answers": { "9.1": { "type": "choice", "choice": "C", "confidence": 0.98, "probabilities": { "B": 0.01, "A": 0.0, "C": 0.99, "D": 0.0 } }, "9.2": { "type": "choice", "choice": "D", "confidence": 0.89, "probabilities": { "D": 0.91, "B": 0.05, "C": 0.01, "A": 0.03 } }, "9.3": { "type": "choice", "choice": "D", "confidence": 0.89, "probabilities": { "C": 0.0, "A": 0.0, "D": 0.92, "B": 0.08 } } }, "usage": { "input_tokens": 886, "output_tokens": 135 } } }, { "exam": "final_fa24", "section": 10, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "10.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct real-valued functions represented by finite integer-coefficient polynomials?", "criteria": { "A": "Countably infinite", "B": "Uncountably infinite", "C": "Finite" } }, "10.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct real-valued functions represented by finite real-coefficient polynomials?", "criteria": { "A": "Uncountably infinite", "B": "Countably infinite", "C": "Finite" } }, "10.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct functions GF(p)->GF(p) represented by finite polynomials for fixed prime p?", "criteria": { "A": "Countably infinite", "B": "Finite", "C": "Uncountably infinite" } } } }, "attempts": [ { "status": 200, "full_response_ms": 317.39345799724106, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"10.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"A\":0.99,\"C\":0.0,\"B\":0.01}},\"10.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.78,\"probabilities\":{\"A\":0.15,\"C\":0.0,\"B\":0.85}},\"10.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.61,\"probabilities\":{\"A\":0.22,\"C\":0.04,\"B\":0.74}}},\"usage\":{\"input_tokens\":629,\"output_tokens\":117}}" } ], "response": { "model": "jev-1.13.0", "answers": { "10.1": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "A": 0.99, "C": 0.0, "B": 0.01 } }, "10.2": { "type": "choice", "choice": "B", "confidence": 0.78, "probabilities": { "A": 0.15, "C": 0.0, "B": 0.85 } }, "10.3": { "type": "choice", "choice": "B", "confidence": 0.61, "probabilities": { "A": 0.22, "C": 0.04, "B": 0.74 } } }, "usage": { "input_tokens": 629, "output_tokens": 117 } } }, { "exam": "final_fa24", "section": 11, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "11.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Fill four blanks: TestHalt(F,a) defines fresh dummy():pass and inner(b): [1];[2], then returns TestCalls([3],[4],a), where TestCalls(P,Q,x) detects whether P(x) ever calls Q.", "criteria": { "A": "(F(a); dummy(); dummy; inner)", "B": "(F(b); dummy(); inner; dummy)", "C": "(F(b); dummy(); F; dummy)", "D": "(dummy(); F(b); inner; dummy)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 357.2974999988219, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"11.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.25,\"probabilities\":{\"D\":0.17,\"A\":0.27,\"B\":0.43,\"C\":0.13}}},\"usage\":{\"input_tokens\":498,\"output_tokens\":48}}" } ], "response": { "model": "jev-1.13.0", "answers": { "11.1": { "type": "choice", "choice": "B", "confidence": 0.25, "probabilities": { "D": 0.17, "A": 0.27, "B": 0.43, "C": 0.13 } } }, "usage": { "input_tokens": 498, "output_tokens": 48 } } }, { "exam": "final_fa24", "section": 12, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "12.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P(A)=1/2,P(B)=1/3,P(C)=1/6,P(A union B)=2/3,P(A given C)=2/3.\nProblem: P(A intersection B)?", "criteria": { "A": "1/6", "B": "1/3", "C": "2/3", "D": "1/9" } }, "12.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P(A)=1/2,P(B)=1/3,P(C)=1/6,P(A union B)=2/3,P(A given C)=2/3.\nProblem: P(B given A)?", "criteria": { "A": "2/3", "B": "1/6", "C": "1/3", "D": "1/2" } }, "12.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P(A)=1/2,P(B)=1/3,P(C)=1/6,P(A union B)=2/3,P(A given C)=2/3.\nProblem: P(C intersection A)?", "criteria": { "A": "2/9", "B": "1/3", "C": "1/9", "D": "1/6" } }, "12.1d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P(A)=1/2,P(B)=1/3,P(C)=1/6,P(A union B)=2/3,P(A given C)=2/3.\nProblem: P(C given A)?", "criteria": { "A": "2/9", "B": "1/9", "C": "1/3", "D": "2/3" } }, "12.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: ID,IE indicate events D,E with P(D)=1/3,P(E)=1/2,P(D intersection E)=1/5. The best affine estimate of ID from IE is a(IE−E[IE])+b.\nProblem: Cov(ID,IE)?", "criteria": { "A": "1/6", "B": "1/5", "C": "−1/30", "D": "1/30" } }, "12.2bi": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: ID,IE indicate events D,E with P(D)=1/3,P(E)=1/2,P(D intersection E)=1/5. The best affine estimate of ID from IE is a(IE−E[IE])+b.\nProblem: Sign of a?", "criteria": { "A": "Positive", "B": "Negative", "C": "Zero" } }, "12.2bii": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: ID,IE indicate events D,E with P(D)=1/3,P(E)=1/2,P(D intersection E)=1/5. The best affine estimate of ID from IE is a(IE−E[IE])+b.\nProblem: Value of a?", "criteria": { "A": "1/3", "B": "1/5", "C": "1/30", "D": "2/15" } }, "12.2biii": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: ID,IE indicate events D,E with P(D)=1/3,P(E)=1/2,P(D intersection E)=1/5. The best affine estimate of ID from IE is a(IE−E[IE])+b.\nProblem: Value of b?", "criteria": { "A": "0", "B": "1/5", "C": "1/3", "D": "1/2" } }, "12.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X,Y have E[X]=2,E[Y]=3,Var(X)=1,Var(Y)=2.\nProblem: E[XY]?", "criteria": { "A": "5", "B": "6", "C": "9", "D": "1" } }, "12.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X,Y have E[X]=2,E[Y]=3,Var(X)=1,Var(Y)=2.\nProblem: E[X²]?", "criteria": { "A": "1", "B": "4", "C": "5", "D": "3" } }, "12.3c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X,Y have E[X]=2,E[Y]=3,Var(X)=1,Var(Y)=2.\nProblem: Var(XY)?", "criteria": { "A": "2", "B": "36", "C": "55", "D": "19" } }, "12.4a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Density f(x)=x on [0,c], zero otherwise. What is c>0?", "criteria": { "A": "1/2", "B": "2", "C": "sqrt(2)", "D": "1" } }, "12.4b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Density f(x)=x on [0,c], zero elsewhere, with c>0 chosen to normalize it. CDF on [0,c]?", "criteria": { "A": "x", "B": "x²", "C": "x²/2", "D": "2x" } } } }, "attempts": [ { "status": 200, "full_response_ms": 326.0857919995033, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"12.1a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.92,\"probabilities\":{\"B\":0.05,\"C\":0.01,\"D\":0.01,\"A\":0.9299999999999999}},\"12.1b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.15,\"probabilities\":{\"B\":0.29,\"C\":0.37,\"D\":0.22,\"A\":0.12}},\"12.1c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.82,\"probabilities\":{\"B\":0.01,\"C\":0.86,\"D\":0.01,\"A\":0.12}},\"12.1d\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.15,\"probabilities\":{\"B\":0.26,\"A\":0.36,\"D\":0.04,\"C\":0.34}},\"12.2a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.72,\"probabilities\":{\"B\":0.01,\"C\":0.19,\"D\":0.79,\"A\":0.01}},\"12.2bi\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.92,\"probabilities\":{\"B\":0.05,\"C\":0.01,\"A\":0.94}},\"12.2bii\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.82,\"probabilities\":{\"B\":0.02,\"A\":0.02,\"D\":0.86,\"C\":0.1}},\"12.2biii\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.77,\"probabilities\":{\"B\":0.06,\"C\":0.84,\"D\":0.05,\"A\":0.05}},\"12.3a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"B\":1.0,\"A\":0.0,\"D\":0.0,\"C\":0.0}},\"12.3b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"C\":1.0,\"D\":0.0,\"A\":0.0}},\"12.3c\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.91,\"probabilities\":{\"B\":0.01,\"C\":0.05,\"D\":0.94,\"A\":0.0}},\"12.4a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.89,\"probabilities\":{\"B\":0.06,\"C\":0.92,\"D\":0.01,\"A\":0.01}},\"12.4b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.46,\"probabilities\":{\"B\":0.6,\"C\":0.36,\"D\":0.03,\"A\":0.01}}},\"usage\":{\"input_tokens\":1993,\"output_tokens\":596}}" } ], "response": { "model": "jev-1.13.0", "answers": { "12.1a": { "type": "choice", "choice": "A", "confidence": 0.92, "probabilities": { "B": 0.05, "C": 0.01, "D": 0.01, "A": 0.9299999999999999 } }, "12.1b": { "type": "choice", "choice": "C", "confidence": 0.15, "probabilities": { "B": 0.29, "C": 0.37, "D": 0.22, "A": 0.12 } }, "12.1c": { "type": "choice", "choice": "C", "confidence": 0.82, "probabilities": { "B": 0.01, "C": 0.86, "D": 0.01, "A": 0.12 } }, "12.1d": { "type": "choice", "choice": "A", "confidence": 0.15, "probabilities": { "B": 0.26, "A": 0.36, "D": 0.04, "C": 0.34 } }, "12.2a": { "type": "choice", "choice": "D", "confidence": 0.72, "probabilities": { "B": 0.01, "C": 0.19, "D": 0.79, "A": 0.01 } }, "12.2bi": { "type": "choice", "choice": "A", "confidence": 0.92, "probabilities": { "B": 0.05, "C": 0.01, "A": 0.94 } }, "12.2bii": { "type": "choice", "choice": "D", "confidence": 0.82, "probabilities": { "B": 0.02, "A": 0.02, "D": 0.86, "C": 0.1 } }, "12.2biii": { "type": "choice", "choice": "C", "confidence": 0.77, "probabilities": { "B": 0.06, "C": 0.84, "D": 0.05, "A": 0.05 } }, "12.3a": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "B": 1.0, "A": 0.0, "D": 0.0, "C": 0.0 } }, "12.3b": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "C": 1.0, "D": 0.0, "A": 0.0 } }, "12.3c": { "type": "choice", "choice": "D", "confidence": 0.91, "probabilities": { "B": 0.01, "C": 0.05, "D": 0.94, "A": 0.0 } }, "12.4a": { "type": "choice", "choice": "C", "confidence": 0.89, "probabilities": { "B": 0.06, "C": 0.92, "D": 0.01, "A": 0.01 } }, "12.4b": { "type": "choice", "choice": "B", "confidence": 0.46, "probabilities": { "B": 0.6, "C": 0.36, "D": 0.03, "A": 0.01 } } }, "usage": { "input_tokens": 1993, "output_tokens": 596 } } }, { "exam": "final_fa24", "section": 13, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "13.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Independent X~Bin(n,p),Y~Bin(n,q). Cov(X,Y)?", "criteria": { "A": "n(p+q)", "B": "np(1−q)", "C": "0", "D": "npq" } }, "13.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Independent X~Bin(n,p),Y~Bin(n,q). Var(X+Y)?", "criteria": { "A": "n²[p(1−p)+q(1−q)]", "B": "npq", "C": "n[p(1−p)+q(1−q)]", "D": "n(p+q)(1−p−q)" } }, "13.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: X~Geom(p) counts trials in {1,2,...}. For integer x>=1, compute E[X−x given X>=x].", "criteria": { "A": "x/p", "B": "0", "C": "(1−p)/p", "D": "1/p" } }, "13.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: X~Poisson(lambda). P(X>0)?", "criteria": { "A": "lambda*exp(−lambda)", "B": "exp(−lambda)", "C": "1/lambda", "D": "1−exp(−lambda)" } }, "13.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Independent Poisson(lambda),Poisson(mu). P(X+Y>0)?", "criteria": { "A": "exp(−lambda−mu)", "B": "1−exp(−lambda)*mu", "C": "(1−exp(−lambda))(1−exp(−mu))", "D": "1−exp(−lambda−mu)" } }, "13.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: PDF of Exp(rate lambda>0) on x>=0?", "criteria": { "A": "exp(−x/lambda)/lambda²", "B": "exp(−lambda*x)", "C": "lambda*exp(−lambda*x)", "D": "lambda*exp(lambda*x)" } }, "13.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: PDF of min(X,Y) for independent Exp(rate lambda) variables, z>=0?", "criteria": { "A": "2lambda*exp(−lambda*z)", "B": "lambda²*exp(−2lambda*z)", "C": "2lambda*exp(−2lambda*z)", "D": "lambda*exp(−lambda*z)" } }, "13.7": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A dart is uniform over a disk of radius r>0. Density of distance x to center?", "criteria": { "A": "1/r on [0,r], zero elsewhere", "B": "x²/r² on [0,r], zero elsewhere", "C": "2x/r² on [0,r], zero elsewhere", "D": "2pi*x on [0,r], zero elsewhere" } } } }, "attempts": [ { "status": 200, "full_response_ms": 349.60312499970314, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"13.1a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"C\":1.0,\"A\":0.0,\"D\":0.0}},\"13.1b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"C\":1.0,\"A\":0.0,\"D\":0.0}},\"13.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.47,\"probabilities\":{\"B\":0.02,\"C\":0.61,\"A\":0.01,\"D\":0.36}},\"13.3\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"A\":0.0,\"C\":0.0,\"D\":1.0}},\"13.4\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"C\":0.0,\"A\":0.0,\"D\":1.0}},\"13.5\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"C\":1.0,\"A\":0.0,\"D\":0.0}},\"13.6\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.97,\"probabilities\":{\"B\":0.0,\"A\":0.01,\"C\":0.98,\"D\":0.01}},\"13.7\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"C\":1.0,\"A\":0.0,\"D\":0.0}}},\"usage\":{\"input_tokens\":1365,\"output_tokens\":365}}" } ], "response": { "model": "jev-1.13.0", "answers": { "13.1a": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "C": 1.0, "A": 0.0, "D": 0.0 } }, "13.1b": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "B": 0.0, "C": 1.0, "A": 0.0, "D": 0.0 } }, "13.2": { "type": "choice", "choice": "C", "confidence": 0.47, "probabilities": { "B": 0.02, "C": 0.61, "A": 0.01, "D": 0.36 } }, "13.3": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "B": 0.0, "A": 0.0, "C": 0.0, "D": 1.0 } }, "13.4": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "A": 0.0, "D": 1.0 } }, "13.5": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "C": 1.0, "A": 0.0, "D": 0.0 } }, "13.6": { "type": "choice", "choice": "C", "confidence": 0.97, "probabilities": { "B": 0.0, "A": 0.01, "C": 0.98, "D": 0.01 } }, "13.7": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "B": 0.0, "C": 1.0, "A": 0.0, "D": 0.0 } } }, "usage": { "input_tokens": 1365, "output_tokens": 365 } } }, { "exam": "final_fa24", "section": 14, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "14.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X,Y~Uniform[0,r],r>0.\nProblem: CDF of max(X,Y)?", "criteria": { "A": "0 below 0; 1−(1−z/r)² on [0,r]; 1 above r", "B": "0 below 0; z/r on [0,r]; 1 above r", "C": "0 below 0; (z/r)² on [0,r]; 1 above r", "D": "0 below 0; 2z/r on [0,r]; 1 above r" } }, "14.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X,Y~Uniform[0,r],r>0.\nProblem: PDF of max(X,Y) on [0,r], zero elsewhere?", "criteria": { "A": "2(r−z)/r²", "B": "2z/r²", "C": "z²/r²", "D": "1/r" } }, "14.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X,Y~Uniform[0,r],r>0.\nProblem: E[max(X,Y)]?", "criteria": { "A": "2r/3", "B": "r", "C": "r/2", "D": "r/3" } }, "14.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X,Y~Uniform[0,r],r>0.\nProblem: E[min(X,Y)]? Hint: avoid integration.", "criteria": { "A": "r/4", "B": "2r/3", "C": "r/3", "D": "r/2" } }, "14.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X,Y~Uniform[0,r],r>0.\nProblem: E[|X−Y|]? Hint: use min, max and linearity.", "criteria": { "A": "2r/3", "B": "0", "C": "r/3", "D": "r/2" } } } }, "attempts": [ { "status": 200, "full_response_ms": 297.1683329997177, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"14.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.97,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":0.02,\"C\":0.98}},\"14.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":1.0,\"A\":0.0,\"C\":0.0}},\"14.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":1.0,\"C\":0.0}},\"14.4\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.91,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":0.06,\"C\":0.9400000000000001}},\"14.5\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.88,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":0.09,\"C\":0.91}}},\"usage\":{\"input_tokens\":993,\"output_tokens\":228}}" } ], "response": { "model": "jev-1.13.0", "answers": { "14.1": { "type": "choice", "choice": "C", "confidence": 0.97, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.02, "C": 0.98 } }, "14.2": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 1.0, "A": 0.0, "C": 0.0 } }, "14.3": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 1.0, "C": 0.0 } }, "14.4": { "type": "choice", "choice": "C", "confidence": 0.91, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.06, "C": 0.9400000000000001 } }, "14.5": { "type": "choice", "choice": "C", "confidence": 0.88, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.09, "C": 0.91 } } }, "usage": { "input_tokens": 993, "output_tokens": 228 } } }, { "exam": "final_fa24", "section": 15, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "15.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A uniform random permutation assigns balls 1..100 to bins 1..100, one each. Ii indicates ball i lands within distance one of bin i, without wraparound.\nProblem: Expected number close to their original bin?", "criteria": { "A": "2", "B": "3", "C": "2.98", "D": "1" } }, "15.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A uniform random permutation assigns balls 1..100 to bins 1..100, one each. Ii indicates ball i lands within distance one of bin i, without wraparound.\nProblem: E[I2*I5]?", "criteria": { "A": "6/(100*99)", "B": "3/100", "C": "9/10000", "D": "9/(100*99)" } }, "15.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A uniform random permutation assigns balls 1..100 to bins 1..100, one each. Ii indicates ball i lands within distance one of bin i, without wraparound.\nProblem: E[I2*I4]?", "criteria": { "A": "8/(100*99)", "B": "9/10000", "C": "9/(100*99)", "D": "6/(100*99)" } }, "15.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A uniform random permutation assigns balls 1..100 to bins 1..100, one each. Ii indicates ball i lands within distance one of bin i, without wraparound.\nProblem: Do there exist distinct i,j for which Ii,Ij are independent?", "criteria": { "A": "Yes", "B": "No" } } } }, "attempts": [ { "status": 200, "full_response_ms": 358.66100000203005, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"15.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.58,\"probabilities\":{\"D\":0.02,\"C\":0.6900000000000001,\"A\":0.18,\"B\":0.11}},\"15.2a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.66,\"probabilities\":{\"D\":0.74,\"C\":0.11,\"A\":0.13,\"B\":0.02}},\"15.2b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.4,\"probabilities\":{\"D\":0.2,\"C\":0.55,\"A\":0.19,\"B\":0.06}},\"15.2c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.58,\"probabilities\":{\"A\":0.79,\"B\":0.21}}},\"usage\":{\"input_tokens\":907,\"output_tokens\":172}}" } ], "response": { "model": "jev-1.13.0", "answers": { "15.1": { "type": "choice", "choice": "C", "confidence": 0.58, "probabilities": { "D": 0.02, "C": 0.6900000000000001, "A": 0.18, "B": 0.11 } }, "15.2a": { "type": "choice", "choice": "D", "confidence": 0.66, "probabilities": { "D": 0.74, "C": 0.11, "A": 0.13, "B": 0.02 } }, "15.2b": { "type": "choice", "choice": "C", "confidence": 0.4, "probabilities": { "D": 0.2, "C": 0.55, "A": 0.19, "B": 0.06 } }, "15.2c": { "type": "choice", "choice": "A", "confidence": 0.58, "probabilities": { "A": 0.79, "B": 0.21 } } }, "usage": { "input_tokens": 907, "output_tokens": 172 } } }, { "exam": "final_fa24", "section": 16, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "16.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Choose one of two 52-card decks with equal probability. Loaded deck:39 red,13 black. Fair deck:26 red,26 black. Draw uniformly without replacement.\nProblem: Given first card red, probability the chosen deck is fair?", "criteria": { "A": "1/2", "B": "3/5", "C": "2/5", "D": "1/4" } }, "16.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Choose one of two 52-card decks with equal probability. Loaded deck:39 red,13 black. Fair deck:26 red,26 black. Draw uniformly without replacement.\nProblem: Given first card red, probability the second is red?", "criteria": { "A": "(1/2)(38/51)+(1/2)(25/51)", "B": "38/51", "C": "(3/5)(38/51)+(2/5)(25/51)", "D": "(3/5)(39/52)+(2/5)(26/52)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 355.7217090019549, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"16.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.67,\"probabilities\":{\"D\":0.04,\"B\":0.18,\"A\":0.02,\"C\":0.76}},\"16.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.45,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":0.41,\"C\":0.59}}},\"usage\":{\"input_tokens\":673,\"output_tokens\":93}}" } ], "response": { "model": "jev-1.13.0", "answers": { "16.1": { "type": "choice", "choice": "C", "confidence": 0.67, "probabilities": { "D": 0.04, "B": 0.18, "A": 0.02, "C": 0.76 } }, "16.2": { "type": "choice", "choice": "C", "confidence": 0.45, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.41, "C": 0.59 } } }, "usage": { "input_tokens": 673, "output_tokens": 93 } } }, { "exam": "final_fa24", "section": 17, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "17.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: X is a sum of indicators, and p=P(X>=1). Prove p<=E[X].", "criteria": { "A": "Every probability is at most one, and every expectation is at least one.", "B": "X is always either zero or one, so equality must hold.", "C": "Indicators are always independent, so multiply their probabilities.", "D": "The union bound gives P(any indicator is one)<=sum of their success probabilities=sum of their expectations=E[X]. No independence needed." } }, "17.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: W counts successes in n independent Bernoulli(p) trials. For k>0, prove P(W>=k)<=np/k.", "criteria": { "A": "Chebyshev gives P(W>=k)=Var(W)/k, which equals np/k.", "B": "W is nonnegative and E[W]=np, so apply Markov’s inequality.", "C": "W is at most np in every outcome.", "D": "Independence makes W deterministic with value np." } }, "17.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Y~Poisson(lambda),lambda>0. Prove P(Y>=10lambda)<=1/(81lambda).", "criteria": { "A": "The event equals |Y−lambda|>=10lambda, giving the stated bound.", "B": "The event implies |Y−lambda|>=9lambda. Chebyshev and Var(Y)=lambda give lambda/(9lambda)²=1/(81lambda).", "C": "Markov gives lambda/(10lambda)=1/(81lambda).", "D": "Poisson variables cannot exceed ten times their means." } } } }, "attempts": [ { "status": 200, "full_response_ms": 296.01220800032024, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"17.1\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"A\":0.0,\"D\":1.0,\"B\":0.0,\"C\":0.0}},\"17.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"A\":0.0,\"D\":0.0,\"B\":1.0,\"C\":0.0}},\"17.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"A\":0.0,\"D\":0.0,\"B\":1.0,\"C\":0.0}}},\"usage\":{\"input_tokens\":854,\"output_tokens\":138}}" } ], "response": { "model": "jev-1.13.0", "answers": { "17.1": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "A": 0.0, "D": 1.0, "B": 0.0, "C": 0.0 } }, "17.2": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "A": 0.0, "D": 0.0, "B": 1.0, "C": 0.0 } }, "17.3": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "A": 0.0, "D": 0.0, "B": 1.0, "C": 0.0 } } }, "usage": { "input_tokens": 854, "output_tokens": 138 } } }, { "exam": "final_fa24", "section": 18, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "18.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For arbitrary A,B,C with positive conditioning probabilities, P(C given A union B)=P(C given A)+P(C given B)−P(C given A intersection B).", "criteria": { "A": "False", "B": "True" } }, "18.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Choose a counterexample and its four probabilities in the order: C|A union B, C|A, C|B, C|A intersection B.", "criteria": { "A": "Two fair independent flips: A=first H,B=second H,C=HH. Values (0,1/2,1/2,1).", "B": "Two fair independent flips: A=first H,B=second H,C=HH. Values (1/3,1/2,1/2,1).", "C": "Set A=B=C to the same positive event; values (1,1,1,1).", "D": "Set C to the empty event; values (0,0,0,0)." } } } }, "attempts": [ { "status": 200, "full_response_ms": 359.9019590001262, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"18.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.89,\"probabilities\":{\"A\":0.95,\"B\":0.05}},\"18.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.68,\"probabilities\":{\"C\":0.01,\"A\":0.21,\"D\":0.01,\"B\":0.77}}},\"usage\":{\"input_tokens\":645,\"output_tokens\":79}}" } ], "response": { "model": "jev-1.13.0", "answers": { "18.1": { "type": "choice", "choice": "A", "confidence": 0.89, "probabilities": { "A": 0.95, "B": 0.05 } }, "18.2": { "type": "choice", "choice": "B", "confidence": 0.68, "probabilities": { "C": 0.01, "A": 0.21, "D": 0.01, "B": 0.77 } } }, "usage": { "input_tokens": 645, "output_tokens": 79 } } }, { "exam": "final_fa24", "section": 19, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "19.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent Xi~Poisson(lambda),lambda>0; An=(X1+...+Xn)/n.\nProblem: E[An]?", "criteria": { "A": "lambda", "B": "n*lambda", "C": "lambda/n", "D": "1/lambda" } }, "19.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent Xi~Poisson(lambda),lambda>0; An=(X1+...+Xn)/n.\nProblem: Smallest integer sample size guaranteed by Chebyshev to make P(|An−E[An]|>=0.1E[An])<=0.05?", "criteria": { "A": "ceil(2000*lambda)", "B": "ceil(400/lambda)", "C": "ceil(2000/lambda)", "D": "ceil(200/lambda)" } }, "19.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent Xi~Poisson(lambda),lambda>0; An=(X1+...+Xn)/n.\nProblem: Instead suppose An is exactly normal with its actual mean and variance, and use P(|Z|>=2)<=0.05. Sufficient integer n from this rule?", "criteria": { "A": "ceil(400*lambda)", "B": "ceil(40/lambda)", "C": "ceil(2000/lambda)", "D": "ceil(400/lambda)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 388.4730829995533, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"19.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":1.0,\"C\":0.0,\"B\":0.0}},\"19.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.33,\"probabilities\":{\"D\":0.44,\"A\":0.02,\"C\":0.49,\"B\":0.05}},\"19.3\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.83,\"probabilities\":{\"D\":0.87,\"A\":0.08,\"C\":0.01,\"B\":0.04}}},\"usage\":{\"input_tokens\":804,\"output_tokens\":138}}" } ], "response": { "model": "jev-1.13.0", "answers": { "19.1": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 1.0, "C": 0.0, "B": 0.0 } }, "19.2": { "type": "choice", "choice": "C", "confidence": 0.33, "probabilities": { "D": 0.44, "A": 0.02, "C": 0.49, "B": 0.05 } }, "19.3": { "type": "choice", "choice": "D", "confidence": 0.83, "probabilities": { "D": 0.87, "A": 0.08, "C": 0.01, "B": 0.04 } } }, "usage": { "input_tokens": 804, "output_tokens": 138 } } }, { "exam": "final_fa24", "section": 20, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "20.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Repeated independent fair four-sided die rolls. State0:last roll not 1 (also start); state1:one trailing 1; state2:at least two trailing 1s.\nProblem: Let b_i be expected future even rolls before first reaching state2 from state i. Choose the first-step equations.", "criteria": { "A": "b0=1/2+3b0/4+b1/4; b1=1/2+3b0/4+b2/4; b2=0", "B": "b0=1/4+3b0/4+b1/4; b1=1/4+3b0/4; b2=0", "C": "b0=1+3b0/4+b1/4; b1=1+3b0/4; b2=0", "D": "b0=1/2+b1/4; b1=1/2+b2/4; b2=0" } }, "20.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Repeated independent fair four-sided die rolls. State0:last roll not 1 (also start); state1:one trailing 1; state2:at least two trailing 1s.\nProblem: Continue rolling even after reaching state2. Is there a unique stationary distribution?", "criteria": { "A": "No", "B": "Yes" } }, "20.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Repeated independent fair four-sided die rolls. State0:last roll not 1 (also start); state1:one trailing 1; state2:at least two trailing 1s.\nProblem: For the continuing chain, choose stationary equations plus normalization.", "criteria": { "A": "p0=3p0/4; p1=p0/4; p2=p1/4+p2; sum p=1", "B": "p0=3(p0+p1)/4; p1=(p0+p2)/4; p2=p1/4; sum p=1", "C": "p0=3(p0+p1+p2)/4; p1=p0/4; p2=(p1+p2)/4; sum p=1", "D": "p0=p1=p2=1/3" } } } }, "attempts": [ { "status": 200, "full_response_ms": 310.453583002527, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"20.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.67,\"probabilities\":{\"B\":0.2,\"A\":0.75,\"D\":0.01,\"C\":0.04}},\"20.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.65,\"probabilities\":{\"B\":0.83,\"A\":0.17}},\"20.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.63,\"probabilities\":{\"B\":0.72,\"A\":0.02,\"D\":0.0,\"C\":0.26}}},\"usage\":{\"input_tokens\":953,\"output_tokens\":124}}" } ], "response": { "model": "jev-1.13.0", "answers": { "20.1": { "type": "choice", "choice": "A", "confidence": 0.67, "probabilities": { "B": 0.2, "A": 0.75, "D": 0.01, "C": 0.04 } }, "20.2": { "type": "choice", "choice": "B", "confidence": 0.65, "probabilities": { "B": 0.83, "A": 0.17 } }, "20.3": { "type": "choice", "choice": "B", "confidence": 0.63, "probabilities": { "B": 0.72, "A": 0.02, "D": 0.0, "C": 0.26 } } }, "usage": { "input_tokens": 953, "output_tokens": 124 } } }, { "exam": "final_fa24", "section": 21, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "21.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Two players start ready and shoot independently with success probability 2/3. Both make: win; both miss: loss. If exactly one misses, next turn that player retrieves while the other shoots. A lone shooter’s success makes both ready next turn; a miss leaves one retrieving. State0=both ready,1=one retrieving,2=win,3=loss. Arrows A:0->2,B:0->1,C:1->0,D:0->3,E:1->1.\nProblem: Give (A,B,C,D,E).", "criteria": { "A": "(4/9,4/9,2/3,1/9,1/3)", "B": "(4/9,4/9,1/3,1/9,2/3)", "C": "(2/3,2/9,1/3,1/3,2/3)", "D": "(4/9,2/9,2/3,1/9,1/3)" } }, "21.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Two players start ready and shoot independently with success probability 2/3. Both make: win; both miss: loss. If exactly one misses, next turn that player retrieves while the other shoots. A lone shooter’s success makes both ready next turn; a miss leaves one retrieving. State0=both ready,1=one retrieving,2=win,3=loss. Arrows A:0->2,B:0->1,C:1->0,D:0->3,E:1->1.\nProblem: Probability of winning from both ready?", "criteria": { "A": "8/9", "B": "2/3", "C": "4/9", "D": "4/5" } }, "21.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Two players start ready and shoot independently with success probability 2/3. Both make: win; both miss: loss. If exactly one misses, next turn that player retrieves while the other shoots. A lone shooter’s success makes both ready next turn; a miss leaves one retrieving. State0=both ready,1=one retrieving,2=win,3=loss. Arrows A:0->2,B:0->1,C:1->0,D:0->3,E:1->1.\nProblem: Expected turns until win or loss from both ready?", "criteria": { "A": "4", "B": "9/5", "C": "5", "D": "3" } } } }, "attempts": [ { "status": 200, "full_response_ms": 378.19520800258033, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"21.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.95,\"probabilities\":{\"A\":0.96,\"B\":0.02,\"C\":0.0,\"D\":0.02}},\"21.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.55,\"probabilities\":{\"A\":0.66,\"B\":0.07,\"C\":0.14,\"D\":0.13}},\"21.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.66,\"probabilities\":{\"A\":0.09,\"B\":0.74,\"C\":0.04,\"D\":0.13}}},\"usage\":{\"input_tokens\":1010,\"output_tokens\":138}}" } ], "response": { "model": "jev-1.13.0", "answers": { "21.1": { "type": "choice", "choice": "A", "confidence": 0.95, "probabilities": { "A": 0.96, "B": 0.02, "C": 0.0, "D": 0.02 } }, "21.2": { "type": "choice", "choice": "A", "confidence": 0.55, "probabilities": { "A": 0.66, "B": 0.07, "C": 0.14, "D": 0.13 } }, "21.3": { "type": "choice", "choice": "B", "confidence": 0.66, "probabilities": { "A": 0.09, "B": 0.74, "C": 0.04, "D": 0.13 } } }, "usage": { "input_tokens": 1010, "output_tokens": 138 } } }, { "exam": "final_sp23", "section": 2, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "2.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: False implies False.", "criteria": { "A": "True", "B": "False" } }, "2.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: False implies True.", "criteria": { "A": "True", "B": "False" } }, "2.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: [NOT exists x P(x)] implies [forall x (P(x) implies Q(x))].", "criteria": { "A": "False", "B": "True" } }, "2.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: [forall x exists y (Q(x,y) implies P(x))] is equivalent to [forall x (NOT P(x) implies NOT forall y Q(x,y))].", "criteria": { "A": "False", "B": "True" } }, "2.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: No stable matching can give every job its least-preferred candidate.", "criteria": { "A": "True", "B": "False" } }, "2.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If job-proposing deferred acceptance finishes on day one, the result is optimal for both jobs and candidates.", "criteria": { "A": "False", "B": "True" } } } }, "attempts": [ { "status": 200, "full_response_ms": 355.03979200075264, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"2.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"A\":1.0}},\"2.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"A\":1.0}},\"2.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.98,\"probabilities\":{\"B\":0.99,\"A\":0.01}},\"2.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.48,\"probabilities\":{\"B\":0.74,\"A\":0.26}},\"2.5\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.41,\"probabilities\":{\"B\":0.29,\"A\":0.71}},\"2.6\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.68,\"probabilities\":{\"B\":0.16,\"A\":0.84}}},\"usage\":{\"input_tokens\":799,\"output_tokens\":183}}" } ], "response": { "model": "jev-1.13.0", "answers": { "2.1": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "A": 1.0 } }, "2.2": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "A": 1.0 } }, "2.3": { "type": "choice", "choice": "B", "confidence": 0.98, "probabilities": { "B": 0.99, "A": 0.01 } }, "2.4": { "type": "choice", "choice": "B", "confidence": 0.48, "probabilities": { "B": 0.74, "A": 0.26 } }, "2.5": { "type": "choice", "choice": "A", "confidence": 0.41, "probabilities": { "B": 0.29, "A": 0.71 } }, "2.6": { "type": "choice", "choice": "A", "confidence": 0.68, "probabilities": { "B": 0.16, "A": 0.84 } } }, "usage": { "input_tokens": 799, "output_tokens": 183 } } }, { "exam": "final_sp23", "section": 3, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "3.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For natural x<=y, choose a valid proof that x<=y/2 or y−x<=y/2.", "criteria": { "A": "Since x<=y, necessarily x<=y/2.", "B": "Every natural y is even, so one term equals y/2.", "C": "If both inequalities fail, adding x>y/2 and y−x>y/2 gives y>y, a contradiction.", "D": "Since y−x>=0, necessarily y−x<=y/2." } }, "3.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: How many ordered natural pairs (a,b) satisfy b²=3a²? Here zero is natural.", "criteria": { "A": "2", "B": "1", "C": "0", "D": "Infinitely many" } }, "3.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Choose a valid argument determining all natural solutions of b²=3a², allowing zero.", "criteria": { "A": "Squares are always even, whereas three times a square is always odd.", "B": "A nonzero square has even exponent of prime3, but 3a² has odd exponent; thus no nonzero solution exists and (0,0) is the only solution.", "C": "Dividing by three implies b=a, hence all pairs with a=b are solutions.", "D": "For a=3k, b=3k also satisfies the equation for every natural k." } } } }, "attempts": [ { "status": 200, "full_response_ms": 618.1105409996235, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"3.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"C\":1.0,\"A\":0.0}},\"3.2a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.65,\"probabilities\":{\"B\":0.74,\"A\":0.21,\"D\":0.03,\"C\":0.02}},\"3.2b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"B\":1.0,\"A\":0.0,\"D\":0.0,\"C\":0.0}}},\"usage\":{\"input_tokens\":804,\"output_tokens\":137}}" } ], "response": { "model": "jev-1.13.0", "answers": { "3.1": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "C": 1.0, "A": 0.0 } }, "3.2a": { "type": "choice", "choice": "B", "confidence": 0.65, "probabilities": { "B": 0.74, "A": 0.21, "D": 0.03, "C": 0.02 } }, "3.2b": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "B": 1.0, "A": 0.0, "D": 0.0, "C": 0.0 } } }, "usage": { "input_tokens": 804, "output_tokens": 137 } } }, { "exam": "final_sp23", "section": 4, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "4.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prime p and nonzero a modulo p: how many distinct residues ax arise as x ranges over 0..p−1?", "criteria": { "A": "p−1", "B": "p", "C": "1", "D": "gcd(a,p)" } }, "4.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Let gcd(a,N)=d. How many distinct residues ax mod N arise for x=0..N−1?", "criteria": { "A": "N", "B": "d", "C": "N/d", "D": "N−d" } }, "4.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For prime p, evaluate 1²·2²···(p−1)² modulo p.", "criteria": { "A": "1", "B": "2", "C": "0", "D": "p−1" } } } }, "attempts": [ { "status": 200, "full_response_ms": 293.14133300067624, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"4.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.78,\"probabilities\":{\"A\":0.16,\"D\":0.0,\"B\":0.84,\"C\":0.0}},\"4.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.98,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"B\":0.01,\"C\":0.99}},\"4.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.77,\"probabilities\":{\"A\":0.83,\"D\":0.17,\"B\":0.0,\"C\":0.0}}},\"usage\":{\"input_tokens\":670,\"output_tokens\":135}}" } ], "response": { "model": "jev-1.13.0", "answers": { "4.1": { "type": "choice", "choice": "B", "confidence": 0.78, "probabilities": { "A": 0.16, "D": 0.0, "B": 0.84, "C": 0.0 } }, "4.2": { "type": "choice", "choice": "C", "confidence": 0.98, "probabilities": { "D": 0.0, "A": 0.0, "B": 0.01, "C": 0.99 } }, "4.3": { "type": "choice", "choice": "A", "confidence": 0.77, "probabilities": { "A": 0.83, "D": 0.17, "B": 0.0, "C": 0.0 } } }, "usage": { "input_tokens": 670, "output_tokens": 135 } } }, { "exam": "final_sp23", "section": 5, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "5.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Choose a fair four-sided or fair six-sided die with equal probability. Roll and observe3. Probability it was the four-sided die?", "criteria": { "A": "3/5", "B": "1/4", "C": "1/2", "D": "2/5" } } } }, "attempts": [ { "status": 200, "full_response_ms": 364.61879200214753, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"5.1\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.51,\"probabilities\":{\"D\":0.64,\"A\":0.25,\"C\":0.07,\"B\":0.04}}},\"usage\":{\"input_tokens\":449,\"output_tokens\":47}}" } ], "response": { "model": "jev-1.13.0", "answers": { "5.1": { "type": "choice", "choice": "D", "confidence": 0.51, "probabilities": { "D": 0.64, "A": 0.25, "C": 0.07, "B": 0.04 } } }, "usage": { "input_tokens": 449, "output_tokens": 47 } } }, { "exam": "final_sp23", "section": 6, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "6.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: People indexed0,...,n−1 arrive in order, n>=8. Person i chooses min(i,7) distinct earlier people as friends. Friendship is undirected.\nProblem: How many friendship edges result?", "criteria": { "A": "C(n,2)", "B": "7n−21", "C": "7n", "D": "7n−28" } }, "6.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: People indexed0,...,n−1 arrive in order, n>=8. Person i chooses min(i,7) distinct earlier people as friends. Friendship is undirected.\nProblem: Maximum possible degree of a person?", "criteria": { "A": "14", "B": "7", "C": "n−7", "D": "n−1" } }, "6.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: People indexed0,...,n−1 arrive in order, n>=8. Person i chooses min(i,7) distinct earlier people as friends. Friendship is undirected.\nProblem: Smallest number of colors sufficient for every possible resulting graph?", "criteria": { "A": "n", "B": "9", "C": "7", "D": "8" } }, "6.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: People indexed0,...,n−1 arrive in order, n>=8. Person i chooses min(i,7) distinct earlier people as friends. Friendship is undirected.\nProblem: Which proves the optimal universal coloring bound?", "criteria": { "A": "The first eight vertices form K8. Color in arrival order; each later vertex has seven earlier neighbors, so eight colors suffice.", "B": "Every new vertex requires a fresh color because it has earlier neighbors.", "C": "All vertices have degree seven, so seven colors always suffice.", "D": "The graph is a tree, so two colors suffice." } }, "6.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: People indexed0,...,n−1 arrive in order, n>=8. Person i chooses min(i,7) distinct earlier people as friends. Friendship is undirected.\nProblem: If each person uniformly chooses the prescribed-size subset of earlier people, expected degree of person0?", "criteria": { "A": "sum(i=1 to n) min(i,7)/i", "B": "sum(i=1 to n−1) min(i,7)/i", "C": "7", "D": "7(n−1)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 334.5862919995852, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"6.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.86,\"probabilities\":{\"C\":0.0,\"B\":0.91,\"A\":0.0,\"D\":0.09}},\"6.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.38,\"probabilities\":{\"C\":0.18,\"A\":0.16,\"B\":0.54,\"D\":0.12}},\"6.3a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.62,\"probabilities\":{\"C\":0.17,\"B\":0.09,\"A\":0.02,\"D\":0.72}},\"6.3b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"C\":0.0,\"A\":1.0,\"B\":0.0,\"D\":0.0}},\"6.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.96,\"probabilities\":{\"C\":0.01,\"A\":0.02,\"B\":0.97,\"D\":0.0}}},\"usage\":{\"input_tokens\":1097,\"output_tokens\":225}}" } ], "response": { "model": "jev-1.13.0", "answers": { "6.1": { "type": "choice", "choice": "B", "confidence": 0.86, "probabilities": { "C": 0.0, "B": 0.91, "A": 0.0, "D": 0.09 } }, "6.2": { "type": "choice", "choice": "B", "confidence": 0.38, "probabilities": { "C": 0.18, "A": 0.16, "B": 0.54, "D": 0.12 } }, "6.3a": { "type": "choice", "choice": "D", "confidence": 0.62, "probabilities": { "C": 0.17, "B": 0.09, "A": 0.02, "D": 0.72 } }, "6.3b": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "C": 0.0, "A": 1.0, "B": 0.0, "D": 0.0 } }, "6.4": { "type": "choice", "choice": "B", "confidence": 0.96, "probabilities": { "C": 0.01, "A": 0.02, "B": 0.97, "D": 0.0 } } }, "usage": { "input_tokens": 1097, "output_tokens": 225 } } }, { "exam": "final_sp23", "section": 7, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "7.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: The three-dimensional cube has more edges than K4.", "criteria": { "A": "False", "B": "True" } }, "7.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Sharp maximum number of vertices in a graph with e edges and c connected components?", "criteria": { "A": "2e+c", "B": "e−c", "C": "e+c−1", "D": "e+c" } }, "7.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A finite undirected graph can have an odd number of odd-degree vertices.", "criteria": { "A": "False", "B": "True" } }, "7.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Average vertex degree in a tree with n vertices?", "criteria": { "A": "2", "B": "n−1", "C": "1−1/n", "D": "2−2/n" } }, "7.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Connected planar embedding has v vertices and average face boundary length s>2. Choose edge count with valid justification.", "criteria": { "A": "e=(s−2)(v−2)/s, by reversing the ratio in the face-count identity.", "B": "e=sv/2, since face length equals average vertex degree.", "C": "e=s(v−2)/(s−2), since sf=2e and v−e+f=2.", "D": "e=s(v−2)/2, since every face contributes two vertices." } } } }, "attempts": [ { "status": 200, "full_response_ms": 480.2326249991893, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"7.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.83,\"probabilities\":{\"B\":0.91,\"A\":0.09}},\"7.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.41,\"probabilities\":{\"B\":0.01,\"C\":0.03,\"D\":0.41,\"A\":0.55}},\"7.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"B\":0.01,\"A\":0.99}},\"7.4\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"C\":0.0,\"D\":1.0,\"A\":0.0}},\"7.5\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"C\":1.0,\"D\":0.0,\"A\":0.0}}},\"usage\":{\"input_tokens\":874,\"output_tokens\":195}}" } ], "response": { "model": "jev-1.13.0", "answers": { "7.1": { "type": "choice", "choice": "B", "confidence": 0.83, "probabilities": { "B": 0.91, "A": 0.09 } }, "7.2": { "type": "choice", "choice": "A", "confidence": 0.41, "probabilities": { "B": 0.01, "C": 0.03, "D": 0.41, "A": 0.55 } }, "7.3": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "B": 0.01, "A": 0.99 } }, "7.4": { "type": "choice", "choice": "D", "confidence": 0.99, "probabilities": { "B": 0.0, "C": 0.0, "D": 1.0, "A": 0.0 } }, "7.5": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "B": 0.0, "C": 1.0, "D": 0.0, "A": 0.0 } } }, "usage": { "input_tokens": 874, "output_tokens": 195 } } }, { "exam": "final_sp23", "section": 8, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "8.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Over GF(7), degree-at-most-two P satisfies P(0)=0 and P(1)=P(6)=2.\nProblem: Remainder when P is divided by x−6?", "criteria": { "A": "2", "B": "6", "C": "1", "D": "0" } }, "8.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Over GF(7), degree-at-most-two P satisfies P(0)=0 and P(1)=P(6)=2.\nProblem: Find P(x).", "criteria": { "A": "x²+1", "B": "2x²+2", "C": "2x²", "D": "2x" } }, "8.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Over GF(7), degree-at-most-two P satisfies P(0)=0 and P(1)=P(6)=2.\nProblem: Which constant polynomial disagrees with exactly one of these three given values?", "criteria": { "A": "0", "B": "1", "C": "2", "D": "6" } }, "8.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Over the reals, degree-at-most-two P has P(0)=0 and P(1)=P(−1)=A. Find P.", "criteria": { "A": "Ax", "B": "Ax²", "C": "A(x²+1)", "D": "A" } }, "8.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Over GF(p), p>=d+1, fixed degree-at-most-d P. How many degree-at-most-d Q agree with P at coordinates1,...,k, where0<=k<=d?", "criteria": { "A": "p^(d−k)", "B": "p^(d+1−k)", "C": "1", "D": "p^k" } } } }, "attempts": [ { "status": 200, "full_response_ms": 288.80108399971505, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"8.1a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":1.0,\"C\":0.0}},\"8.1b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.22,\"probabilities\":{\"D\":0.42,\"B\":0.09,\"A\":0.14,\"C\":0.35}},\"8.1c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.27,\"probabilities\":{\"D\":0.06,\"B\":0.2,\"A\":0.28,\"C\":0.46}},\"8.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.98,\"probabilities\":{\"D\":0.01,\"B\":0.99,\"A\":0.0,\"C\":0.0}},\"8.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.35,\"probabilities\":{\"D\":0.0,\"B\":0.49,\"A\":0.51,\"C\":0.0}}},\"usage\":{\"input_tokens\":969,\"output_tokens\":226}}" } ], "response": { "model": "jev-1.13.0", "answers": { "8.1a": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "D": 0.0, "B": 0.0, "A": 1.0, "C": 0.0 } }, "8.1b": { "type": "choice", "choice": "D", "confidence": 0.22, "probabilities": { "D": 0.42, "B": 0.09, "A": 0.14, "C": 0.35 } }, "8.1c": { "type": "choice", "choice": "C", "confidence": 0.27, "probabilities": { "D": 0.06, "B": 0.2, "A": 0.28, "C": 0.46 } }, "8.2": { "type": "choice", "choice": "B", "confidence": 0.98, "probabilities": { "D": 0.01, "B": 0.99, "A": 0.0, "C": 0.0 } }, "8.3": { "type": "choice", "choice": "A", "confidence": 0.35, "probabilities": { "D": 0.0, "B": 0.49, "A": 0.51, "C": 0.0 } } }, "usage": { "input_tokens": 969, "output_tokens": 226 } } }, { "exam": "final_sp23", "section": 9, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "9.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Between any two distinct reals in [0,1] there is a rational.", "criteria": { "A": "False", "B": "True" } }, "9.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: The rationals and [0,1]×[0,1] have the same cardinality.", "criteria": { "A": "True", "B": "False" } }, "9.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: The set of finite programs is countable.", "criteria": { "A": "False", "B": "True" } }, "9.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: All outputs generated by finite programs on finite inputs form a countable set, even allowing infinite output sequences.", "criteria": { "A": "False", "B": "True" } }, "9.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: There exists a program P(Q,x) that halts exactly when Q(x) does not halt.", "criteria": { "A": "False", "B": "True" } } } }, "attempts": [ { "status": 200, "full_response_ms": 383.6813330017321, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"9.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.97,\"probabilities\":{\"B\":0.99,\"A\":0.01}},\"9.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.28,\"probabilities\":{\"B\":0.64,\"A\":0.36}},\"9.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.97,\"probabilities\":{\"B\":0.98,\"A\":0.02}},\"9.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.63,\"probabilities\":{\"B\":0.82,\"A\":0.18}},\"9.5\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.92,\"probabilities\":{\"B\":0.04,\"A\":0.96}}},\"usage\":{\"input_tokens\":730,\"output_tokens\":153}}" } ], "response": { "model": "jev-1.13.0", "answers": { "9.1": { "type": "choice", "choice": "B", "confidence": 0.97, "probabilities": { "B": 0.99, "A": 0.01 } }, "9.2": { "type": "choice", "choice": "B", "confidence": 0.28, "probabilities": { "B": 0.64, "A": 0.36 } }, "9.3": { "type": "choice", "choice": "B", "confidence": 0.97, "probabilities": { "B": 0.98, "A": 0.02 } }, "9.4": { "type": "choice", "choice": "B", "confidence": 0.63, "probabilities": { "B": 0.82, "A": 0.18 } }, "9.5": { "type": "choice", "choice": "A", "confidence": 0.92, "probabilities": { "B": 0.04, "A": 0.96 } } }, "usage": { "input_tokens": 730, "output_tokens": 153 } } }, { "exam": "final_sp23", "section": 10, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "10.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of two-element subsets of an n-element set?", "criteria": { "A": "C(n,2)", "B": "n²", "C": "2^n", "D": "n(n−1)" } }, "10.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of nonempty subsets of an n-element set?", "criteria": { "A": "n!", "B": "2^n", "C": "2^n−1", "D": "n−1" } }, "10.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Orderings of k distinct elements?", "criteria": { "A": "k^k", "B": "2^k", "C": "k", "D": "k!" } }, "10.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Hamiltonian paths in K_n, counted as ordered vertex sequences (reversals distinct), n>=3?", "criteria": { "A": "C(n,2)", "B": "(n−1)!", "C": "n!", "D": "n!/2" } }, "10.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Hamiltonian cycles in K_n, rotations identified but reverse orientations distinct, n>=3?", "criteria": { "A": "(n−1)!/2", "B": "(n−1)!", "C": "n!", "D": "2^n" } } } }, "attempts": [ { "status": 200, "full_response_ms": 428.4737080015475, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"10.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"A\":1.0,\"D\":0.0,\"C\":0.0}},\"10.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"A\":0.0,\"D\":0.0,\"C\":1.0}},\"10.3\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"A\":0.0,\"D\":1.0,\"C\":0.0}},\"10.4\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.94,\"probabilities\":{\"B\":0.02,\"A\":0.0,\"D\":0.02,\"C\":0.96}},\"10.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.55,\"probabilities\":{\"A\":0.34,\"B\":0.66,\"C\":0.0,\"D\":0.0}}},\"usage\":{\"input_tokens\":879,\"output_tokens\":228}}" } ], "response": { "model": "jev-1.13.0", "answers": { "10.1": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "A": 1.0, "D": 0.0, "C": 0.0 } }, "10.2": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "A": 0.0, "D": 0.0, "C": 1.0 } }, "10.3": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "B": 0.0, "A": 0.0, "D": 1.0, "C": 0.0 } }, "10.4": { "type": "choice", "choice": "C", "confidence": 0.94, "probabilities": { "B": 0.02, "A": 0.0, "D": 0.02, "C": 0.96 } }, "10.5": { "type": "choice", "choice": "B", "confidence": 0.55, "probabilities": { "A": 0.34, "B": 0.66, "C": 0.0, "D": 0.0 } } }, "usage": { "input_tokens": 879, "output_tokens": 228 } } }, { "exam": "final_sp23", "section": 11, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "11.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: S(n,k) counts partitions of n labeled objects into k nonempty unlabeled blocks.\nProblem: S(n,n−1), n>=2?", "criteria": { "A": "n!", "B": "n−1", "C": "C(n,2)", "D": "2^(n−1)" } }, "11.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: S(n,k) counts partitions of n labeled objects into k nonempty unlabeled blocks.\nProblem: S(n,2), n>=2?", "criteria": { "A": "2^n−1", "B": "C(n,2)", "C": "2^(n−1)−1", "D": "2^(n−1)" } }, "11.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: S(n,k) counts partitions of n labeled objects into k nonempty unlabeled blocks.\nProblem: Choose the correct recurrence with combinatorial justification.", "criteria": { "A": "S(n+1,k)=kS(n,k)+S(n,k−1): the new object joins one of k old blocks or forms a singleton.", "B": "S(n+1,k)=S(n,k)+kS(n,k−1): a singleton has k distinct labels.", "C": "S(n+1,k)=(k+1)S(n,k): every partition can place the object in k+1 existing blocks.", "D": "S(n+1,k)=nS(n,k): distinguish which existing object the newcomer follows." } }, "11.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: S(n,k) counts partitions of n labeled objects into k nonempty unlabeled blocks.\nProblem: Multisets of size n from values1..m containing exactly k distinct values,1<=k<=min(n,m)?", "criteria": { "A": "C(n−1,k−1)C(m,k)", "B": "m^n", "C": "C(n+k−1,k−1)C(m,k)", "D": "S(n,k)C(m,k)" } }, "11.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: S(n,k) counts partitions of n labeled objects into k nonempty unlabeled blocks.\nProblem: Ordered length-n sequences from1..m using exactly k distinct values?", "criteria": { "A": "S(n,k)C(m,k)", "B": "S(n,k)k!C(m,k)", "C": "k^n C(m,k)", "D": "C(n−1,k−1)C(m,k)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 357.245041999704, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"11.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"C\":1.0,\"B\":0.0,\"A\":0.0}},\"11.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.93,\"probabilities\":{\"D\":0.05,\"C\":0.95,\"B\":0.0,\"A\":0.0}},\"11.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"C\":0.0,\"B\":0.0,\"A\":1.0}},\"11.4\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.48,\"probabilities\":{\"D\":0.61,\"C\":0.09,\"B\":0.0,\"A\":0.3}},\"11.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.9,\"probabilities\":{\"D\":0.0,\"C\":0.01,\"B\":0.92,\"A\":0.07}}},\"usage\":{\"input_tokens\":1116,\"output_tokens\":228}}" } ], "response": { "model": "jev-1.13.0", "answers": { "11.1": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "D": 0.0, "C": 1.0, "B": 0.0, "A": 0.0 } }, "11.2": { "type": "choice", "choice": "C", "confidence": 0.93, "probabilities": { "D": 0.05, "C": 0.95, "B": 0.0, "A": 0.0 } }, "11.3": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "C": 0.0, "B": 0.0, "A": 1.0 } }, "11.4": { "type": "choice", "choice": "D", "confidence": 0.48, "probabilities": { "D": 0.61, "C": 0.09, "B": 0.0, "A": 0.3 } }, "11.5": { "type": "choice", "choice": "B", "confidence": 0.9, "probabilities": { "D": 0.0, "C": 0.01, "B": 0.92, "A": 0.07 } } }, "usage": { "input_tokens": 1116, "output_tokens": 228 } } }, { "exam": "final_sp23", "section": 12, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "12.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Flip a fair coin independently100 times. S=number of heads minus number of tails.\nProblem: Exact P(S>=20)?", "criteria": { "A": "2^(−100) sum(h=20..100) C(100,h)", "B": "2^(−100) C(100,60)", "C": "2^(−100) sum(h=60..100) C(100,h)", "D": "2^(−100) sum(h=70..100) C(100,h)" } }, "12.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Flip a fair coin independently100 times. S=number of heads minus number of tails.\nProblem: Apply Markov to the nonnegative number of heads to bound P(S>=20).", "criteria": { "A": "0", "B": "1/2", "C": "1/5", "D": "5/6" } }, "12.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Flip a fair coin independently100 times. S=number of heads minus number of tails.\nProblem: What bound comes from P(S>=20)<=P(|S|>=20) followed directly by two-sided Chebyshev, without symmetry refinement?", "criteria": { "A": "1/2", "B": "1/20", "C": "1/4", "D": "1/8" } }, "12.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Flip a fair coin independently100 times. S=number of heads minus number of tails.\nProblem: Use the CLT without continuity correction to approximate P(S>=20), writing Phi for standard Normal CDF.", "criteria": { "A": "Phi(2)", "B": "1−Phi(2)", "C": "1−Phi(0.2)", "D": "1−Phi(20)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 356.9667500014475, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"12.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.86,\"probabilities\":{\"B\":0.0,\"D\":0.08,\"A\":0.02,\"C\":0.9}},\"12.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.34,\"probabilities\":{\"B\":0.25,\"D\":0.07,\"A\":0.17,\"C\":0.51}},\"12.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.16,\"probabilities\":{\"B\":0.37,\"D\":0.27,\"A\":0.28,\"C\":0.08}},\"12.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.96,\"probabilities\":{\"B\":0.97,\"D\":0.0,\"A\":0.01,\"C\":0.02}}},\"usage\":{\"input_tokens\":952,\"output_tokens\":183}}" } ], "response": { "model": "jev-1.13.0", "answers": { "12.1": { "type": "choice", "choice": "C", "confidence": 0.86, "probabilities": { "B": 0.0, "D": 0.08, "A": 0.02, "C": 0.9 } }, "12.2": { "type": "choice", "choice": "C", "confidence": 0.34, "probabilities": { "B": 0.25, "D": 0.07, "A": 0.17, "C": 0.51 } }, "12.3": { "type": "choice", "choice": "B", "confidence": 0.16, "probabilities": { "B": 0.37, "D": 0.27, "A": 0.28, "C": 0.08 } }, "12.4": { "type": "choice", "choice": "B", "confidence": 0.96, "probabilities": { "B": 0.97, "D": 0.0, "A": 0.01, "C": 0.02 } } }, "usage": { "input_tokens": 952, "output_tokens": 183 } } }, { "exam": "final_sp23", "section": 13, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "13.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: R is uniform on [1/4,3/4]. Conditional on R=r, flip independently with heads probability r until first heads. J counts flips including the final heads.\nProblem: E[R]?", "criteria": { "A": "1/4", "B": "1/2", "C": "3/4", "D": "1" } }, "13.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: R is uniform on [1/4,3/4]. Conditional on R=r, flip independently with heads probability r until first heads. J counts flips including the final heads.\nProblem: E[J]?", "criteria": { "A": "4 ln(3)", "B": "2", "C": "2 ln(3)", "D": "ln(3)" } }, "13.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: R is uniform on [1/4,3/4]. Conditional on R=r, flip independently with heads probability r until first heads. J counts flips including the final heads.\nProblem: P(J>70 given R=r)?", "criteria": { "A": "(1−r)^70", "B": "r^70", "C": "(1−r)^69", "D": "1−(1−r)^70" } }, "13.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: R is uniform on [1/4,3/4]. Conditional on R=r, flip independently with heads probability r until first heads. J counts flips including the final heads.\nProblem: P(J>70)?", "criteria": { "A": "2[(3/4)^71−(1/4)^71]/71", "B": "(1/2)^70", "C": "[(3/4)^71−(1/4)^71]/71", "D": "2[(3/4)^70−(1/4)^70]/70" } } } }, "attempts": [ { "status": 200, "full_response_ms": 473.30512500047917, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"13.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"C\":0.0,\"B\":1.0}},\"13.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.6,\"probabilities\":{\"D\":0.07,\"A\":0.12,\"C\":0.7,\"B\":0.11}},\"13.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.8,\"probabilities\":{\"D\":0.0,\"A\":0.85,\"C\":0.15,\"B\":0.0}},\"13.4\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.26,\"probabilities\":{\"D\":0.08,\"A\":0.4,\"C\":0.45,\"B\":0.07}}},\"usage\":{\"input_tokens\":932,\"output_tokens\":183}}" } ], "response": { "model": "jev-1.13.0", "answers": { "13.1": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.0, "B": 1.0 } }, "13.2": { "type": "choice", "choice": "C", "confidence": 0.6, "probabilities": { "D": 0.07, "A": 0.12, "C": 0.7, "B": 0.11 } }, "13.3": { "type": "choice", "choice": "A", "confidence": 0.8, "probabilities": { "D": 0.0, "A": 0.85, "C": 0.15, "B": 0.0 } }, "13.4": { "type": "choice", "choice": "C", "confidence": 0.26, "probabilities": { "D": 0.08, "A": 0.4, "C": 0.45, "B": 0.07 } } }, "usage": { "input_tokens": 932, "output_tokens": 183 } } }, { "exam": "final_sp23", "section": 14, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "14.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: N~Poisson(mu) is independent of iid lifetimes Xi~Exponential(rate lambda). T=sum(i=1..N)Xi, empty sum zero. E[T]?", "criteria": { "A": "mu*lambda", "B": "lambda/mu", "C": "mu/lambda", "D": "1/lambda" } } } }, "attempts": [ { "status": 200, "full_response_ms": 367.11920900052064, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"14.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"C\":1.0,\"A\":0.0,\"D\":0.0}}},\"usage\":{\"input_tokens\":464,\"output_tokens\":48}}" } ], "response": { "model": "jev-1.13.0", "answers": { "14.1": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "B": 0.0, "C": 1.0, "A": 0.0, "D": 0.0 } } }, "usage": { "input_tokens": 464, "output_tokens": 48 } } }, { "exam": "final_sp23", "section": 15, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "15.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: X~Normal(mu,sigma²), where sigma² denotes the variance.\nProblem: E[X]?", "criteria": { "A": "mu²", "B": "mu", "C": "0", "D": "sigma" } }, "15.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: X~Normal(mu,sigma²), where sigma² denotes the variance.\nProblem: E[X²]?", "criteria": { "A": "sigma²", "B": "mu²+sigma²", "C": "mu²", "D": "mu+sigma²" } }, "15.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: X~Normal(mu,sigma²), where sigma² denotes the variance.\nProblem: E[X³]?", "criteria": { "A": "mu³+3mu²*sigma", "B": "mu³+sigma³", "C": "mu³+3mu*sigma²", "D": "mu³" } } } }, "attempts": [ { "status": 200, "full_response_ms": 302.3862919981184, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"15.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"B\":1.0,\"C\":0.0}},\"15.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"B\":1.0,\"C\":0.0}},\"15.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":0.0,\"C\":1.0}}},\"usage\":{\"input_tokens\":671,\"output_tokens\":138}}" } ], "response": { "model": "jev-1.13.0", "answers": { "15.1": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 1.0, "C": 0.0 } }, "15.2": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 1.0, "C": 0.0 } }, "15.3": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.0, "C": 1.0 } } }, "usage": { "input_tokens": 671, "output_tokens": 138 } } }, { "exam": "final_sp23", "section": 16, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "16.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Joint probabilities, columns S=1,2,3: row P=1:(.05,.10,0); row P=2:(.10,.05,.15); row P=3:(.05,.10,.40).\nProblem: Marginals (S=1,2,3) and (P=1,2,3)?", "criteria": { "A": "S:(1/3,1/3,1/3); P:(1/3,1/3,1/3)", "B": "S:(.25,.20,.55); P:(.30,.15,.55)", "C": "S:(.20,.25,.55); P:(.15,.30,.55)", "D": "S:(.15,.30,.55); P:(.20,.25,.55)" } }, "16.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Joint probabilities, columns S=1,2,3: row P=1:(.05,.10,0); row P=2:(.10,.05,.15); row P=3:(.05,.10,.40).\nProblem: (E[S],E[P])?", "criteria": { "A": "(2.40,2.35)", "B": "(.55,.55)", "C": "(2.35,2.40)", "D": "(2,2)" } }, "16.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Joint probabilities, columns S=1,2,3: row P=1:(.05,.10,0); row P=2:(.10,.05,.15); row P=3:(.05,.10,.40).\nProblem: P(S=P)?", "criteria": { "A": "1/3", "B": "2/5", "C": "1/2", "D": "3/4" } }, "16.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Joint probabilities, columns S=1,2,3: row P=1:(.05,.10,0); row P=2:(.10,.05,.15); row P=3:(.05,.10,.40).\nProblem: P(S=3 given S>=P)?", "criteria": { "A": "4/5", "B": "11/20", "C": "3/5", "D": "11/15" } } } }, "attempts": [ { "status": 200, "full_response_ms": 320.69995799975004, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"16.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.95,\"probabilities\":{\"C\":0.97,\"B\":0.0,\"A\":0.0,\"D\":0.03}},\"16.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.46,\"probabilities\":{\"C\":0.36,\"A\":0.59,\"B\":0.0,\"D\":0.05}},\"16.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.36,\"probabilities\":{\"C\":0.52,\"B\":0.38,\"A\":0.05,\"D\":0.05}},\"16.4\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.32,\"probabilities\":{\"C\":0.21,\"B\":0.11,\"A\":0.49,\"D\":0.19}}},\"usage\":{\"input_tokens\":1080,\"output_tokens\":183}}" } ], "response": { "model": "jev-1.13.0", "answers": { "16.1": { "type": "choice", "choice": "C", "confidence": 0.95, "probabilities": { "C": 0.97, "B": 0.0, "A": 0.0, "D": 0.03 } }, "16.2": { "type": "choice", "choice": "A", "confidence": 0.46, "probabilities": { "C": 0.36, "A": 0.59, "B": 0.0, "D": 0.05 } }, "16.3": { "type": "choice", "choice": "C", "confidence": 0.36, "probabilities": { "C": 0.52, "B": 0.38, "A": 0.05, "D": 0.05 } }, "16.4": { "type": "choice", "choice": "A", "confidence": 0.32, "probabilities": { "C": 0.21, "B": 0.11, "A": 0.49, "D": 0.19 } } }, "usage": { "input_tokens": 1080, "output_tokens": 183 } } }, { "exam": "final_sp23", "section": 17, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "17.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: I_A is1 when event A occurs and0 otherwise. Events need not be independent.\nProblem: Why does E[I_A]=P(A)?", "criteria": { "A": "The expectation is1*P(A)+0*P(not A).", "B": "All indicators are fair Bernoulli variables.", "C": "The expectation of every indicator equals its variance.", "D": "The expectation sums P(A) and P(not A)." } }, "17.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: I_A is1 when event A occurs and0 otherwise. Events need not be independent.\nProblem: Choose the valid pointwise identity and justification.", "criteria": { "A": "I_A I_B=I_A+I_B, because indicators only take0 and1.", "B": "min(I_A,I_B)=I_(A union B), because minimum detects either event.", "C": "I_A I_B=I_(A intersection B)=min(I_A,I_B), since all three are1 exactly when both events occur.", "D": "I_A I_B=I_(A union B), since multiplication represents either event." } }, "17.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: I_A is1 when event A occurs and0 otherwise. Events need not be independent.\nProblem: Choose the valid union identity and justification.", "criteria": { "A": "I_(A union B)=I_A I_B because the events are independent.", "B": "I_(A union B)=I_A+I_B without any subtraction because indicators are disjoint.", "C": "I_(A union B)=I_A+I_B−I_(A intersection B)=max(I_A,I_B); the overlap term corrects double counting.", "D": "I_(A union B)=min(I_A,I_B) because a union requires both events." } }, "17.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: I_A is1 when event A occurs and0 otherwise. Events need not be independent.\nProblem: Why are I_A,I_B uncorrelated exactly when A,B are independent?", "criteria": { "A": "Every pair of indicators has zero covariance.", "B": "All uncorrelated random variables are independent, regardless of their distributions.", "C": "Zero covariance means A and B cannot both occur.", "D": "Cov(I_A,I_B)=P(A intersection B)−P(A)P(B); zero covariance is exactly the independence criterion for two events." } }, "17.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: I_A is1 when event A occurs and0 otherwise. Events need not be independent.\nProblem: For real alpha_i of either sign and arbitrary events A_i, why is sum(i,j) alpha_i alpha_j P(A_i intersection A_j)>=0? Hint: consider the square of sum_i alpha_i I_(A_i).", "criteria": { "A": "Each summand is nonnegative because every product alpha_i alpha_j is nonnegative.", "B": "The sum equals E[(sum_i alpha_i I_(A_i))²], an expectation of a nonnegative quantity.", "C": "The sum equals (sum_i alpha_i P(A_i))² for all events.", "D": "Distinct indicators are independent, so every cross term vanishes." } } } }, "attempts": [ { "status": 200, "full_response_ms": 380.83770900266245, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"17.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":0.0,\"D\":0.0,\"A\":1.0}},\"17.2a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"C\":1.0,\"D\":0.0,\"B\":0.0,\"A\":0.0}},\"17.2b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"C\":1.0,\"D\":0.0,\"B\":0.0,\"A\":0.0}},\"17.2c\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"D\":1.0,\"B\":0.0,\"A\":0.0}},\"17.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"D\":0.0,\"B\":1.0,\"A\":0.0}}},\"usage\":{\"input_tokens\":1264,\"output_tokens\":231}}" } ], "response": { "model": "jev-1.13.0", "answers": { "17.1": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "D": 0.0, "A": 1.0 } }, "17.2a": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "C": 1.0, "D": 0.0, "B": 0.0, "A": 0.0 } }, "17.2b": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "C": 1.0, "D": 0.0, "B": 0.0, "A": 0.0 } }, "17.2c": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "C": 0.0, "D": 1.0, "B": 0.0, "A": 0.0 } }, "17.3": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "C": 0.0, "D": 0.0, "B": 1.0, "A": 0.0 } } }, "usage": { "input_tokens": 1264, "output_tokens": 231 } } }, { "exam": "final_sp23", "section": 18, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "18.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P(M=m)=alpha*m for integers1<=m<=n, zero otherwise. Useful sums: sum(m=1..n)m=n(n+1)/2; sum(m=1..n)m²=n(n+1)(2n+1)/6.\nProblem: Normalization alpha?", "criteria": { "A": "1/[n(n+1)]", "B": "1/n", "C": "2/[n(n+1)]", "D": "6/[n(n+1)(2n+1)]" } }, "18.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P(M=m)=alpha*m for integers1<=m<=n, zero otherwise. Useful sums: sum(m=1..n)m=n(n+1)/2; sum(m=1..n)m²=n(n+1)(2n+1)/6.\nProblem: E[M], expressed in terms of alpha?", "criteria": { "A": "alpha²*n(n+1)(2n+1)/6", "B": "alpha*n(n+1)(2n+1)/6", "C": "alpha*n(n+1)/2", "D": "alpha*n²" } }, "18.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P(M=m)=alpha*m for integers1<=m<=n, zero otherwise. Useful sums: sum(m=1..n)m=n(n+1)/2; sum(m=1..n)m²=n(n+1)(2n+1)/6.\nProblem: CDF at integer m?", "criteria": { "A": "0 if m<1; alpha*m² if1<=m<=n;1 if m>n", "B": "0 if m<1; alpha*m(m+1)/2 if1<=m<=n;1 if m>n", "C": "0 if m<1; alpha*m(m−1)/2 if1<=m<=n;1 if m>n", "D": "0 if m<1; alpha*m if1<=m<=n;1 if m>n" } } } }, "attempts": [ { "status": 200, "full_response_ms": 332.5494159980735, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"18.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"C\":1.0,\"A\":0.0}},\"18.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.64,\"probabilities\":{\"B\":0.73,\"D\":0.0,\"C\":0.02,\"A\":0.25}},\"18.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"B\":0.99,\"D\":0.0,\"C\":0.01,\"A\":0.0}}},\"usage\":{\"input_tokens\":908,\"output_tokens\":138}}" } ], "response": { "model": "jev-1.13.0", "answers": { "18.1": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "C": 1.0, "A": 0.0 } }, "18.2": { "type": "choice", "choice": "B", "confidence": 0.64, "probabilities": { "B": 0.73, "D": 0.0, "C": 0.02, "A": 0.25 } }, "18.3": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "B": 0.99, "D": 0.0, "C": 0.01, "A": 0.0 } } }, "usage": { "input_tokens": 908, "output_tokens": 138 } } }, { "exam": "final_sp23", "section": 19, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "19.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A bidirectional laser is anchored at (−1,0). Its angle Theta from the positive x-axis is uniform on (−pi/2,pi/2). Y is its vertical coordinate where the beam intersects x=0.\nProblem: Y as a function of Theta?", "criteria": { "A": "tan(Theta)", "B": "cos(Theta)", "C": "sin(Theta)", "D": "1/tan(Theta)" } }, "19.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A bidirectional laser is anchored at (−1,0). Its angle Theta from the positive x-axis is uniform on (−pi/2,pi/2). Y is its vertical coordinate where the beam intersects x=0.\nProblem: CDF F_Y(y) for real y?", "criteria": { "A": "1/2+arctan(y)/pi", "B": "1/2+tan(y)/pi", "C": "arctan(y)/pi", "D": "1/[pi(1+y²)]" } }, "19.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A bidirectional laser is anchored at (−1,0). Its angle Theta from the positive x-axis is uniform on (−pi/2,pi/2). Y is its vertical coordinate where the beam intersects x=0.\nProblem: Density f_Y(y) for real y?", "criteria": { "A": "1/[pi(1+y²)]", "B": "1/2+arctan(y)/pi", "C": "exp(−y²/2)/sqrt(2pi)", "D": "1/[pi(1+y)]" } } } }, "attempts": [ { "status": 200, "full_response_ms": 335.57733300040127, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"19.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"C\":0.0,\"D\":0.0,\"A\":1.0,\"B\":0.0}},\"19.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"C\":0.0,\"D\":0.0,\"A\":1.0,\"B\":0.0}},\"19.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"D\":0.0,\"A\":1.0,\"B\":0.0}}},\"usage\":{\"input_tokens\":836,\"output_tokens\":138}}" } ], "response": { "model": "jev-1.13.0", "answers": { "19.1": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "C": 0.0, "D": 0.0, "A": 1.0, "B": 0.0 } }, "19.2": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "C": 0.0, "D": 0.0, "A": 1.0, "B": 0.0 } }, "19.3": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "C": 0.0, "D": 0.0, "A": 1.0, "B": 0.0 } } }, "usage": { "input_tokens": 836, "output_tokens": 138 } } }, { "exam": "final_sp23", "section": 20, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "20.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Four guests receive their four distinct umbrellas in a uniformly random permutation. C_i means guest i receives their own umbrella.\nProblem: P(C_1)?", "criteria": { "A": "1/24", "B": "1/4", "C": "3/4", "D": "1/2" } }, "20.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Four guests receive their four distinct umbrellas in a uniformly random permutation. C_i means guest i receives their own umbrella.\nProblem: Why is P(C_i)=P(C_j) for every i,j?", "criteria": { "A": "Symmetry", "B": "Transitivity", "C": "Independence", "D": "Disjointness" } }, "20.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Four guests receive their four distinct umbrellas in a uniformly random permutation. C_i means guest i receives their own umbrella.\nProblem: Number of distinct pairwise-intersection terms in inclusion-exclusion?", "criteria": { "A": "16", "B": "12", "C": "6", "D": "4" } }, "20.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Four guests receive their four distinct umbrellas in a uniformly random permutation. C_i means guest i receives their own umbrella.\nProblem: Probability at least one guest receives their own umbrella?", "criteria": { "A": "9/24", "B": "1−(3/4)^4", "C": "15/24", "D": "1/4" } } } }, "attempts": [ { "status": 200, "full_response_ms": 433.86820800151327, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"20.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"A\":0.01,\"C\":0.0,\"B\":0.99,\"D\":0.0}},\"20.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"A\":1.0,\"C\":0.0,\"B\":0.0,\"D\":0.0}},\"20.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.98,\"probabilities\":{\"A\":0.0,\"C\":0.99,\"B\":0.01,\"D\":0.0}},\"20.4\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.9,\"probabilities\":{\"A\":0.06,\"C\":0.93,\"B\":0.01,\"D\":0.0}}},\"usage\":{\"input_tokens\":851,\"output_tokens\":183}}" } ], "response": { "model": "jev-1.13.0", "answers": { "20.1": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "A": 0.01, "C": 0.0, "B": 0.99, "D": 0.0 } }, "20.2": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "A": 1.0, "C": 0.0, "B": 0.0, "D": 0.0 } }, "20.3": { "type": "choice", "choice": "C", "confidence": 0.98, "probabilities": { "A": 0.0, "C": 0.99, "B": 0.01, "D": 0.0 } }, "20.4": { "type": "choice", "choice": "C", "confidence": 0.9, "probabilities": { "A": 0.06, "C": 0.93, "B": 0.01, "D": 0.0 } } }, "usage": { "input_tokens": 851, "output_tokens": 183 } } }, { "exam": "final_sp23", "section": 21, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "21.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: n>=4 distinct balls independently choose uniformly among n labeled bins. X counts indices i=1,...,n−1 for which bins i and i+1 are both empty.\nProblem: E[X]?", "criteria": { "A": "(n−1)((n−2)/n)^n", "B": "n((n−2)/n)^n", "C": "((n−2)/n)^n", "D": "(n−1)((n−1)/n)^(2n)" } }, "21.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: n>=4 distinct balls independently choose uniformly among n labeled bins. X counts indices i=1,...,n−1 for which bins i and i+1 are both empty.\nProblem: Define a=((n−2)/n)^n, b=((n−3)/n)^n, c=((n−4)/n)^n. Var(X)?", "criteria": { "A": "(n−1)a+2(n−2)c+(n−2)(n−3)b−(n−1)²a²", "B": "(n−1)a(1−a)", "C": "(n−1)a+2(n−2)b+(n−2)(n−3)c−(n−1)²a²", "D": "(n−1)a+(n−2)b+(n−2)(n−3)c−(n−1)²a²" } } } }, "attempts": [ { "status": 200, "full_response_ms": 463.3912919998693, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"21.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.97,\"probabilities\":{\"B\":0.0,\"D\":0.01,\"C\":0.01,\"A\":0.98}},\"21.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.75,\"probabilities\":{\"B\":0.01,\"D\":0.05,\"C\":0.81,\"A\":0.13}}},\"usage\":{\"input_tokens\":733,\"output_tokens\":93}}" } ], "response": { "model": "jev-1.13.0", "answers": { "21.1": { "type": "choice", "choice": "A", "confidence": 0.97, "probabilities": { "B": 0.0, "D": 0.01, "C": 0.01, "A": 0.98 } }, "21.2": { "type": "choice", "choice": "C", "confidence": 0.75, "probabilities": { "B": 0.01, "D": 0.05, "C": 0.81, "A": 0.13 } } }, "usage": { "input_tokens": 733, "output_tokens": 93 } } }, { "exam": "final_sp24", "section": 1, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "1.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Suppose P(x) is false for every x>=0. Then for every x>=0, P(x) implies there exists y>=x with Q(y).", "criteria": { "A": "False", "B": "True" } }, "1.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of propositional forms on n variables up to logical equivalence, with AND, OR, NOT and no quantifiers?", "criteria": { "A": "n", "B": "2n", "C": "n²", "D": "2^n", "E": "2^(2^n)" } }, "1.c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Odd prime p and integer n satisfy n=−n mod p. What residue must n have?", "criteria": { "A": "1", "B": "0", "C": "2", "D": "p/2", "E": "p−1" } }, "1.d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A total program can decide for arbitrary P,input x and line number n whether that line is ever executed in P(x).", "criteria": { "A": "True", "B": "False" } }, "1.e": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For fixed nonnegative k, number of integer-coefficient polynomials of degree at most k?", "criteria": { "A": "Countably infinite", "B": "Uncountably infinite", "C": "Finite" } }, "1.f": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Ten message symbols in 0..15, correct up to six corrupted transmitted symbols with Reed–Solomon. Smallest prime field size?", "criteria": { "A": "11", "B": "17", "C": "23", "D": "29" } }, "1.g": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A graph on n vertices is a disjoint union of k trees. Edge count?", "criteria": { "A": "k(n−1)", "B": "n−k+1", "C": "n−1", "D": "n−k", "E": "Not determined" } }, "1.h": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Positive n,m,k with k<=n,m. Evaluate sum(i=0..k) C(n,i)C(m,k−i).", "criteria": { "A": "C(n+m,k)", "B": "C(nm,k)", "C": "n!m!/(k!)²", "D": "2^(n+m)", "E": "None of these" } }, "1.i": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Shortest paths in the 7-cube from 0000000 to 1011001?", "criteria": { "A": "128", "B": "1", "C": "16", "D": "24", "E": "4" } }, "1.j1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: There exists a simple undirected graph with 7 vertices and 7 edges containing an Eulerian tour.", "criteria": { "A": "True", "B": "False" } }, "1.j2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: There exists a simple undirected graph with 4 vertices and 5 edges containing an Eulerian tour.", "criteria": { "A": "False", "B": "True" } }, "1.j3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: There exists a simple undirected graph with 5 vertices and 7 edges containing an Eulerian tour.", "criteria": { "A": "True", "B": "False" } } } }, "attempts": [ { "status": 200, "full_response_ms": 319.60733300002175, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"1.a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.97,\"probabilities\":{\"A\":0.01,\"B\":0.99}},\"1.b\":{\"type\":\"choice\",\"choice\":\"E\",\"confidence\":1.0,\"probabilities\":{\"E\":1.0,\"D\":0.0,\"A\":0.0,\"C\":0.0,\"B\":0.0}},\"1.c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.97,\"probabilities\":{\"E\":0.0,\"D\":0.02,\"A\":0.0,\"C\":0.0,\"B\":0.98}},\"1.d\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.98,\"probabilities\":{\"A\":0.01,\"B\":0.99}},\"1.e\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"A\":1.0,\"C\":0.0,\"B\":0.0}},\"1.f\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.64,\"probabilities\":{\"D\":0.08,\"A\":0.07,\"C\":0.12,\"B\":0.73}},\"1.g\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"E\":0.0,\"A\":0.0,\"C\":0.0,\"B\":0.0}},\"1.h\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"E\":0.0,\"D\":0.0,\"A\":1.0,\"C\":0.0,\"B\":0.0}},\"1.i\":{\"type\":\"choice\",\"choice\":\"E\",\"confidence\":0.98,\"probabilities\":{\"D\":0.0,\"E\":0.98,\"A\":0.01,\"C\":0.01,\"B\":0.0}},\"1.j1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.35,\"probabilities\":{\"A\":0.67,\"B\":0.33}},\"1.j2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.52,\"probabilities\":{\"A\":0.76,\"B\":0.24}},\"1.j3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.46,\"probabilities\":{\"A\":0.27,\"B\":0.73}}},\"usage\":{\"input_tokens\":1654,\"output_tokens\":480}}" } ], "response": { "model": "jev-1.13.0", "answers": { "1.a": { "type": "choice", "choice": "B", "confidence": 0.97, "probabilities": { "A": 0.01, "B": 0.99 } }, "1.b": { "type": "choice", "choice": "E", "confidence": 1.0, "probabilities": { "E": 1.0, "D": 0.0, "A": 0.0, "C": 0.0, "B": 0.0 } }, "1.c": { "type": "choice", "choice": "B", "confidence": 0.97, "probabilities": { "E": 0.0, "D": 0.02, "A": 0.0, "C": 0.0, "B": 0.98 } }, "1.d": { "type": "choice", "choice": "B", "confidence": 0.98, "probabilities": { "A": 0.01, "B": 0.99 } }, "1.e": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "A": 1.0, "C": 0.0, "B": 0.0 } }, "1.f": { "type": "choice", "choice": "B", "confidence": 0.64, "probabilities": { "D": 0.08, "A": 0.07, "C": 0.12, "B": 0.73 } }, "1.g": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "E": 0.0, "A": 0.0, "C": 0.0, "B": 0.0 } }, "1.h": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "E": 0.0, "D": 0.0, "A": 1.0, "C": 0.0, "B": 0.0 } }, "1.i": { "type": "choice", "choice": "E", "confidence": 0.98, "probabilities": { "D": 0.0, "E": 0.98, "A": 0.01, "C": 0.01, "B": 0.0 } }, "1.j1": { "type": "choice", "choice": "A", "confidence": 0.35, "probabilities": { "A": 0.67, "B": 0.33 } }, "1.j2": { "type": "choice", "choice": "A", "confidence": 0.52, "probabilities": { "A": 0.76, "B": 0.24 } }, "1.j3": { "type": "choice", "choice": "B", "confidence": 0.46, "probabilities": { "A": 0.27, "B": 0.73 } } }, "usage": { "input_tokens": 1654, "output_tokens": 480 } } }, { "exam": "final_sp24", "section": 2, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "2.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: 3^42 modulo 55?", "criteria": { "A": "27", "B": "1", "C": "9", "D": "3" } }, "2.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Solve 17x=7 mod 42, x in 0..41.", "criteria": { "A": "7", "B": "35", "C": "17", "D": "5" } }, "2.c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: GF(5), threshold three shares. P(1)=1,P(2)=1,P(4)=2; degree at most two. Secret P(0)?", "criteria": { "A": "1", "B": "2", "C": "4", "D": "3" } }, "2.d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Is f(x)=x² mod 7 a bijection on 0..6? Give a witness or inverse.", "criteria": { "A": "Yes: its inverse is x^4 mod 7.", "B": "Yes: its inverse is itself.", "C": "No: f(1)=f(6)=1.", "D": "No: f(0)=f(1)=0." } } } }, "attempts": [ { "status": 200, "full_response_ms": 396.38312500028405, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"2.a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.53,\"probabilities\":{\"D\":0.09,\"A\":0.65,\"C\":0.14,\"B\":0.12}},\"2.b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.23,\"probabilities\":{\"D\":0.36,\"A\":0.15,\"C\":0.07,\"B\":0.42}},\"2.c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.06,\"probabilities\":{\"D\":0.28,\"B\":0.3,\"C\":0.28,\"A\":0.14}},\"2.d\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"C\":1.0,\"B\":0.0}}},\"usage\":{\"input_tokens\":803,\"output_tokens\":175}}" } ], "response": { "model": "jev-1.13.0", "answers": { "2.a": { "type": "choice", "choice": "A", "confidence": 0.53, "probabilities": { "D": 0.09, "A": 0.65, "C": 0.14, "B": 0.12 } }, "2.b": { "type": "choice", "choice": "B", "confidence": 0.23, "probabilities": { "D": 0.36, "A": 0.15, "C": 0.07, "B": 0.42 } }, "2.c": { "type": "choice", "choice": "B", "confidence": 0.06, "probabilities": { "D": 0.28, "B": 0.3, "C": 0.28, "A": 0.14 } }, "2.d": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "D": 0.0, "A": 0.0, "C": 1.0, "B": 0.0 } } }, "usage": { "input_tokens": 803, "output_tokens": 175 } } }, { "exam": "final_sp24", "section": 3, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "3.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For x≠1, prove (x^n−1)/(x−1)=sum(i=0..n−1)x^i by induction on n>=1.", "criteria": { "A": "Assume x=1, divide both sides by x−1, and then extend to all x.", "B": "Base n=1 is 1=1. If the identity holds at n=k, then x^(k+1)−1=x(x^k−1)+(x−1)=(x−1)[x sum(i=0..k−1)x^i+1]=(x−1)sum(i=0..k)x^i; divide by nonzero x−1.", "C": "Base n=1 holds. Replace n by n+1 in the desired identity, which proves the induction step without using the hypothesis.", "D": "The geometric series formula proves itself, so no base or induction step is needed." } }, "3.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prove 2^m−1 prime implies m prime for natural m. Hint: factor a geometric sum.", "criteria": { "A": "m=0,1 do not yield primes. If m=ab with a,b>1, then 2^m−1=(2^a−1)[1+2^a+...+(2^a)^(b−1)], a product of two integers greater than one.", "B": "If m=ab, then 2^m−1=(2^a−1)(2^b−1).", "C": "A prime exponent always makes a Mersenne prime, and reversing that statement proves the result.", "D": "Every odd number is prime, so m must be odd and hence prime." } } } }, "attempts": [ { "status": 200, "full_response_ms": 329.00541700291797, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"3.a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"B\":1.0,\"C\":0.0}},\"3.b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":1.0,\"B\":0.0,\"C\":0.0}}},\"usage\":{\"input_tokens\":814,\"output_tokens\":89}}" } ], "response": { "model": "jev-1.13.0", "answers": { "3.a": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 1.0, "C": 0.0 } }, "3.b": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 1.0, "B": 0.0, "C": 0.0 } } }, "usage": { "input_tokens": 814, "output_tokens": 89 } } }, { "exam": "final_sp24", "section": 4, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "4.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Jobs 1:A>B>C>D,2:B>C>D>A,3:C>D>A>B,4:B>A>C>D. Candidates A:2>3>1>4,B:3>1>2>4,C:1>2>4>3,D:2>4>3>1.\nProblem: Classify (1,B),(2,C),(3,A),(4,D).", "criteria": { "A": "Stable and job-optimal", "B": "Unstable; blocking pair (1,A)", "C": "Unstable; blocking pair (4,C)", "D": "Stable and candidate-optimal" } }, "4.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Jobs 1:A>B>C>D,2:B>C>D>A,3:C>D>A>B,4:B>A>C>D. Candidates A:2>3>1>4,B:3>1>2>4,C:1>2>4>3,D:2>4>3>1.\nProblem: Classify (1,A),(2,B),(3,C),(4,D).", "criteria": { "A": "Unstable; blocking pair (4,C)", "B": "Stable and job-optimal", "C": "Stable and candidate-optimal", "D": "Unstable; blocking pair (1,B)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 423.2227499996952, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"4.a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.16,\"probabilities\":{\"B\":0.22,\"C\":0.15,\"A\":0.37,\"D\":0.26}},\"4.b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.38,\"probabilities\":{\"B\":0.54,\"C\":0.32,\"A\":0.07,\"D\":0.07}}},\"usage\":{\"input_tokens\":714,\"output_tokens\":89}}" } ], "response": { "model": "jev-1.13.0", "answers": { "4.a": { "type": "choice", "choice": "A", "confidence": 0.16, "probabilities": { "B": 0.22, "C": 0.15, "A": 0.37, "D": 0.26 } }, "4.b": { "type": "choice", "choice": "B", "confidence": 0.38, "probabilities": { "B": 0.54, "C": 0.32, "A": 0.07, "D": 0.07 } } }, "usage": { "input_tokens": 714, "output_tokens": 89 } } }, { "exam": "final_sp24", "section": 5, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "5.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Uniformly choose one fair four-sided die: faces {1,2,3,4} or faces {1,2,2,4}. Roll the chosen die twice independently conditional on identity.\nProblem: Given first roll 3, probability second roll 2?", "criteria": { "A": "5/12", "B": "3/8", "C": "1/4", "D": "1/2" } }, "5.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Uniformly choose one fair four-sided die: faces {1,2,3,4} or faces {1,2,2,4}. Roll the chosen die twice independently conditional on identity.\nProblem: Given first roll 2, probability second roll 2?", "criteria": { "A": "1/4", "B": "3/8", "C": "5/12", "D": "1/2" } } } }, "attempts": [ { "status": 200, "full_response_ms": 413.6264589978964, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"5.a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.13,\"probabilities\":{\"D\":0.3,\"C\":0.21,\"A\":0.33999999999999997,\"B\":0.15}},\"5.b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.21,\"probabilities\":{\"D\":0.28,\"A\":0.03,\"C\":0.41,\"B\":0.28}}},\"usage\":{\"input_tokens\":624,\"output_tokens\":89}}" } ], "response": { "model": "jev-1.13.0", "answers": { "5.a": { "type": "choice", "choice": "A", "confidence": 0.13, "probabilities": { "D": 0.3, "C": 0.21, "A": 0.33999999999999997, "B": 0.15 } }, "5.b": { "type": "choice", "choice": "C", "confidence": 0.21, "probabilities": { "D": 0.28, "A": 0.03, "C": 0.41, "B": 0.28 } } }, "usage": { "input_tokens": 624, "output_tokens": 89 } } }, { "exam": "final_sp24", "section": 6, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "6.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P,Q have degree at most d with every coefficient independently uniform in GF(p), prime p>d.\nProblem: Probability P(x)=Q(x) for every field element?", "criteria": { "A": "1/p^d", "B": "1/p^(d+1)", "C": "1/(d+1)", "D": "1/p" } }, "6.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P,Q have degree at most d with every coefficient independently uniform in GF(p), prime p>d.\nProblem: Probability P(0)=Q(0)?", "criteria": { "A": "1/p^(d+1)", "B": "1/p²", "C": "1/p", "D": "1/2" } }, "6.c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P,Q have degree at most d with every coefficient independently uniform in GF(p), prime p>d.\nProblem: For 0<=k<=d−1, probability P(i)=Q(i) for every i=0,...,k?", "criteria": { "A": "(k+1)/p", "B": "1/p^k", "C": "1/p^(k+1)", "D": "1/p^(d−k)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 368.765582999913, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"6.a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.95,\"probabilities\":{\"C\":0.0,\"A\":0.03,\"D\":0.01,\"B\":0.96}},\"6.b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.98,\"probabilities\":{\"C\":0.98,\"A\":0.01,\"D\":0.0,\"B\":0.01}},\"6.c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.86,\"probabilities\":{\"C\":0.89,\"A\":0.06,\"D\":0.02,\"B\":0.03}}},\"usage\":{\"input_tokens\":735,\"output_tokens\":132}}" } ], "response": { "model": "jev-1.13.0", "answers": { "6.a": { "type": "choice", "choice": "B", "confidence": 0.95, "probabilities": { "C": 0.0, "A": 0.03, "D": 0.01, "B": 0.96 } }, "6.b": { "type": "choice", "choice": "C", "confidence": 0.98, "probabilities": { "C": 0.98, "A": 0.01, "D": 0.0, "B": 0.01 } }, "6.c": { "type": "choice", "choice": "C", "confidence": 0.86, "probabilities": { "C": 0.89, "A": 0.06, "D": 0.02, "B": 0.03 } } }, "usage": { "input_tokens": 735, "output_tokens": 132 } } }, { "exam": "final_sp24", "section": 7, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "7.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: N independent node lifetimes are Exp(rate lambda). Divide into ell groups of k=N/ell nodes. A group fails when any node fails; system fails when all groups fail. P(system fails by t>=0)?", "criteria": { "A": "(1−exp(−lambda*t))^ell", "B": "(1−exp(−k*lambda*t))^ell", "C": "1−exp(−N*lambda*t)", "D": "1−(1−exp(−lambda*t))^N" } } } }, "attempts": [ { "status": 200, "full_response_ms": 364.40666599810356, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"7.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.67,\"probabilities\":{\"B\":0.76,\"D\":0.01,\"C\":0.01,\"A\":0.22}}},\"usage\":{\"input_tokens\":503,\"output_tokens\":47}}" } ], "response": { "model": "jev-1.13.0", "answers": { "7.1": { "type": "choice", "choice": "B", "confidence": 0.67, "probabilities": { "B": 0.76, "D": 0.01, "C": 0.01, "A": 0.22 } } }, "usage": { "input_tokens": 503, "output_tokens": 47 } } }, { "exam": "final_sp24", "section": 8, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "8.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Density f(x)=alpha*x on [0,1], zero elsewhere; mu=E[X].\nProblem: Normalizing alpha?", "criteria": { "A": "1", "B": "1/2", "C": "2", "D": "3" } }, "8.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Density f(x)=alpha*x on [0,1], zero elsewhere; mu=E[X].\nProblem: P(X<=1/2), expressed in alpha?", "criteria": { "A": "alpha/8", "B": "alpha/4", "C": "alpha/2", "D": "alpha/16" } }, "8.c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Density f(x)=alpha*x on [0,1], zero elsewhere; mu=E[X].\nProblem: E[X], expressed in alpha?", "criteria": { "A": "alpha/4", "B": "alpha/3", "C": "alpha/2", "D": "alpha²/3" } }, "8.d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Density f(x)=alpha*x on [0,1], zero elsewhere; mu=E[X].\nProblem: Var(X), expressed in alpha and mu?", "criteria": { "A": "alpha/3−mu²", "B": "alpha/4−mu", "C": "alpha/4+mu²", "D": "alpha/4−mu²" } } } }, "attempts": [ { "status": 200, "full_response_ms": 304.48183399857953, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"8.a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"A\":0.0,\"B\":0.0,\"C\":1.0,\"D\":0.0}},\"8.b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.29,\"probabilities\":{\"A\":0.47,\"B\":0.46,\"C\":0.01,\"D\":0.06}},\"8.c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.64,\"probabilities\":{\"A\":0.08,\"B\":0.73,\"C\":0.13,\"D\":0.06}},\"8.d\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.7,\"probabilities\":{\"A\":0.21,\"B\":0.01,\"C\":0.0,\"D\":0.78}}},\"usage\":{\"input_tokens\":819,\"output_tokens\":175}}" } ], "response": { "model": "jev-1.13.0", "answers": { "8.a": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "A": 0.0, "B": 0.0, "C": 1.0, "D": 0.0 } }, "8.b": { "type": "choice", "choice": "A", "confidence": 0.29, "probabilities": { "A": 0.47, "B": 0.46, "C": 0.01, "D": 0.06 } }, "8.c": { "type": "choice", "choice": "B", "confidence": 0.64, "probabilities": { "A": 0.08, "B": 0.73, "C": 0.13, "D": 0.06 } }, "8.d": { "type": "choice", "choice": "D", "confidence": 0.7, "probabilities": { "A": 0.21, "B": 0.01, "C": 0.0, "D": 0.78 } } }, "usage": { "input_tokens": 819, "output_tokens": 175 } } }, { "exam": "final_sp24", "section": 9, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "9.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.\nProblem: Under the given assumptions, X and Z are independent.", "criteria": { "A": "Could be either", "B": "True", "C": "False" } }, "9.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.\nProblem: Under the given assumptions, E[X+2Y−Z]=0.", "criteria": { "A": "Could be either", "B": "True", "C": "False" } }, "9.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.\nProblem: Under the given assumptions, P(Y>=10)<=1/10.", "criteria": { "A": "True", "B": "False", "C": "Could be either" } }, "9.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.\nProblem: Under the given assumptions, E[XY]=2.", "criteria": { "A": "Could be either", "B": "False", "C": "True" } }, "9.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.\nProblem: Under the given assumptions, Var(X−Y)=20.", "criteria": { "A": "Could be either", "B": "True", "C": "False" } }, "9.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.\nProblem: Under the given assumptions, Var(2X)=20.", "criteria": { "A": "True", "B": "Could be either", "C": "False" } }, "9.7": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.\nProblem: Under the given assumptions, P(|X|>=5)<=2/5.", "criteria": { "A": "True", "B": "False", "C": "Could be either" } }, "9.8": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.\nProblem: Under the given assumptions, P(|X|>=sqrt(10))>0.", "criteria": { "A": "True", "B": "Could be either", "C": "False" } } } }, "attempts": [ { "status": 200, "full_response_ms": 398.80608299790765, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"9.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.51,\"probabilities\":{\"C\":0.32,\"B\":0.01,\"A\":0.67}},\"9.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.27,\"probabilities\":{\"C\":0.46,\"B\":0.51,\"A\":0.03}},\"9.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.43,\"probabilities\":{\"C\":0.22,\"B\":0.16,\"A\":0.62}},\"9.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.93,\"probabilities\":{\"C\":0.03,\"B\":0.95,\"A\":0.02}},\"9.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.87,\"probabilities\":{\"C\":0.04,\"B\":0.91,\"A\":0.05}},\"9.6\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.97,\"probabilities\":{\"C\":0.01,\"B\":0.01,\"A\":0.98}},\"9.7\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.69,\"probabilities\":{\"C\":0.1,\"B\":0.1,\"A\":0.8}},\"9.8\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.28,\"probabilities\":{\"C\":0.05,\"B\":0.52,\"A\":0.43}}},\"usage\":{\"input_tokens\":1289,\"output_tokens\":299}}" } ], "response": { "model": "jev-1.13.0", "answers": { "9.1": { "type": "choice", "choice": "A", "confidence": 0.51, "probabilities": { "C": 0.32, "B": 0.01, "A": 0.67 } }, "9.2": { "type": "choice", "choice": "B", "confidence": 0.27, "probabilities": { "C": 0.46, "B": 0.51, "A": 0.03 } }, "9.3": { "type": "choice", "choice": "A", "confidence": 0.43, "probabilities": { "C": 0.22, "B": 0.16, "A": 0.62 } }, "9.4": { "type": "choice", "choice": "B", "confidence": 0.93, "probabilities": { "C": 0.03, "B": 0.95, "A": 0.02 } }, "9.5": { "type": "choice", "choice": "B", "confidence": 0.87, "probabilities": { "C": 0.04, "B": 0.91, "A": 0.05 } }, "9.6": { "type": "choice", "choice": "A", "confidence": 0.97, "probabilities": { "C": 0.01, "B": 0.01, "A": 0.98 } }, "9.7": { "type": "choice", "choice": "A", "confidence": 0.69, "probabilities": { "C": 0.1, "B": 0.1, "A": 0.8 } }, "9.8": { "type": "choice", "choice": "B", "confidence": 0.28, "probabilities": { "C": 0.05, "B": 0.52, "A": 0.43 } } }, "usage": { "input_tokens": 1289, "output_tokens": 299 } } }, { "exam": "final_sp24", "section": 10, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "10.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Xi are iid uniform on {1,2,3}; Sn=sum Xi. Choose a,b so (Sn−a)/b converges to N(0,1).", "criteria": { "A": "a=n,b=sqrt(n)", "B": "a=2n,b=2n/3", "C": "a=2,b=sqrt(2/3)", "D": "a=2n,b=sqrt(2n/3)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 334.2922499978158, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"10.1\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"A\":0.0,\"D\":1.0,\"B\":0.0}}},\"usage\":{\"input_tokens\":487,\"output_tokens\":48}}" } ], "response": { "model": "jev-1.13.0", "answers": { "10.1": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "C": 0.0, "A": 0.0, "D": 1.0, "B": 0.0 } } }, "usage": { "input_tokens": 487, "output_tokens": 48 } } }, { "exam": "final_sp24", "section": 11, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "11.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Piano has 52 white and 36 black keys. Chords are unordered sets of 2..10 distinct keys. Melodies are ordered sequences, repetition allowed unless constrained.\nProblem: Five-key chords with exactly two white and three black?", "criteria": { "A": "C(88,5)", "B": "C(52,3)C(36,2)", "C": "52²*36³", "D": "C(52,2)C(36,3)" } }, "11.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Piano has 52 white and 36 black keys. Chords are unordered sets of 2..10 distinct keys. Melodies are ordered sequences, repetition allowed unless constrained.\nProblem: Chords from a fixed set of ten keys?", "criteria": { "A": "C(10,2)", "B": "2^10−11", "C": "2^10", "D": "2^10−1" } }, "11.c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Piano has 52 white and 36 black keys. Chords are unordered sets of 2..10 distinct keys. Melodies are ordered sequences, repetition allowed unless constrained.\nProblem: Melodies of exactly twelve keys?", "criteria": { "A": "12^88", "B": "88!/76!", "C": "C(88,12)", "D": "88^12" } }, "11.d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Piano has 52 white and 36 black keys. Chords are unordered sets of 2..10 distinct keys. Melodies are ordered sequences, repetition allowed unless constrained.\nProblem: Melodies with 40 white and 10 black notes, no adjacent black notes?", "criteria": { "A": "52^40*36^10*C(41,10)", "B": "C(52,40)C(36,10)", "C": "52^40*36^10*C(39,10)", "D": "52^40*36^10*C(50,10)" } }, "11.e": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Piano has 52 white and 36 black keys. Chords are unordered sets of 2..10 distinct keys. Melodies are ordered sequences, repetition allowed unless constrained.\nProblem: Choose 200 keys as a multiset, unlimited repetition?", "criteria": { "A": "C(200,88)", "B": "C(287,87)", "C": "C(288,88)", "D": "88^200" } }, "11.f": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Piano has 52 white and 36 black keys. Chords are unordered sets of 2..10 distinct keys. Melodies are ordered sequences, repetition allowed unless constrained.\nProblem: Split a fixed 200-note melody into four consecutive parts, at least twenty notes each. Count?", "criteria": { "A": "C(199,3)", "B": "C(120,3)", "C": "C(123,3)", "D": "C(124,4)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 356.3250830011384, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"11.a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"C\":0.0,\"D\":1.0,\"A\":0.0}},\"11.b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.95,\"probabilities\":{\"B\":0.96,\"C\":0.0,\"A\":0.01,\"D\":0.03}},\"11.c\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.97,\"probabilities\":{\"B\":0.0,\"C\":0.0,\"D\":0.98,\"A\":0.02}},\"11.d\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.94,\"probabilities\":{\"B\":0.0,\"C\":0.03,\"D\":0.02,\"A\":0.95}},\"11.e\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.19,\"probabilities\":{\"B\":0.32,\"C\":0.39,\"D\":0.25,\"A\":0.04}},\"11.f\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.52,\"probabilities\":{\"B\":0.14,\"C\":0.2,\"A\":0.64,\"D\":0.02}}},\"usage\":{\"input_tokens\":1353,\"output_tokens\":267}}" } ], "response": { "model": "jev-1.13.0", "answers": { "11.a": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "D": 1.0, "A": 0.0 } }, "11.b": { "type": "choice", "choice": "B", "confidence": 0.95, "probabilities": { "B": 0.96, "C": 0.0, "A": 0.01, "D": 0.03 } }, "11.c": { "type": "choice", "choice": "D", "confidence": 0.97, "probabilities": { "B": 0.0, "C": 0.0, "D": 0.98, "A": 0.02 } }, "11.d": { "type": "choice", "choice": "A", "confidence": 0.94, "probabilities": { "B": 0.0, "C": 0.03, "D": 0.02, "A": 0.95 } }, "11.e": { "type": "choice", "choice": "C", "confidence": 0.19, "probabilities": { "B": 0.32, "C": 0.39, "D": 0.25, "A": 0.04 } }, "11.f": { "type": "choice", "choice": "A", "confidence": 0.52, "probabilities": { "B": 0.14, "C": 0.2, "A": 0.64, "D": 0.02 } } }, "usage": { "input_tokens": 1353, "output_tokens": 267 } } }, { "exam": "final_sp24", "section": 12, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "12.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.\nProblem: Probability a fixed vertex is isolated?", "criteria": { "A": "1/8", "B": "1/4", "C": "1/n", "D": "1/2" } }, "12.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.\nProblem: E[X]?", "criteria": { "A": "n/2", "B": "1/4", "C": "n/8", "D": "n/4" } }, "12.c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.\nProblem: Probability adjacent vertices are both isolated?", "criteria": { "A": "1/4", "B": "1/2", "C": "1/16", "D": "1/8" } }, "12.d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.\nProblem: For two nonadjacent vertices (when present), probability both isolated?", "criteria": { "A": "1/4", "B": "1/16", "C": "1/32", "D": "1/8" } }, "12.e": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.\nProblem: Var(X)?", "criteria": { "A": "5n/16", "B": "3n/16", "C": "n/4", "D": "n²/16" } }, "12.f": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.\nProblem: Chebyshev upper bound on P(X>=n/2), clipped to probability range?", "criteria": { "A": "min(1,5/(4n))", "B": "min(1,5/n)", "C": "min(1,20/n)", "D": "min(1,3/n)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 408.9225840034487, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"12.a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.96,\"probabilities\":{\"B\":0.97,\"C\":0.0,\"A\":0.02,\"D\":0.01}},\"12.b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.7,\"probabilities\":{\"B\":0.0,\"C\":0.03,\"A\":0.19,\"D\":0.78}},\"12.c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.34,\"probabilities\":{\"B\":0.01,\"C\":0.51,\"A\":0.18,\"D\":0.3}},\"12.d\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.64,\"probabilities\":{\"B\":0.74,\"C\":0.09,\"A\":0.07,\"D\":0.1}},\"12.e\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.46,\"probabilities\":{\"B\":0.6,\"C\":0.11,\"A\":0.28,\"D\":0.01}},\"12.f\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.22,\"probabilities\":{\"B\":0.3,\"C\":0.42,\"A\":0.17,\"D\":0.11}}},\"usage\":{\"input_tokens\":1127,\"output_tokens\":267}}" } ], "response": { "model": "jev-1.13.0", "answers": { "12.a": { "type": "choice", "choice": "B", "confidence": 0.96, "probabilities": { "B": 0.97, "C": 0.0, "A": 0.02, "D": 0.01 } }, "12.b": { "type": "choice", "choice": "D", "confidence": 0.7, "probabilities": { "B": 0.0, "C": 0.03, "A": 0.19, "D": 0.78 } }, "12.c": { "type": "choice", "choice": "C", "confidence": 0.34, "probabilities": { "B": 0.01, "C": 0.51, "A": 0.18, "D": 0.3 } }, "12.d": { "type": "choice", "choice": "B", "confidence": 0.64, "probabilities": { "B": 0.74, "C": 0.09, "A": 0.07, "D": 0.1 } }, "12.e": { "type": "choice", "choice": "B", "confidence": 0.46, "probabilities": { "B": 0.6, "C": 0.11, "A": 0.28, "D": 0.01 } }, "12.f": { "type": "choice", "choice": "C", "confidence": 0.22, "probabilities": { "B": 0.3, "C": 0.42, "A": 0.17, "D": 0.11 } } }, "usage": { "input_tokens": 1127, "output_tokens": 267 } } }, { "exam": "final_sp24", "section": 13, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "13.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Dolphins arrive as a homogeneous Poisson process with mean spacing 20 minutes. Each dolphin performs a trick independently with probability p each minute, then leaves at that minute’s end independently with probability q>0. All behaviors independent across dolphins and time.\nProblem: Distribution of arrivals in one hour?", "criteria": { "A": "Poisson(3)", "B": "Poisson(20)", "C": "Bin(60,1/20)", "D": "Geom(1/20)" } }, "13.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Dolphins arrive as a homogeneous Poisson process with mean spacing 20 minutes. Each dolphin performs a trick independently with probability p each minute, then leaves at that minute’s end independently with probability q>0. All behaviors independent across dolphins and time.\nProblem: Distribution of minutes a dolphin stays, counting its first minute?", "criteria": { "A": "Geom(p) on 1,2,...", "B": "Geom(q) on 1,2,...", "C": "Exp(rate q)", "D": "Poisson(1/q)" } }, "13.c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Dolphins arrive as a homogeneous Poisson process with mean spacing 20 minutes. Each dolphin performs a trick independently with probability p each minute, then leaves at that minute’s end independently with probability q>0. All behaviors independent across dolphins and time.\nProblem: Probability a dolphin stays exactly five minutes?", "criteria": { "A": "(1−q)^4*q", "B": "(1−q)^4", "C": "(1−q)^5*q", "D": "q^4*(1−q)" } }, "13.d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Dolphins arrive as a homogeneous Poisson process with mean spacing 20 minutes. Each dolphin performs a trick independently with probability p each minute, then leaves at that minute’s end independently with probability q>0. All behaviors independent across dolphins and time.\nProblem: Expected minutes a dolphin stays?", "criteria": { "A": "q", "B": "1/p", "C": "1/q", "D": "(1−q)/q" } }, "13.e": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Dolphins arrive as a homogeneous Poisson process with mean spacing 20 minutes. Each dolphin performs a trick independently with probability p each minute, then leaves at that minute’s end independently with probability q>0. All behaviors independent across dolphins and time.\nProblem: Given a twenty-minute stay, probability of exactly k tricks?", "criteria": { "A": "C(20,k)q^k(1−q)^(20−k)", "B": "p^k(1−p)^(20−k)", "C": "C(20,k)p^k*q", "D": "C(20,k)p^k(1−p)^(20−k)" } }, "13.f": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Dolphins arrive as a homogeneous Poisson process with mean spacing 20 minutes. Each dolphin performs a trick independently with probability p each minute, then leaves at that minute’s end independently with probability q>0. All behaviors independent across dolphins and time.\nProblem: Given a twenty-minute stay, expected tricks?", "criteria": { "A": "20q", "B": "20p", "C": "20/p", "D": "p/q" } }, "13.g": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Dolphins arrive as a homogeneous Poisson process with mean spacing 20 minutes. Each dolphin performs a trick independently with probability p each minute, then leaves at that minute’s end independently with probability q>0. All behaviors independent across dolphins and time.\nProblem: Unconditional expected total tricks per dolphin?", "criteria": { "A": "1/(pq)", "B": "p/(1−q)", "C": "q/p", "D": "p/q" } } } }, "attempts": [ { "status": 200, "full_response_ms": 310.4866249996121, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"13.a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"A\":1.0,\"D\":0.0,\"B\":0.0}},\"13.b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.89,\"probabilities\":{\"C\":0.0,\"A\":0.08,\"D\":0.0,\"B\":0.92}},\"13.c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.93,\"probabilities\":{\"D\":0.0,\"A\":0.95,\"C\":0.04,\"B\":0.01}},\"13.d\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.89,\"probabilities\":{\"C\":0.91,\"A\":0.01,\"D\":0.06,\"B\":0.02}},\"13.e\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.78,\"probabilities\":{\"D\":0.84,\"A\":0.1,\"C\":0.02,\"B\":0.04}},\"13.f\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.64,\"probabilities\":{\"C\":0.01,\"A\":0.05,\"D\":0.2,\"B\":0.74}},\"13.g\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.5,\"probabilities\":{\"C\":0.0,\"A\":0.02,\"D\":0.62,\"B\":0.36}}},\"usage\":{\"input_tokens\":1501,\"output_tokens\":311}}" } ], "response": { "model": "jev-1.13.0", "answers": { "13.a": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "C": 0.0, "A": 1.0, "D": 0.0, "B": 0.0 } }, "13.b": { "type": "choice", "choice": "B", "confidence": 0.89, "probabilities": { "C": 0.0, "A": 0.08, "D": 0.0, "B": 0.92 } }, "13.c": { "type": "choice", "choice": "A", "confidence": 0.93, "probabilities": { "D": 0.0, "A": 0.95, "C": 0.04, "B": 0.01 } }, "13.d": { "type": "choice", "choice": "C", "confidence": 0.89, "probabilities": { "C": 0.91, "A": 0.01, "D": 0.06, "B": 0.02 } }, "13.e": { "type": "choice", "choice": "D", "confidence": 0.78, "probabilities": { "D": 0.84, "A": 0.1, "C": 0.02, "B": 0.04 } }, "13.f": { "type": "choice", "choice": "B", "confidence": 0.64, "probabilities": { "C": 0.01, "A": 0.05, "D": 0.2, "B": 0.74 } }, "13.g": { "type": "choice", "choice": "D", "confidence": 0.5, "probabilities": { "C": 0.0, "A": 0.02, "D": 0.62, "B": 0.36 } } }, "usage": { "input_tokens": 1501, "output_tokens": 311 } } }, { "exam": "final_sp24", "section": 14, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "14.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Markov chain: 1->2 with probability1; 2->3 with1; 3->1 and 3->4 each1/2; 4->5 with1; 5->2 with1.\nProblem: Is it irreducible? Justify.", "criteria": { "A": "No; no self-loops exist.", "B": "No; transitions from states1,2,4,5 are deterministic.", "C": "Yes; every state can reach state2, and state2 can reach every state.", "D": "Yes; every state has the same number of outgoing edges." } }, "14.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Markov chain: 1->2 with probability1; 2->3 with1; 3->1 and 3->4 each1/2; 4->5 with1; 5->2 with1.\nProblem: Is it aperiodic? Justify.", "criteria": { "A": "Yes; all states have period0.", "B": "No; no self-loop exists, so period is2.", "C": "No; a five-state chain always has period5.", "D": "Yes; state2 has return paths of lengths3 and4, whose gcd is1." } }, "14.c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Markov chain: 1->2 with probability1; 2->3 with1; 3->1 and 3->4 each1/2; 4->5 with1; 5->2 with1.\nProblem: Stationary pi1=pi5=1/7. Find pi2.", "criteria": { "A": "3/7", "B": "1/7", "C": "2/7", "D": "1/5" } }, "14.d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Markov chain: 1->2 with probability1; 2->3 with1; 3->1 and 3->4 each1/2; 4->5 with1; 5->2 with1.\nProblem: bi is expected hitting time of state1 from i. Given b3=5, find b5.", "criteria": { "A": "5", "B": "10", "C": "7", "D": "6" } }, "14.e": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Markov chain: 1->2 with probability1; 2->3 with1; 3->1 and 3->4 each1/2; 4->5 with1; 5->2 with1.\nProblem: Replace 4->5 by a probability-one self-loop at4. Expected hitting time of state1 from5?", "criteria": { "A": "14", "B": "Infinity", "C": "5", "D": "7" } } } }, "attempts": [ { "status": 200, "full_response_ms": 488.92054200041457, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"14.a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.95,\"probabilities\":{\"B\":0.04,\"C\":0.96,\"A\":0.0,\"D\":0.0}},\"14.b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"C\":0.0,\"A\":0.0,\"D\":1.0}},\"14.c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.58,\"probabilities\":{\"B\":0.09,\"C\":0.6900000000000001,\"D\":0.0,\"A\":0.22}},\"14.d\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.21,\"probabilities\":{\"B\":0.1,\"C\":0.38,\"A\":0.11,\"D\":0.41}},\"14.e\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.9,\"probabilities\":{\"B\":0.93,\"C\":0.02,\"A\":0.02,\"D\":0.03}}},\"usage\":{\"input_tokens\":1172,\"output_tokens\":223}}" } ], "response": { "model": "jev-1.13.0", "answers": { "14.a": { "type": "choice", "choice": "C", "confidence": 0.95, "probabilities": { "B": 0.04, "C": 0.96, "A": 0.0, "D": 0.0 } }, "14.b": { "type": "choice", "choice": "D", "confidence": 0.99, "probabilities": { "B": 0.0, "C": 0.0, "A": 0.0, "D": 1.0 } }, "14.c": { "type": "choice", "choice": "C", "confidence": 0.58, "probabilities": { "B": 0.09, "C": 0.6900000000000001, "D": 0.0, "A": 0.22 } }, "14.d": { "type": "choice", "choice": "D", "confidence": 0.21, "probabilities": { "B": 0.1, "C": 0.38, "A": 0.11, "D": 0.41 } }, "14.e": { "type": "choice", "choice": "B", "confidence": 0.9, "probabilities": { "B": 0.93, "C": 0.02, "A": 0.02, "D": 0.03 } } }, "usage": { "input_tokens": 1172, "output_tokens": 223 } } }, { "exam": "final_sp25", "section": 2, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "2.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: The last question of this same exam states: it is Jay’s birthday today (May 16).\nProblem: Which TA has a birthday on May 16?", "criteria": { "A": "Andy", "B": "Malia", "C": "Evelyn", "D": "Jay" } } } }, "attempts": [ { "status": 200, "full_response_ms": 300.96354100169265, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"2.1\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.99,\"probabilities\":{\"D\":1.0,\"A\":0.0,\"B\":0.0,\"C\":0.0}}},\"usage\":{\"input_tokens\":448,\"output_tokens\":47}}" } ], "response": { "model": "jev-1.13.0", "answers": { "2.1": { "type": "choice", "choice": "D", "confidence": 0.99, "probabilities": { "D": 1.0, "A": 0.0, "B": 0.0, "C": 0.0 } } }, "usage": { "input_tokens": 448, "output_tokens": 47 } } }, { "exam": "final_sp25", "section": 3, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "3.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Complete: (P implies NOT Q) is equivalent to (NOT P OR __).", "criteria": { "A": "Q", "B": "P", "C": "NOT Q", "D": "True" } }, "3.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: P AND (Q OR R) is equivalent to (P OR Q) AND (P OR R).", "criteria": { "A": "False", "B": "True" } }, "3.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Truth status of P AND (NOT P OR Q) AND (NOT P OR NOT Q)?", "criteria": { "A": "Could be either", "B": "False", "C": "True" } }, "3.2ai": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: N includes zero. For every n, P(n)=>Q(n+1) and Q(n)=>P(n+1). Also P(0) holds.\nProblem: P(n) holds for every even n.", "criteria": { "A": "True", "B": "False" } }, "3.2aii": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: N includes zero. For every n, P(n)=>Q(n+1) and Q(n)=>P(n+1). Also P(0) holds.\nProblem: For every n, NOT Q(n+1) implies NOT P(n).", "criteria": { "A": "False", "B": "True" } }, "3.2aiii": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: N includes zero. For every n, P(n)=>Q(n+1) and Q(n)=>P(n+1). Also P(0) holds.\nProblem: Q(0) implies Q(n) for every n.", "criteria": { "A": "False", "B": "True" } }, "3.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: NOT forall n [P(n) OR Q(n)] implies exists n NOT P(n).", "criteria": { "A": "True", "B": "False" } } } }, "attempts": [ { "status": 200, "full_response_ms": 429.8258330018143, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"3.1a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"C\":1.0,\"B\":0.0}},\"3.1b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.52,\"probabilities\":{\"A\":0.76,\"B\":0.24}},\"3.1c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.89,\"probabilities\":{\"A\":0.07,\"C\":0.01,\"B\":0.92}},\"3.2ai\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.92,\"probabilities\":{\"A\":0.96,\"B\":0.04}},\"3.2aii\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.95,\"probabilities\":{\"A\":0.03,\"B\":0.97}},\"3.2aiii\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.43,\"probabilities\":{\"A\":0.29,\"B\":0.71}},\"3.2b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"A\":0.99,\"B\":0.01}}},\"usage\":{\"input_tokens\":1005,\"output_tokens\":243}}" } ], "response": { "model": "jev-1.13.0", "answers": { "3.1a": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "D": 0.0, "A": 0.0, "C": 1.0, "B": 0.0 } }, "3.1b": { "type": "choice", "choice": "A", "confidence": 0.52, "probabilities": { "A": 0.76, "B": 0.24 } }, "3.1c": { "type": "choice", "choice": "B", "confidence": 0.89, "probabilities": { "A": 0.07, "C": 0.01, "B": 0.92 } }, "3.2ai": { "type": "choice", "choice": "A", "confidence": 0.92, "probabilities": { "A": 0.96, "B": 0.04 } }, "3.2aii": { "type": "choice", "choice": "B", "confidence": 0.95, "probabilities": { "A": 0.03, "B": 0.97 } }, "3.2aiii": { "type": "choice", "choice": "B", "confidence": 0.43, "probabilities": { "A": 0.29, "B": 0.71 } }, "3.2b": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "A": 0.99, "B": 0.01 } } }, "usage": { "input_tokens": 1005, "output_tokens": 243 } } }, { "exam": "final_sp25", "section": 4, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "4.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct primes p,q>2 and positive e,d satisfy ed=1 mod (p−1)(q−1). Prove x^(ed)=x mod pq for every integer x, without merely citing RSA correctness.", "criteria": { "A": "Cancel x from the congruence for all x, including multiples of p and q, then use its inverse.", "B": "Fermat applies modulo pq directly with exponent pq−1, and ed=pq−1.", "C": "Modulo each prime, if x is zero the equality is immediate; otherwise Fermat applies because ed−1 is a multiple of that prime minus one. CRT then combines the equalities modulo p and q.", "D": "Since ed=1 modulo the totient, ed=1 as integers, so the equality is exact." } }, "4.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Up to isomorphism, give all trees on four vertices as edge sets on {1,2,3,4}.", "criteria": { "A": "{12,23,34} and {12,23,34,41}", "B": "Only {12,23,34}", "C": "{12,23,31} and {12,23,34}", "D": "{12,23,34} and {12,13,14}" } }, "4.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Every three-vertex tree is a subgraph of K3 (not necessarily induced).", "criteria": { "A": "True", "B": "False" } }, "4.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prove a simple graph of minimum degree d contains every k-vertex tree for 1<=k<=d.", "criteria": { "A": "Any k vertices of the graph form a tree when k<=d.", "B": "Induct on tree size. Remove a leaf, embed the smaller tree, and map the leaf to an unused neighbor of its parent’s image. At most k−2 other embedded vertices can occupy neighbors, fewer than d. The single-vertex base is immediate.", "C": "Every degree-d vertex has all its neighbors mutually adjacent, so it contains every tree directly.", "D": "Induct by adding arbitrary edges to the target tree; every missing edge must already appear in the host graph." } } } }, "attempts": [ { "status": 200, "full_response_ms": 314.1008749989851, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"4.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"A\":0.0,\"B\":0.0,\"C\":1.0,\"D\":0.0}},\"4.2a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.97,\"probabilities\":{\"A\":0.0,\"B\":0.02,\"C\":0.0,\"D\":0.98}},\"4.2b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.51,\"probabilities\":{\"A\":0.75,\"B\":0.25}},\"4.2c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"A\":0.0,\"B\":1.0,\"C\":0.0,\"D\":0.0}}},\"usage\":{\"input_tokens\":1041,\"output_tokens\":168}}" } ], "response": { "model": "jev-1.13.0", "answers": { "4.1": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "A": 0.0, "B": 0.0, "C": 1.0, "D": 0.0 } }, "4.2a": { "type": "choice", "choice": "D", "confidence": 0.97, "probabilities": { "A": 0.0, "B": 0.02, "C": 0.0, "D": 0.98 } }, "4.2b": { "type": "choice", "choice": "A", "confidence": 0.51, "probabilities": { "A": 0.75, "B": 0.25 } }, "4.2c": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "A": 0.0, "B": 1.0, "C": 0.0, "D": 0.0 } } }, "usage": { "input_tokens": 1041, "output_tokens": 168 } } }, { "exam": "final_sp25", "section": 5, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "5.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: All n jobs have the same ranking. In synchronous job-proposing deferred acceptance, counting the initial proposal day as day 1, how many days until all are held?", "criteria": { "A": "n−1", "B": "n", "C": "1", "D": "n²" } }, "5.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If all jobs have the same strict ranking, the stable matching is unique.", "criteria": { "A": "True", "B": "False" } }, "5.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If all candidates have the same strict ranking, the stable matching is unique.", "criteria": { "A": "True", "B": "False" } }, "5.4a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: All preference lists are independently uniform permutations, n jobs and n candidates. Expected number of candidates receiving exactly i first-day proposals?", "criteria": { "A": "n*(1−1/n)^i", "B": "n/i", "C": "n*C(n,i)*(1/n)^i*(1−1/n)^(n−i)", "D": "C(n,i)*(1/n)^i*(1−1/n)^(n−i)" } }, "5.4b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For n=2 with independently uniform strict preference lists, probability both possible matchings are stable?", "criteria": { "A": "1/16", "B": "1/4", "C": "1/8", "D": "1/2" } } } }, "attempts": [ { "status": 200, "full_response_ms": 312.6039579983626, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"5.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.22,\"probabilities\":{\"D\":0.04,\"A\":0.36,\"C\":0.19,\"B\":0.41}},\"5.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.77,\"probabilities\":{\"A\":0.88,\"B\":0.12}},\"5.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.86,\"probabilities\":{\"A\":0.93,\"B\":0.07}},\"5.4a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.68,\"probabilities\":{\"D\":0.19,\"A\":0.04,\"C\":0.76,\"B\":0.01}},\"5.4b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.17,\"probabilities\":{\"D\":0.39,\"A\":0.05,\"C\":0.24,\"B\":0.32}}},\"usage\":{\"input_tokens\":877,\"output_tokens\":197}}" } ], "response": { "model": "jev-1.13.0", "answers": { "5.1": { "type": "choice", "choice": "B", "confidence": 0.22, "probabilities": { "D": 0.04, "A": 0.36, "C": 0.19, "B": 0.41 } }, "5.2": { "type": "choice", "choice": "A", "confidence": 0.77, "probabilities": { "A": 0.88, "B": 0.12 } }, "5.3": { "type": "choice", "choice": "A", "confidence": 0.86, "probabilities": { "A": 0.93, "B": 0.07 } }, "5.4a": { "type": "choice", "choice": "C", "confidence": 0.68, "probabilities": { "D": 0.19, "A": 0.04, "C": 0.76, "B": 0.01 } }, "5.4b": { "type": "choice", "choice": "D", "confidence": 0.17, "probabilities": { "D": 0.39, "A": 0.05, "C": 0.24, "B": 0.32 } } }, "usage": { "input_tokens": 877, "output_tokens": 197 } } }, { "exam": "final_sp25", "section": 6, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "6.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A connected simple planar embedding has n vertices and every face boundary has four edge-sides. Edge count?", "criteria": { "A": "2n−2", "B": "3n−6", "C": "n−1", "D": "2n−4" } }, "6.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: G has a connected simple planar embedding with v vertices,e edges,f faces, all face boundaries simple cycles. F keeps G, adds one vertex per face and joins it to every vertex on that face, including the outer face. F is planar.\nProblem: Number of vertices of F?", "criteria": { "A": "v+f+e", "B": "2v", "C": "v+f", "D": "v+e" } }, "6.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: G has a connected simple planar embedding with v vertices,e edges,f faces, all face boundaries simple cycles. F keeps G, adds one vertex per face and joins it to every vertex on that face, including the outer face. F is planar.\nProblem: Number of edges of F?", "criteria": { "A": "3(v+f)−6", "B": "3v−6", "C": "2e", "D": "e+f" } }, "6.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Remove one edge from a tree. How many connected components result?", "criteria": { "A": "2", "B": "1", "C": "3", "D": "0" } }, "6.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Minimum edges removed from K(2n) to make it bipartite?", "criteria": { "A": "n−1", "B": "n²", "C": "n(n−1)", "D": "2n(n−1)" } }, "6.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Minimum vertex colors for a tree with at least one edge?", "criteria": { "A": "1", "B": "d", "C": "2", "D": "d+1" } }, "6.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Minimum edge colors for a tree of maximum degree d?", "criteria": { "A": "2d", "B": "d", "C": "d+1", "D": "2" } }, "6.7": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Maximum edges in a simple n-vertex graph, n>=3, where every cycle has length at least n?", "criteria": { "A": "2n−2", "B": "n", "C": "n−1", "D": "n+1" } }, "6.8": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Length in edges of an Eulerian tour in a graph with n vertices,m edges?", "criteria": { "A": "n", "B": "n−1", "C": "m", "D": "2m" } }, "6.9a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Delete the edges of a simple k-cycle from a connected graph, retaining all vertices. What universal lower bound on the remaining component count is attainable for some such graphs?", "criteria": { "A": "k", "B": "1", "C": "k−1", "D": "2" } }, "6.9b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Delete the edges of a simple k-cycle from a connected graph, retaining all vertices. Maximum possible remaining component count?", "criteria": { "A": "k+1", "B": "2k", "C": "k", "D": "k−1" } } } }, "attempts": [ { "status": 200, "full_response_ms": 414.6086250002554, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"6.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.42,\"probabilities\":{\"C\":0.0,\"B\":0.0,\"A\":0.57,\"D\":0.43}},\"6.2a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"C\":1.0,\"B\":0.0,\"A\":0.0,\"D\":0.0}},\"6.2b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.61,\"probabilities\":{\"C\":0.2,\"B\":0.01,\"A\":0.7,\"D\":0.09}},\"6.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":0.0,\"A\":1.0,\"D\":0.0}},\"6.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.54,\"probabilities\":{\"C\":0.33,\"B\":0.65,\"A\":0.01,\"D\":0.01}},\"6.5\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"C\":1.0,\"B\":0.0,\"A\":0.0,\"D\":0.0}},\"6.6\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.92,\"probabilities\":{\"C\":0.04,\"B\":0.94,\"A\":0.0,\"D\":0.02}},\"6.7\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.15,\"probabilities\":{\"C\":0.24,\"B\":0.34,\"A\":0.37,\"D\":0.05}},\"6.8\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.86,\"probabilities\":{\"C\":0.89,\"B\":0.0,\"A\":0.0,\"D\":0.11}},\"6.9a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.22,\"probabilities\":{\"C\":0.11,\"B\":0.26,\"A\":0.42,\"D\":0.21}},\"6.9b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.52,\"probabilities\":{\"C\":0.63,\"B\":0.04,\"A\":0.25,\"D\":0.08}}},\"usage\":{\"input_tokens\":1626,\"output_tokens\":491}}" } ], "response": { "model": "jev-1.13.0", "answers": { "6.1": { "type": "choice", "choice": "A", "confidence": 0.42, "probabilities": { "C": 0.0, "B": 0.0, "A": 0.57, "D": 0.43 } }, "6.2a": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "C": 1.0, "B": 0.0, "A": 0.0, "D": 0.0 } }, "6.2b": { "type": "choice", "choice": "A", "confidence": 0.61, "probabilities": { "C": 0.2, "B": 0.01, "A": 0.7, "D": 0.09 } }, "6.3": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 1.0, "D": 0.0 } }, "6.4": { "type": "choice", "choice": "B", "confidence": 0.54, "probabilities": { "C": 0.33, "B": 0.65, "A": 0.01, "D": 0.01 } }, "6.5": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "C": 1.0, "B": 0.0, "A": 0.0, "D": 0.0 } }, "6.6": { "type": "choice", "choice": "B", "confidence": 0.92, "probabilities": { "C": 0.04, "B": 0.94, "A": 0.0, "D": 0.02 } }, "6.7": { "type": "choice", "choice": "A", "confidence": 0.15, "probabilities": { "C": 0.24, "B": 0.34, "A": 0.37, "D": 0.05 } }, "6.8": { "type": "choice", "choice": "C", "confidence": 0.86, "probabilities": { "C": 0.89, "B": 0.0, "A": 0.0, "D": 0.11 } }, "6.9a": { "type": "choice", "choice": "A", "confidence": 0.22, "probabilities": { "C": 0.11, "B": 0.26, "A": 0.42, "D": 0.21 } }, "6.9b": { "type": "choice", "choice": "C", "confidence": 0.52, "probabilities": { "C": 0.63, "B": 0.04, "A": 0.25, "D": 0.08 } } }, "usage": { "input_tokens": 1626, "output_tokens": 491 } } }, { "exam": "final_sp25", "section": 7, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "7.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: 2^36 mod 7?", "criteria": { "A": "1", "B": "2", "C": "4", "D": "0" } }, "7.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: 2^36 mod 35?", "criteria": { "A": "4", "B": "16", "C": "1", "D": "8" } }, "7.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: gcd(385,70)?", "criteria": { "A": "5", "B": "35", "C": "70", "D": "7" } }, "7.7a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of distinct square residues modulo 3, including zero?", "criteria": { "A": "2", "B": "3", "C": "1", "D": "0" } }, "7.7c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of distinct square residues modulo 15, including zero?", "criteria": { "A": "6", "B": "8", "C": "4", "D": "7" } }, "7.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If x=0 mod d, then bx−kd=0 mod d for all integers b,k.", "criteria": { "A": "True", "B": "False" } }, "7.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If x=1 mod d, then bx−kd=1 mod d for all integers b,k.", "criteria": { "A": "False", "B": "True" } }, "7.6a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: gcd(a,m)=d and 2a=b mod m, with positive modulus m.\nProblem: Assume d>1. Give the least positive t, nonzero modulo m, with (2+t)a=b mod m.", "criteria": { "A": "m/d", "B": "d", "C": "a/d", "D": "m" } }, "7.6b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: gcd(a,m)=d and 2a=b mod m, with positive modulus m.\nProblem: b must be even.", "criteria": { "A": "True", "B": "False" } }, "7.6c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: gcd(a,m)=d and 2a=b mod m, with positive modulus m.\nProblem: d divides b.", "criteria": { "A": "True", "B": "False" } }, "7.6d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: gcd(a,m)=d and 2a=b mod m, with positive modulus m.\nProblem: How many distinct solutions modulo m does ax=b mod m have, given it is solvable?", "criteria": { "A": "d", "B": "m/d", "C": "1", "D": "m" } }, "7.7b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prime p>2: count square residues including zero.", "criteria": { "A": "(p−1)/2", "B": "p", "C": "(p+1)/2", "D": "p−1" } }, "7.7d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct primes p,q>2: count square residues modulo pq, including zero. Hint: CRT.", "criteria": { "A": "(p−1)(q−1)/4", "B": "(p+q)/2", "C": "(pq+1)/2", "D": "(p+1)(q+1)/4" } } } }, "attempts": [ { "status": 200, "full_response_ms": 435.76137499985634, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"7.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.21,\"probabilities\":{\"B\":0.33,\"D\":0.0,\"C\":0.25,\"A\":0.42}},\"7.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.37,\"probabilities\":{\"B\":0.53,\"D\":0.07,\"C\":0.18,\"A\":0.22}},\"7.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.93,\"probabilities\":{\"B\":0.95,\"D\":0.04,\"C\":0.0,\"A\":0.01}},\"7.7a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.93,\"probabilities\":{\"B\":0.04,\"D\":0.0,\"C\":0.01,\"A\":0.95}},\"7.7c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.56,\"probabilities\":{\"B\":0.05,\"D\":0.12,\"C\":0.16,\"A\":0.67}},\"7.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"A\":1.0}},\"7.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.49,\"probabilities\":{\"B\":0.74,\"A\":0.26}},\"7.6a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.79,\"probabilities\":{\"B\":0.13,\"D\":0.02,\"C\":0.01,\"A\":0.84}},\"7.6b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.46,\"probabilities\":{\"B\":0.73,\"A\":0.27}},\"7.6c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.91,\"probabilities\":{\"B\":0.05,\"A\":0.95}},\"7.6d\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.75,\"probabilities\":{\"B\":0.16,\"D\":0.0,\"C\":0.03,\"A\":0.81}},\"7.7b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"C\":1.0,\"A\":0.0}},\"7.7d\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"D\":1.0,\"C\":0.0,\"A\":0.0}}},\"usage\":{\"input_tokens\":1639,\"output_tokens\":527}}" } ], "response": { "model": "jev-1.13.0", "answers": { "7.1": { "type": "choice", "choice": "A", "confidence": 0.21, "probabilities": { "B": 0.33, "D": 0.0, "C": 0.25, "A": 0.42 } }, "7.2": { "type": "choice", "choice": "B", "confidence": 0.37, "probabilities": { "B": 0.53, "D": 0.07, "C": 0.18, "A": 0.22 } }, "7.5": { "type": "choice", "choice": "B", "confidence": 0.93, "probabilities": { "B": 0.95, "D": 0.04, "C": 0.0, "A": 0.01 } }, "7.7a": { "type": "choice", "choice": "A", "confidence": 0.93, "probabilities": { "B": 0.04, "D": 0.0, "C": 0.01, "A": 0.95 } }, "7.7c": { "type": "choice", "choice": "A", "confidence": 0.56, "probabilities": { "B": 0.05, "D": 0.12, "C": 0.16, "A": 0.67 } }, "7.3": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "B": 0.0, "A": 1.0 } }, "7.4": { "type": "choice", "choice": "B", "confidence": 0.49, "probabilities": { "B": 0.74, "A": 0.26 } }, "7.6a": { "type": "choice", "choice": "A", "confidence": 0.79, "probabilities": { "B": 0.13, "D": 0.02, "C": 0.01, "A": 0.84 } }, "7.6b": { "type": "choice", "choice": "B", "confidence": 0.46, "probabilities": { "B": 0.73, "A": 0.27 } }, "7.6c": { "type": "choice", "choice": "A", "confidence": 0.91, "probabilities": { "B": 0.05, "A": 0.95 } }, "7.6d": { "type": "choice", "choice": "A", "confidence": 0.75, "probabilities": { "B": 0.16, "D": 0.0, "C": 0.03, "A": 0.81 } }, "7.7b": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "B": 0.0, "D": 0.0, "C": 1.0, "A": 0.0 } }, "7.7d": { "type": "choice", "choice": "D", "confidence": 0.99, "probabilities": { "B": 0.0, "D": 1.0, "C": 0.0, "A": 0.0 } } }, "usage": { "input_tokens": 1639, "output_tokens": 527 } } }, { "exam": "final_sp25", "section": 8, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "8.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: GF(5) interpolation yields P=x²+4x+1. One normalized Lagrange basis polynomial is Delta=x²+3x+2. Recover the three interpolation points.", "criteria": { "A": "{(1,2),(3,0),(4,0)}", "B": "{(1,1),(3,2),(4,3)}", "C": "{(0,1),(3,2),(4,3)}", "D": "{(1,1),(2,3),(4,3)}" } }, "8.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Every polynomial of degree exactly two is a bijection over every prime field.", "criteria": { "A": "True", "B": "False" } }, "8.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Every polynomial of degree exactly one is a bijection over a prime field.", "criteria": { "A": "True", "B": "False" } }, "8.4a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Where do 2x+3 and 3x+2 intersect over GF(5)?", "criteria": { "A": "4", "B": "0", "C": "1", "D": "2" } }, "8.4b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct formal polynomials of degrees dP,dQ over GF(p), p>max(dP,dQ): tight general upper bound on intersection coordinates?", "criteria": { "A": "dP+dQ", "B": "p", "C": "min(dP,dQ)", "D": "max(dP,dQ)" } }, "8.4c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Over GF(p),p>3, fixed P. Count polynomials Q of degree at most three with Q(1)=P(1),Q(2)=P(2).", "criteria": { "A": "(p−1)p", "B": "p²", "C": "1", "D": "p" } }, "8.5a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Encode n message symbols as polynomial coefficients. Degree bound of message P?", "criteria": { "A": "At most n+2k", "B": "At most n−1", "C": "Exactly n", "D": "Exactly k" } }, "8.5b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: How many distinct evaluations suffice to correct k corruptions in n message symbols?", "criteria": { "A": "n+2k−1", "B": "2n+k", "C": "n+k", "D": "n+2k" } }, "8.5c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Why does E(i)(P(i)−ri)=0 at each transmitted coordinate, where E vanishes at error locations?", "criteria": { "A": "At a corrupted coordinate E(i)=0; at an uncorrupted one P(i)−ri=0. Thus at least one factor vanishes everywhere.", "B": "Both factors vanish at every coordinate.", "C": "Every field product with an error polynomial is zero.", "D": "The product has degree zero, so it is identically zero." } } } }, "attempts": [ { "status": 200, "full_response_ms": 399.2643340025097, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"8.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.45,\"probabilities\":{\"B\":0.58,\"C\":0.12,\"D\":0.07,\"A\":0.23}},\"8.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"B\":0.99,\"A\":0.01}},\"8.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.89,\"probabilities\":{\"B\":0.05,\"A\":0.95}},\"8.4a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.18,\"probabilities\":{\"B\":0.07,\"A\":0.39,\"D\":0.16,\"C\":0.38}},\"8.4b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.32,\"probabilities\":{\"B\":0.0,\"C\":0.49,\"D\":0.41,\"A\":0.1}},\"8.4c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.81,\"probabilities\":{\"B\":0.86,\"A\":0.07,\"D\":0.07,\"C\":0.0}},\"8.5a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"B\":1.0,\"C\":0.0,\"D\":0.0,\"A\":0.0}},\"8.5b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.82,\"probabilities\":{\"B\":0.0,\"A\":0.05,\"D\":0.87,\"C\":0.08}},\"8.5c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"C\":0.0,\"D\":0.0,\"A\":1.0}}},\"usage\":{\"input_tokens\":1422,\"output_tokens\":377}}" } ], "response": { "model": "jev-1.13.0", "answers": { "8.1": { "type": "choice", "choice": "B", "confidence": 0.45, "probabilities": { "B": 0.58, "C": 0.12, "D": 0.07, "A": 0.23 } }, "8.2": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "B": 0.99, "A": 0.01 } }, "8.3": { "type": "choice", "choice": "A", "confidence": 0.89, "probabilities": { "B": 0.05, "A": 0.95 } }, "8.4a": { "type": "choice", "choice": "A", "confidence": 0.18, "probabilities": { "B": 0.07, "A": 0.39, "D": 0.16, "C": 0.38 } }, "8.4b": { "type": "choice", "choice": "C", "confidence": 0.32, "probabilities": { "B": 0.0, "C": 0.49, "D": 0.41, "A": 0.1 } }, "8.4c": { "type": "choice", "choice": "B", "confidence": 0.81, "probabilities": { "B": 0.86, "A": 0.07, "D": 0.07, "C": 0.0 } }, "8.5a": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "B": 1.0, "C": 0.0, "D": 0.0, "A": 0.0 } }, "8.5b": { "type": "choice", "choice": "D", "confidence": 0.82, "probabilities": { "B": 0.0, "A": 0.05, "D": 0.87, "C": 0.08 } }, "8.5c": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "D": 0.0, "A": 1.0 } } }, "usage": { "input_tokens": 1422, "output_tokens": 377 } } }, { "exam": "final_sp25", "section": 9, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "9.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct arrangements of BAA?", "criteria": { "A": "1", "B": "3", "C": "6", "D": "2" } }, "9.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct arrangements of 126?", "criteria": { "A": "1", "B": "3", "C": "9", "D": "6" } }, "9.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: n distinct bears, k distinct children,n>k. Assign one bear each, without reuse. Count?", "criteria": { "A": "n^k", "B": "k^n", "C": "C(n,k)", "D": "n!/(n−k)!" } }, "9.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Bipartite labeled graph with fixed parts each of size n and exactly m edges. Count?", "criteria": { "A": "2^(n²)", "B": "C(n²,m)", "C": "n^(2m)", "D": "C(2n,m)" } }, "9.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of functions from p residues to p residues?", "criteria": { "A": "p^p", "B": "p!", "C": "2^p", "D": "p²" } }, "9.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of bijections from p residues to themselves?", "criteria": { "A": "p²", "B": "p^p", "C": "p", "D": "p!" } }, "9.7": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Form three named teams of four distinct players from 15 people, no overlap. Count?", "criteria": { "A": "15!/(4!4!4!3!3!)", "B": "C(15,12)", "C": "C(15,4)^3", "D": "C(15,4)C(11,4)C(7,4)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 311.1654579988681, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"9.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.84,\"probabilities\":{\"B\":0.88,\"D\":0.11,\"A\":0.01,\"C\":0.0}},\"9.2\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.31,\"probabilities\":{\"D\":0.48,\"B\":0.11,\"A\":0.34,\"C\":0.07}},\"9.3\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"B\":0.0,\"A\":0.0,\"C\":0.0}},\"9.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":1.0,\"A\":0.0,\"C\":0.0}},\"9.5\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":1.0,\"C\":0.0}},\"9.6\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":1.0,\"A\":0.0,\"C\":0.0}},\"9.7\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.8,\"probabilities\":{\"D\":0.86,\"B\":0.0,\"A\":0.14,\"C\":0.0}}},\"usage\":{\"input_tokens\":1112,\"output_tokens\":311}}" } ], "response": { "model": "jev-1.13.0", "answers": { "9.1": { "type": "choice", "choice": "B", "confidence": 0.84, "probabilities": { "B": 0.88, "D": 0.11, "A": 0.01, "C": 0.0 } }, "9.2": { "type": "choice", "choice": "D", "confidence": 0.31, "probabilities": { "D": 0.48, "B": 0.11, "A": 0.34, "C": 0.07 } }, "9.3": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "A": 0.0, "C": 0.0 } }, "9.4": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 1.0, "A": 0.0, "C": 0.0 } }, "9.5": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 1.0, "C": 0.0 } }, "9.6": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 1.0, "A": 0.0, "C": 0.0 } }, "9.7": { "type": "choice", "choice": "D", "confidence": 0.8, "probabilities": { "D": 0.86, "B": 0.0, "A": 0.14, "C": 0.0 } } }, "usage": { "input_tokens": 1112, "output_tokens": 311 } } }, { "exam": "final_sp25", "section": 10, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "10.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Give a combinatorial proof of sum(i=0..n) C(n,i)C(n,n−i)=C(2n,n).", "criteria": { "A": "Choose n objects from two disjoint n-object groups. Classify each choice by how many i come from the first group; exactly n−i come from the second.", "B": "Choose the same subset twice, which counts every n-subset of 2n once.", "C": "Choose i from both groups, always obtaining n total objects regardless of i.", "D": "Every binomial coefficient equals the number of permutations of its top argument." } }, "10.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Given sum(i=0..n) C(n,i)C(n,n−i)=C(2n,n), why does sum(i=0..n) C(n,i)²=C(2n,n) follow?", "criteria": { "A": "All summands are equal to C(2n,n)/n.", "B": "C(n,n−i)=C(n,i) by taking complements of subsets.", "C": "Every binomial coefficient equals its own square.", "D": "C(n,n−i)=n−i." } } } }, "attempts": [ { "status": 200, "full_response_ms": 733.361208000133, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"10.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":1.0,\"C\":0.0}},\"10.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":1.0,\"A\":0.0,\"C\":0.0}}},\"usage\":{\"input_tokens\":702,\"output_tokens\":93}}" } ], "response": { "model": "jev-1.13.0", "answers": { "10.1": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 1.0, "C": 0.0 } }, "10.2": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 1.0, "A": 0.0, "C": 0.0 } } }, "usage": { "input_tokens": 702, "output_tokens": 93 } } }, { "exam": "final_sp25", "section": 11, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "11.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: The power set of N has the same cardinality as [0,1].", "criteria": { "A": "False", "B": "True" } }, "11.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A finite Cartesian product of countable sets is countable.", "criteria": { "A": "False", "B": "True" } }, "11.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A=Z×Z; f:A->C is onto and g:C->B is injective.\nProblem: A is countable.", "criteria": { "A": "True", "B": "False" } }, "11.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A=Z×Z; f:A->C is onto and g:C->B is injective.\nProblem: These assumptions imply |A| is strictly smaller than |B|.", "criteria": { "A": "False", "B": "True" } }, "11.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Give a bijection (0,1]->[1,infinity).", "criteria": { "A": "f(x)=1/x", "B": "f(x)=1/(1−x)", "C": "f(x)=x+1", "D": "f(x)=−ln(x)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 717.5805830011086, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"11.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.9,\"probabilities\":{\"B\":0.95,\"A\":0.05}},\"11.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.97,\"probabilities\":{\"B\":0.99,\"A\":0.01}},\"11.3a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"A\":1.0}},\"11.3b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"B\":0.01,\"A\":0.99}},\"11.4\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"C\":0.0,\"D\":0.0,\"A\":1.0}}},\"usage\":{\"input_tokens\":794,\"output_tokens\":174}}" } ], "response": { "model": "jev-1.13.0", "answers": { "11.1": { "type": "choice", "choice": "B", "confidence": 0.9, "probabilities": { "B": 0.95, "A": 0.05 } }, "11.2": { "type": "choice", "choice": "B", "confidence": 0.97, "probabilities": { "B": 0.99, "A": 0.01 } }, "11.3a": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "A": 1.0 } }, "11.3b": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "B": 0.01, "A": 0.99 } }, "11.4": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "B": 0.0, "C": 0.0, "D": 0.0, "A": 1.0 } } }, "usage": { "input_tokens": 794, "output_tokens": 174 } } }, { "exam": "final_sp25", "section": 12, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "12.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: There is an algorithm deciding whether P(x) halts within |x|^k steps, for arbitrary P, nonempty finite x and nonnegative integer k.", "criteria": { "A": "False", "B": "True" } }, "12.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Hypothetical HaltsOnEmpty decides whether a program halts on empty input. Fill: Halt(P,x) defines inner(y): __1__; then returns HaltsOnEmpty(__2__).", "criteria": { "A": "(return True; inner)", "B": "(P(x); inner)", "C": "(P(x); x)", "D": "(P(y); P)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 326.3699160015676, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"12.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.44,\"probabilities\":{\"B\":0.72,\"A\":0.28}},\"12.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.59,\"probabilities\":{\"B\":0.7,\"D\":0.12,\"A\":0.02,\"C\":0.16}}},\"usage\":{\"input_tokens\":569,\"output_tokens\":79}}" } ], "response": { "model": "jev-1.13.0", "answers": { "12.1": { "type": "choice", "choice": "B", "confidence": 0.44, "probabilities": { "B": 0.72, "A": 0.28 } }, "12.2": { "type": "choice", "choice": "B", "confidence": 0.59, "probabilities": { "B": 0.7, "D": 0.12, "A": 0.02, "C": 0.16 } } }, "usage": { "input_tokens": 569, "output_tokens": 79 } } }, { "exam": "final_sp25", "section": 13, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "13.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Sharp upper bound on P(A union B) using only a=P(A),b=P(B)?", "criteria": { "A": "a+b", "B": "a*b", "C": "max(a,b)", "D": "min(1,a+b)" } }, "13.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Sharp lower bound on P(A union B) using only a=P(A),b=P(B)?", "criteria": { "A": "a+b", "B": "max(a,b)", "C": "min(a,b)", "D": "a*b" } }, "13.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A is a subset of B and both have positive probability. P(A given B)?", "criteria": { "A": "1", "B": "P(B)/P(A)", "C": "P(A)/P(B)", "D": "P(A)P(B)" } }, "13.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A is a subset of B and P(A)>0. P(B given A)?", "criteria": { "A": "0", "B": "P(B)", "C": "P(A)/P(B)", "D": "1" } }, "13.5a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Events have probabilities a=P(A),b=P(B),c=P(C), with P(A intersection B)=P(B intersection C)=alpha. Assume these quantities are feasible.\nProblem: Sharp upper bound on P(A intersection C)?", "criteria": { "A": "a+c", "B": "min(a,b,c)", "C": "alpha", "D": "min(a,c)" } }, "13.5b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Events have probabilities a=P(A),b=P(B),c=P(C), with P(A intersection B)=P(B intersection C)=alpha. Assume these quantities are feasible.\nProblem: Sharp lower bound on P(A intersection C) using all the given quantities?", "criteria": { "A": "max(0,2alpha−b)+max(0,a+c−2alpha+b−1)", "B": "max(0,a+c−1)", "C": "max(0,2alpha−b)", "D": "alpha" } }, "13.5c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Events have probabilities a=P(A),b=P(B),c=P(C), with P(A intersection B)=P(B intersection C)=alpha. Assume these quantities are feasible.\nProblem: Justify a sharp lower bound using the partition into B and its complement.", "criteria": { "A": "The overlap is always exactly alpha, regardless of a,b,c.", "B": "The two known intersections are equal, so A and C must be identical.", "C": "Ignore the complement of B, because neither A nor C can extend beyond B.", "D": "Inside B, overlap is at least max(0,2alpha−b). Outside B, the masses a−alpha and c−alpha share a region of mass 1−b, forcing max(0,a+c−2alpha+b−1). Add the independently attainable minima." } } } }, "attempts": [ { "status": 200, "full_response_ms": 297.2415420008474, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"13.1\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.98,\"probabilities\":{\"B\":0.0,\"D\":0.98,\"C\":0.0,\"A\":0.02}},\"13.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"B\":1.0,\"D\":0.0,\"C\":0.0,\"A\":0.0}},\"13.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"C\":1.0,\"A\":0.0}},\"13.4\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":1.0,\"C\":0.0,\"A\":0.0}},\"13.5a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.66,\"probabilities\":{\"B\":0.08,\"D\":0.75,\"C\":0.16,\"A\":0.01}},\"13.5b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.9,\"probabilities\":{\"B\":0.02,\"D\":0.0,\"C\":0.05,\"A\":0.93}},\"13.5c\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":1.0,\"C\":0.0,\"A\":0.0}}},\"usage\":{\"input_tokens\":1325,\"output_tokens\":321}}" } ], "response": { "model": "jev-1.13.0", "answers": { "13.1": { "type": "choice", "choice": "D", "confidence": 0.98, "probabilities": { "B": 0.0, "D": 0.98, "C": 0.0, "A": 0.02 } }, "13.2": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "B": 1.0, "D": 0.0, "C": 0.0, "A": 0.0 } }, "13.3": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "B": 0.0, "D": 0.0, "C": 1.0, "A": 0.0 } }, "13.4": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 1.0, "C": 0.0, "A": 0.0 } }, "13.5a": { "type": "choice", "choice": "D", "confidence": 0.66, "probabilities": { "B": 0.08, "D": 0.75, "C": 0.16, "A": 0.01 } }, "13.5b": { "type": "choice", "choice": "A", "confidence": 0.9, "probabilities": { "B": 0.02, "D": 0.0, "C": 0.05, "A": 0.93 } }, "13.5c": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 1.0, "C": 0.0, "A": 0.0 } } }, "usage": { "input_tokens": 1325, "output_tokens": 321 } } }, { "exam": "final_sp25", "section": 14, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "14.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Choose a fair four-sided or fair six-sided die with equal probability, then roll the chosen die twice independently conditional on its identity. A:first roll 1; B:second roll 1; C:four-sided die selected.\nProblem: P(A given B)?", "criteria": { "A": "5/24", "B": "13/60", "C": "1/6", "D": "1/4" } }, "14.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Choose a fair four-sided or fair six-sided die with equal probability, then roll the chosen die twice independently conditional on its identity. A:first roll 1; B:second roll 1; C:four-sided die selected.\nProblem: P(C given A)?", "criteria": { "A": "2/5", "B": "1/4", "C": "1/2", "D": "3/5" } } } }, "attempts": [ { "status": 200, "full_response_ms": 372.60020800022176, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"14.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.46,\"probabilities\":{\"B\":0.08,\"C\":0.59,\"A\":0.1,\"D\":0.23}},\"14.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.34,\"probabilities\":{\"C\":0.14,\"B\":0.02,\"A\":0.51,\"D\":0.33}}},\"usage\":{\"input_tokens\":621,\"output_tokens\":93}}" } ], "response": { "model": "jev-1.13.0", "answers": { "14.1": { "type": "choice", "choice": "C", "confidence": 0.46, "probabilities": { "B": 0.08, "C": 0.59, "A": 0.1, "D": 0.23 } }, "14.2": { "type": "choice", "choice": "A", "confidence": 0.34, "probabilities": { "C": 0.14, "B": 0.02, "A": 0.51, "D": 0.33 } } }, "usage": { "input_tokens": 621, "output_tokens": 93 } } }, { "exam": "final_sp25", "section": 15, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "15.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Independent Bernoulli X,Y each have mean 1/2. P(X=Y)?", "criteria": { "A": "1", "B": "1/4", "C": "1/2", "D": "3/4" } }, "15.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Bernoulli X,Y each have mean 1/2 and correlation 1/2.\nProblem: E[XY]?", "criteria": { "A": "1/4", "B": "3/8", "C": "1/2", "D": "1/8" } }, "15.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Bernoulli X,Y each have mean 1/2 and correlation 1/2.\nProblem: P(X=Y)?", "criteria": { "A": "1/2", "B": "3/4", "C": "3/8", "D": "1/4" } } } }, "attempts": [ { "status": 200, "full_response_ms": 371.270874999027, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"15.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.98,\"probabilities\":{\"D\":0.01,\"A\":0.0,\"B\":0.0,\"C\":0.99}},\"15.2a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.88,\"probabilities\":{\"D\":0.02,\"A\":0.01,\"B\":0.91,\"C\":0.06}},\"15.2b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.53,\"probabilities\":{\"D\":0.02,\"A\":0.27,\"B\":0.66,\"C\":0.05}}},\"usage\":{\"input_tokens\":672,\"output_tokens\":140}}" } ], "response": { "model": "jev-1.13.0", "answers": { "15.1": { "type": "choice", "choice": "C", "confidence": 0.98, "probabilities": { "D": 0.01, "A": 0.0, "B": 0.0, "C": 0.99 } }, "15.2a": { "type": "choice", "choice": "B", "confidence": 0.88, "probabilities": { "D": 0.02, "A": 0.01, "B": 0.91, "C": 0.06 } }, "15.2b": { "type": "choice", "choice": "B", "confidence": 0.53, "probabilities": { "D": 0.02, "A": 0.27, "B": 0.66, "C": 0.05 } } }, "usage": { "input_tokens": 672, "output_tokens": 140 } } }, { "exam": "final_sp25", "section": 16, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "16.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Sample n people independently from a population with positive-test probability p. Xi indicates a positive test.\nProblem: E[Xi]?", "criteria": { "A": "p(1−p)", "B": "1−p", "C": "p", "D": "np" } }, "16.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Sample n people independently from a population with positive-test probability p. Xi indicates a positive test.\nProblem: Sharp p-independent upper bound on Var(Xi)?", "criteria": { "A": "1", "B": "p", "C": "1/4", "D": "1/2" } }, "16.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Sample n people independently from a population with positive-test probability p. Xi indicates a positive test.\nProblem: 50 of 100 independently sampled people test positive. Using Chebyshev and the worst-case Bernoulli variance, give a 95% confidence interval for p.", "criteria": { "A": "[1/2−1/20,1/2+1/20]", "B": "[1/2−1/sqrt(20),1/2+1/sqrt(20)]", "C": "[1/2−1/10,1/2+1/10]", "D": "[1/2−1/sqrt(100),1/2+1/sqrt(100)]" } } } }, "attempts": [ { "status": 200, "full_response_ms": 288.18266700181994, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"16.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"C\":1.0,\"A\":0.0,\"B\":0.0,\"D\":0.0}},\"16.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.87,\"probabilities\":{\"C\":0.91,\"A\":0.02,\"B\":0.07,\"D\":0.0}},\"16.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.53,\"probabilities\":{\"C\":0.26,\"A\":0.06,\"B\":0.65,\"D\":0.03}}},\"usage\":{\"input_tokens\":785,\"output_tokens\":138}}" } ], "response": { "model": "jev-1.13.0", "answers": { "16.1": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "C": 1.0, "A": 0.0, "B": 0.0, "D": 0.0 } }, "16.2": { "type": "choice", "choice": "C", "confidence": 0.87, "probabilities": { "C": 0.91, "A": 0.02, "B": 0.07, "D": 0.0 } }, "16.3": { "type": "choice", "choice": "B", "confidence": 0.53, "probabilities": { "C": 0.26, "A": 0.06, "B": 0.65, "D": 0.03 } } }, "usage": { "input_tokens": 785, "output_tokens": 138 } } }, { "exam": "final_sp25", "section": 17, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "17.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prove every tournament (one directed edge per unordered vertex pair) has a Hamiltonian path.", "criteria": { "A": "Every tournament has a vertex pointing to all others, so choose it first recursively.", "B": "All tournaments are acyclic, so a topological sort always gives a path.", "C": "Follow outgoing edges greedily; a visited vertex can never be encountered again.", "D": "Induct including zero/one-vertex bases. Pick v; split remaining vertices into those pointing to v and those v points to. Each induced tournament has a Hamiltonian path; concatenate the first path, v, then the second." } }, "17.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Orient every edge of a complete n-vertex graph independently with a fair coin. Number of possible tournaments?", "criteria": { "A": "C(n,2)", "B": "2^C(n,2)", "C": "2^(n²)", "D": "n!" } }, "17.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Orient every edge of a complete n-vertex graph independently with a fair coin. Expected Hamiltonian paths in the resulting tournament?", "criteria": { "A": "n!/2^(n−1)", "B": "n!/2^C(n,2)", "C": "2^(n−1)", "D": "n/2" } } } }, "attempts": [ { "status": 200, "full_response_ms": 435.24579100267147, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"17.1\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"A\":0.0,\"C\":0.0,\"B\":0.0}},\"17.2a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"C\":0.0,\"B\":1.0}},\"17.2b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.88,\"probabilities\":{\"D\":0.0,\"A\":0.91,\"C\":0.0,\"B\":0.09}}},\"usage\":{\"input_tokens\":790,\"output_tokens\":140}}" } ], "response": { "model": "jev-1.13.0", "answers": { "17.1": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "A": 0.0, "C": 0.0, "B": 0.0 } }, "17.2a": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.0, "B": 1.0 } }, "17.2b": { "type": "choice", "choice": "A", "confidence": 0.88, "probabilities": { "D": 0.0, "A": 0.91, "C": 0.0, "B": 0.09 } } }, "usage": { "input_tokens": 790, "output_tokens": 140 } } }, { "exam": "final_sp25", "section": 18, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "18.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Random variables have finite second moments; Var(X)>0.\nProblem: The best affine least-squares estimator of Y from X passes through the origin iff E[Y] equals what?", "criteria": { "A": "Cov(X,Y)/E[X]", "B": "0", "C": "Cov(X,Y)*E[X]/Var(X)", "D": "E[X]" } }, "18.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Random variables have finite second moments; Var(X)>0.\nProblem: E[X²] in terms of variance and mean?", "criteria": { "A": "Var(X)−E[X]²", "B": "E[X]²", "C": "Var(X)+E[X]²", "D": "Var(X)²+E[X]" } }, "18.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Random variables have finite second moments; Var(X)>0.\nProblem: If E[X]=E[Y]=t, what t makes Cov(X,Y)=E[XY]?", "criteria": { "A": "0", "B": "1/2", "C": "−1", "D": "1" } }, "18.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Random variables have finite second moments; Var(X)>0.\nProblem: If X,Y are independent, best affine least-squares estimator of Y given X?", "criteria": { "A": "E[XY]", "B": "E[X]", "C": "E[Y]", "D": "X" } } } }, "attempts": [ { "status": 200, "full_response_ms": 337.2207080028602, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"18.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.84,\"probabilities\":{\"B\":0.88,\"D\":0.0,\"A\":0.04,\"C\":0.08}},\"18.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":0.0,\"C\":1.0}},\"18.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":1.0,\"C\":0.0}},\"18.4\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":0.0,\"C\":1.0}}},\"usage\":{\"input_tokens\":838,\"output_tokens\":183}}" } ], "response": { "model": "jev-1.13.0", "answers": { "18.1": { "type": "choice", "choice": "B", "confidence": 0.84, "probabilities": { "B": 0.88, "D": 0.0, "A": 0.04, "C": 0.08 } }, "18.2": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.0, "C": 1.0 } }, "18.3": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 1.0, "C": 0.0 } }, "18.4": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.0, "C": 1.0 } } }, "usage": { "input_tokens": 838, "output_tokens": 183 } } }, { "exam": "final_sp25", "section": 19, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "19.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Independent X,Y~Bin(n,p). Distribution of X+Y?", "criteria": { "A": "Poisson(2np)", "B": "Bin(n,2p)", "C": "Bin(2n,p)", "D": "Bin(2n,p²)" } }, "19.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Independent X,Y~Geom(p), counting trials from 1. Distribution of min(X,Y)?", "criteria": { "A": "Geom(2p−p²)", "B": "Geom(p²)", "C": "Geom(p)", "D": "Geom(2p)" } }, "19.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Fixed lambda>0 and nonnegative integer i. Limit as n->infinity of C(n,i)(lambda/n)^i(1−lambda/n)^(n−i)?", "criteria": { "A": "exp(−lambda)", "B": "exp(−i)*i^lambda/lambda!", "C": "lambda^i/i!", "D": "exp(−lambda)*lambda^i/i!" } }, "19.4a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: First X~Uniform(0,1]; conditional on X=x, draw Y~Uniform[0,x].\nProblem: Marginal density of X?", "criteria": { "A": "−ln(x) on (0,1], zero elsewhere", "B": "1 on (0,1], zero elsewhere", "C": "1/x on (0,1], zero elsewhere", "D": "2x on (0,1], zero elsewhere" } }, "19.4b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: First X~Uniform(0,1]; conditional on X=x, draw Y~Uniform[0,x].\nProblem: Describe the joint support region, replacing the requested shading.", "criteria": { "A": "0<=x,y<=1, the full square", "B": "0=0", "D": "0=a)?", "criteria": { "A": "[F(b)−F(a)]/F(a) for all a,b", "B": "F(b)/(1−F(a)) for all a,b", "C": "0 if b=s)?", "criteria": { "A": "(t−s)/(b−s)", "B": "(t−s)/(t−a)", "C": "(t−s)/(b−a)", "D": "(t−a)/(b−a)" } }, "20.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A nondegenerate bounded uniform distribution is memoryless.", "criteria": { "A": "True", "B": "False" } }, "20.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: X is the minimum of m independent Uniform[0,1] draws. Its density?", "criteria": { "A": "1/m on [0,1], zero elsewhere", "B": "(1−x)^m on [0,1], zero elsewhere", "C": "m*x^(m−1) on [0,1], zero elsewhere", "D": "m(1−x)^(m−1) on [0,1], zero elsewhere" } }, "20.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Expected minimum of m independent Uniform[0,1] draws?", "criteria": { "A": "1/(m+1)", "B": "m/(m+1)", "C": "1/m", "D": "1/2" } } } }, "attempts": [ { "status": 200, "full_response_ms": 393.2557079970138, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"20.1a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":1.0,\"A\":0.0,\"C\":0.0}},\"20.1b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":1.0,\"A\":0.0,\"C\":0.0}},\"20.1c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":0.0,\"C\":1.0}},\"20.2a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":1.0,\"C\":0.0}},\"20.2b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"B\":1.0,\"A\":0.0}},\"20.3a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.98,\"probabilities\":{\"B\":0.0,\"D\":0.99,\"A\":0.0,\"C\":0.01}},\"20.3b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":1.0,\"C\":0.0}}},\"usage\":{\"input_tokens\":1232,\"output_tokens\":311}}" } ], "response": { "model": "jev-1.13.0", "answers": { "20.1a": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 1.0, "A": 0.0, "C": 0.0 } }, "20.1b": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 1.0, "A": 0.0, "C": 0.0 } }, "20.1c": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 } }, "20.2a": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "B": 0.0, "D": 0.0, "A": 1.0, "C": 0.0 } }, "20.2b": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "B": 1.0, "A": 0.0 } }, "20.3a": { "type": "choice", "choice": "D", "confidence": 0.98, "probabilities": { "B": 0.0, "D": 0.99, "A": 0.0, "C": 0.01 } }, "20.3b": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 1.0, "C": 0.0 } } }, "usage": { "input_tokens": 1232, "output_tokens": 311 } } }, { "exam": "final_sp25", "section": 21, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "21.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: States 0..29. From i, move to i+3 or i+5 modulo 30, each with probability 1/2.\nProblem: Is this chain irreducible?", "criteria": { "A": "Yes", "B": "No" } }, "21.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: States 0..29. From i, move to i+3 or i+5 modulo 30, each with probability 1/2.\nProblem: Period of state 0?", "criteria": { "A": "10", "B": "1", "C": "2", "D": "6" } }, "21.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: States 0..29. From i, move to i+3 or i+5 modulo 30, each with probability 1/2.\nProblem: The uniform distribution on these states is invariant.", "criteria": { "A": "True", "B": "False" } }, "21.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Start at Evelyn. Each person chooses uniformly among outgoing neighbors: Evelyn->{Andy,Jay}; Andy->{Malia,Jay}; Jay->{Malia}; Malia->{Gavin}; Gavin->{Evelyn}. One transition per minute.\nProblem: Expected minutes until first reaching Andy?", "criteria": { "A": "4", "B": "8", "C": "5", "D": "2" } }, "21.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Start at Evelyn. Each person chooses uniformly among outgoing neighbors: Evelyn->{Andy,Jay}; Andy->{Malia,Jay}; Jay->{Malia}; Malia->{Gavin}; Gavin->{Evelyn}. One transition per minute.\nProblem: Probability of reaching Jay before Gavin?", "criteria": { "A": "2/3", "B": "1/4", "C": "3/4", "D": "1/2" } } } }, "attempts": [ { "status": 200, "full_response_ms": 344.0762909995101, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"21.1a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.73,\"probabilities\":{\"B\":0.14,\"A\":0.86}},\"21.1b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.36,\"probabilities\":{\"B\":0.52,\"D\":0.06,\"C\":0.19,\"A\":0.23}},\"21.1c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.83,\"probabilities\":{\"B\":0.08,\"A\":0.92}},\"21.2a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.27,\"probabilities\":{\"B\":0.1,\"D\":0.45,\"C\":0.13,\"A\":0.32}},\"21.2b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.36,\"probabilities\":{\"B\":0.04,\"D\":0.12,\"C\":0.32,\"A\":0.52}}},\"usage\":{\"input_tokens\":940,\"output_tokens\":205}}" } ], "response": { "model": "jev-1.13.0", "answers": { "21.1a": { "type": "choice", "choice": "A", "confidence": 0.73, "probabilities": { "B": 0.14, "A": 0.86 } }, "21.1b": { "type": "choice", "choice": "B", "confidence": 0.36, "probabilities": { "B": 0.52, "D": 0.06, "C": 0.19, "A": 0.23 } }, "21.1c": { "type": "choice", "choice": "A", "confidence": 0.83, "probabilities": { "B": 0.08, "A": 0.92 } }, "21.2a": { "type": "choice", "choice": "D", "confidence": 0.27, "probabilities": { "B": 0.1, "D": 0.45, "C": 0.13, "A": 0.32 } }, "21.2b": { "type": "choice", "choice": "A", "confidence": 0.36, "probabilities": { "B": 0.04, "D": 0.12, "C": 0.32, "A": 0.52 } } }, "usage": { "input_tokens": 940, "output_tokens": 205 } } }, { "exam": "final_su25", "section": 3, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "3.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: (NOT P implies P) is equivalent to P.", "criteria": { "A": "False", "B": "True" } }, "3.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: [(P OR Q) AND (P implies R) AND (Q implies R)] is equivalent to R.", "criteria": { "A": "True", "B": "False" } }, "3.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: exists x [P(x) AND NOT Q(x)] is equivalent to forall x [NOT P(x) implies Q(x)].", "criteria": { "A": "False", "B": "True" } }, "3.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: exists x [P(x) AND NOT Q(x)] is equivalent to NOT forall x [P(x) implies Q(x)].", "criteria": { "A": "True", "B": "False" } }, "3.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Simplify (P AND Q) OR (P AND NOT Q).", "criteria": { "A": "Q", "B": "P OR Q", "C": "False", "D": "P" } }, "3.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Simplify P AND (P implies Q).", "criteria": { "A": "P AND Q", "B": "Q", "C": "P OR Q", "D": "P" } } } }, "attempts": [ { "status": 200, "full_response_ms": 397.77895800216356, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"3.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.87,\"probabilities\":{\"A\":0.07,\"B\":0.93}},\"3.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.16,\"probabilities\":{\"A\":0.58,\"B\":0.42}},\"3.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.57,\"probabilities\":{\"A\":0.78,\"B\":0.22}},\"3.4\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"A\":0.99,\"B\":0.01}},\"3.5\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"C\":0.0,\"A\":0.0,\"B\":0.0}},\"3.6\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.97,\"probabilities\":{\"D\":0.02,\"C\":0.0,\"A\":0.97,\"B\":0.01}}},\"usage\":{\"input_tokens\":862,\"output_tokens\":211}}" } ], "response": { "model": "jev-1.13.0", "answers": { "3.1": { "type": "choice", "choice": "B", "confidence": 0.87, "probabilities": { "A": 0.07, "B": 0.93 } }, "3.2": { "type": "choice", "choice": "A", "confidence": 0.16, "probabilities": { "A": 0.58, "B": 0.42 } }, "3.3": { "type": "choice", "choice": "A", "confidence": 0.57, "probabilities": { "A": 0.78, "B": 0.22 } }, "3.4": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "A": 0.99, "B": 0.01 } }, "3.5": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "C": 0.0, "A": 0.0, "B": 0.0 } }, "3.6": { "type": "choice", "choice": "A", "confidence": 0.97, "probabilities": { "D": 0.02, "C": 0.0, "A": 0.97, "B": 0.01 } } }, "usage": { "input_tokens": 862, "output_tokens": 211 } } }, { "exam": "final_su25", "section": 4, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "4.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Which proof technique uses [P(1) AND ... AND P(k)] implies P(k+1)?", "criteria": { "A": "Contraposition", "B": "Strong induction", "C": "Proof by cases", "D": "Diagonalization" } }, "4.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Real x,y have (x+y)/2=a. Prove x<=a or y<=a.", "criteria": { "A": "If both x>a and y>a, their average would exceed a, a contradiction.", "B": "The average is always less than each input.", "C": "Since the average is a, both x and y must equal a.", "D": "Assume x<=a; this assumption proves the desired statement for every x,y." } } } }, "attempts": [ { "status": 200, "full_response_ms": 351.2803330013412, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"4.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"B\":1.0,\"C\":0.0,\"D\":0.0,\"A\":0.0}},\"4.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"C\":0.0,\"D\":0.0,\"A\":1.0}}},\"usage\":{\"input_tokens\":605,\"output_tokens\":91}}" } ], "response": { "model": "jev-1.13.0", "answers": { "4.1": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "B": 1.0, "C": 0.0, "D": 0.0, "A": 0.0 } }, "4.2": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "D": 0.0, "A": 1.0 } } }, "usage": { "input_tokens": 605, "output_tokens": 91 } } }, { "exam": "final_su25", "section": 5, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "5.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Every graph with maximum degree at most two is planar.", "criteria": { "A": "False", "B": "True" } }, "5.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A graph containing a vertex of degree six cannot be planar.", "criteria": { "A": "True", "B": "False" } }, "5.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A simple cycle on n vertices is bipartite iff n is odd.", "criteria": { "A": "False", "B": "True" } }, "5.4a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Minimum vertex colors for a d-dimensional hypercube, d>=1?", "criteria": { "A": "d+1", "B": "2", "C": "d", "D": "2^d" } }, "5.4b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A connected simple planar graph with at least two edges has no triangles. Give and justify an edge bound in terms of n vertices.", "criteria": { "A": "e<=2n−4: every face has at least four edge-sides, so 4f<=2e; combine with n−e+f=2.", "B": "e<=2n: Euler’s formula is n+e=f+2.", "C": "e<=n−1: no triangles means there are no cycles.", "D": "e<=3n−6 and this is tight even without triangles; every face remains triangular." } }, "5.4c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prove the four-dimensional hypercube is nonplanar.", "criteria": { "A": "It has degree four at every vertex, and any such graph is nonplanar.", "B": "It has an odd cycle, which all planar graphs forbid.", "C": "It has 16 vertices, exceeding the maximum number permitted in a planar graph.", "D": "It has 16 vertices and 32 edges and is bipartite; a triangle-free planar graph on 16 vertices has at most 28 edges." } } } }, "attempts": [ { "status": 200, "full_response_ms": 301.9209169979149, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"5.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.93,\"probabilities\":{\"B\":0.97,\"A\":0.03}},\"5.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.97,\"probabilities\":{\"B\":0.98,\"A\":0.02}},\"5.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"B\":0.01,\"A\":0.99}},\"5.4a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"B\":1.0,\"C\":0.0,\"D\":0.0,\"A\":0.0}},\"5.4b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"C\":0.0,\"D\":0.0,\"A\":1.0}},\"5.4c\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"C\":0.0,\"D\":1.0,\"A\":0.0}}},\"usage\":{\"input_tokens\":1031,\"output_tokens\":228}}" } ], "response": { "model": "jev-1.13.0", "answers": { "5.1": { "type": "choice", "choice": "B", "confidence": 0.93, "probabilities": { "B": 0.97, "A": 0.03 } }, "5.2": { "type": "choice", "choice": "B", "confidence": 0.97, "probabilities": { "B": 0.98, "A": 0.02 } }, "5.3": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "B": 0.01, "A": 0.99 } }, "5.4a": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "B": 1.0, "C": 0.0, "D": 0.0, "A": 0.0 } }, "5.4b": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "D": 0.0, "A": 1.0 } }, "5.4c": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "D": 1.0, "A": 0.0 } } }, "usage": { "input_tokens": 1031, "output_tokens": 228 } } }, { "exam": "final_su25", "section": 6, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "6.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If all candidates rank the same job last, job-proposing deferred acceptance cannot finish in one day.", "criteria": { "A": "True", "B": "False" } }, "6.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If all jobs have the same strict ranking of candidates, the stable matching is unique.", "criteria": { "A": "True", "B": "False" } }, "6.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: It is impossible for every candidate to get its favorite job in a job-optimal stable matching.", "criteria": { "A": "False", "B": "True" } }, "6.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A job and candidate who rank each other first are paired in every stable matching.", "criteria": { "A": "False", "B": "True" } } } }, "attempts": [ { "status": 200, "full_response_ms": 298.6532079994504, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"6.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.72,\"probabilities\":{\"A\":0.14,\"B\":0.86}},\"6.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.85,\"probabilities\":{\"A\":0.93,\"B\":0.07}},\"6.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.3,\"probabilities\":{\"A\":0.65,\"B\":0.35}},\"6.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.88,\"probabilities\":{\"A\":0.06,\"B\":0.94}}},\"usage\":{\"input_tokens\":649,\"output_tokens\":123}}" } ], "response": { "model": "jev-1.13.0", "answers": { "6.1": { "type": "choice", "choice": "B", "confidence": 0.72, "probabilities": { "A": 0.14, "B": 0.86 } }, "6.2": { "type": "choice", "choice": "A", "confidence": 0.85, "probabilities": { "A": 0.93, "B": 0.07 } }, "6.3": { "type": "choice", "choice": "A", "confidence": 0.3, "probabilities": { "A": 0.65, "B": 0.35 } }, "6.4": { "type": "choice", "choice": "B", "confidence": 0.88, "probabilities": { "A": 0.06, "B": 0.94 } } }, "usage": { "input_tokens": 649, "output_tokens": 123 } } }, { "exam": "final_su25", "section": 7, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "7.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: There exists integer x with x=1 mod 10 and x=2 mod 15.", "criteria": { "A": "True", "B": "False" } }, "7.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Why are x=1 mod 10 and x=2 mod 15 inconsistent?", "criteria": { "A": "Their product is even, which prevents integer solutions.", "B": "Neither 10 nor 15 is prime, so no system using them has a solution.", "C": "The residues 1 and 2 must be identical as integers in any CRT system.", "D": "Reducing both modulo 5 requires x=1 and x=2 mod 5 simultaneously." } }, "7.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: There exists integer x with x=6 mod 10 and x=1 mod 15.", "criteria": { "A": "False", "B": "True" } }, "7.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Give a witness satisfying x=6 mod 10 and x=1 mod 15.", "criteria": { "A": "31", "B": "16", "C": "46+1", "D": "6" } }, "7.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prove x²=(m−x)² modulo positive m.", "criteria": { "A": "The equality holds as integers for all x.", "B": "Both squares always have residue one.", "C": "Their difference is m²−2mx=m(m−2x), a multiple of m.", "D": "Squaring changes every residue to zero." } }, "7.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For positive even n, prove sum(k=1..n) (−1)^k k² is divisible by n+1.", "criteria": { "A": "There are an even number of terms, so every such sum is divisible by n+1.", "B": "Pair adjacent terms, whose differences are all exactly n+1.", "C": "Every individual squared term is divisible by n+1.", "D": "Pair k with n+1−k. Their squares agree modulo n+1 and their signs are opposite since n is even, so every pair cancels." } } } }, "attempts": [ { "status": 200, "full_response_ms": 391.2162079977861, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"7.1a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.22,\"probabilities\":{\"A\":0.61,\"B\":0.39}},\"7.1b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"A\":0.0,\"B\":0.0,\"D\":1.0}},\"7.2a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.19,\"probabilities\":{\"A\":0.6,\"B\":0.4}},\"7.2b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.54,\"probabilities\":{\"D\":0.01,\"A\":0.66,\"B\":0.29,\"C\":0.04}},\"7.3a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"C\":1.0,\"A\":0.0,\"B\":0.0,\"D\":0.0}},\"7.3b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.95,\"probabilities\":{\"C\":0.0,\"A\":0.01,\"D\":0.96,\"B\":0.03}}},\"usage\":{\"input_tokens\":1095,\"output_tokens\":245}}" } ], "response": { "model": "jev-1.13.0", "answers": { "7.1a": { "type": "choice", "choice": "A", "confidence": 0.22, "probabilities": { "A": 0.61, "B": 0.39 } }, "7.1b": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "C": 0.0, "A": 0.0, "B": 0.0, "D": 1.0 } }, "7.2a": { "type": "choice", "choice": "A", "confidence": 0.19, "probabilities": { "A": 0.6, "B": 0.4 } }, "7.2b": { "type": "choice", "choice": "A", "confidence": 0.54, "probabilities": { "D": 0.01, "A": 0.66, "B": 0.29, "C": 0.04 } }, "7.3a": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "C": 1.0, "A": 0.0, "B": 0.0, "D": 0.0 } }, "7.3b": { "type": "choice", "choice": "D", "confidence": 0.95, "probabilities": { "C": 0.0, "A": 0.01, "D": 0.96, "B": 0.03 } } }, "usage": { "input_tokens": 1095, "output_tokens": 245 } } }, { "exam": "final_su25", "section": 8, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "8.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Over GF(5), how many formal polynomials of degree at most 3 pass through one specified point?", "criteria": { "A": "25", "B": "625", "C": "125", "D": "100" } }, "8.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A Shamir polynomial of degree at most two over GF(5) gives shares (1,3),(2,4),(4,2). Secret is P(0).\nProblem: Find the secret.", "criteria": { "A": "1", "B": "4", "C": "2", "D": "3" } }, "8.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A Shamir polynomial of degree at most two over GF(5) gives shares (1,3),(2,4),(4,2). Secret is P(0).\nProblem: What value belongs in a share at x=3?", "criteria": { "A": "1", "B": "2", "C": "4", "D": "3" } }, "8.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Eight Reed–Solomon evaluations are transmitted over GF(11), correcting up to two corrupted values. Maximum message length in field symbols?", "criteria": { "A": "6", "B": "2", "C": "4", "D": "8" } }, "8.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Minimal error locator is E(x)=x²−6x+8 over GF(11). Which evaluation coordinates are corrupted?", "criteria": { "A": "6 and 8", "B": "1 and 8", "C": "2 and 4", "D": "3 and 5" } } } }, "attempts": [ { "status": 200, "full_response_ms": 340.02358299767366, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"8.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.78,\"probabilities\":{\"D\":0.03,\"A\":0.03,\"B\":0.1,\"C\":0.84}},\"8.2a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.13,\"probabilities\":{\"D\":0.27,\"A\":0.33999999999999997,\"B\":0.24,\"C\":0.15}},\"8.2b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.26,\"probabilities\":{\"D\":0.27,\"A\":0.44,\"B\":0.12,\"C\":0.17}},\"8.3a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.62,\"probabilities\":{\"D\":0.0,\"A\":0.27,\"B\":0.01,\"C\":0.72}},\"8.3b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.96,\"probabilities\":{\"D\":0.01,\"A\":0.01,\"B\":0.01,\"C\":0.97}}},\"usage\":{\"input_tokens\":937,\"output_tokens\":227}}" } ], "response": { "model": "jev-1.13.0", "answers": { "8.1": { "type": "choice", "choice": "C", "confidence": 0.78, "probabilities": { "D": 0.03, "A": 0.03, "B": 0.1, "C": 0.84 } }, "8.2a": { "type": "choice", "choice": "A", "confidence": 0.13, "probabilities": { "D": 0.27, "A": 0.33999999999999997, "B": 0.24, "C": 0.15 } }, "8.2b": { "type": "choice", "choice": "A", "confidence": 0.26, "probabilities": { "D": 0.27, "A": 0.44, "B": 0.12, "C": 0.17 } }, "8.3a": { "type": "choice", "choice": "C", "confidence": 0.62, "probabilities": { "D": 0.0, "A": 0.27, "B": 0.01, "C": 0.72 } }, "8.3b": { "type": "choice", "choice": "C", "confidence": 0.96, "probabilities": { "D": 0.01, "A": 0.01, "B": 0.01, "C": 0.97 } } }, "usage": { "input_tokens": 937, "output_tokens": 227 } } }, { "exam": "final_su25", "section": 9, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "9.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Ten rounds give +30 for a win, −30 for a loss, starting at zero. Number of ordered outcomes ending at +60?", "criteria": { "A": "C(10,4)", "B": "2^10", "C": "C(10,2)", "D": "C(10,5)" } }, "9.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Ten-digit IDs allow 0..9 in every position, with no equal adjacent digits. Count?", "criteria": { "A": "9*10^9", "B": "10*9^9", "C": "10^10", "D": "10!" } }, "9.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distribute eight integer points into three named stats; first nonnegative, others strictly positive. Count?", "criteria": { "A": "3^8", "B": "C(8,2)", "C": "C(7,2)", "D": "C(10,2)" } }, "9.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Choose distinct captain and lieutenant from eleven people. Count?", "criteria": { "A": "121", "B": "110", "C": "55", "D": "22" } }, "9.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Seven distinct collectible items; choose a subset with at least two. Count?", "criteria": { "A": "120", "B": "21", "C": "127", "D": "128" } }, "9.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Two islands each have n named checkpoints, n>=2. Travel only between islands. Starting at one fixed checkpoint, count directed tours visiting every checkpoint exactly once before returning.", "criteria": { "A": "n!(n−1)!/2", "B": "n!(n−1)!", "C": "(2n−1)!", "D": "(n!)²" } }, "9.7": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of functions from an n-element set to itself that are not bijections?", "criteria": { "A": "n^n−n!", "B": "n^n−n", "C": "n!−n", "D": "2^n−n!" } } } }, "attempts": [ { "status": 200, "full_response_ms": 287.7190840008552, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"9.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.32,\"probabilities\":{\"C\":0.32,\"B\":0.01,\"A\":0.49,\"D\":0.18}},\"9.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"C\":0.0,\"B\":0.99,\"A\":0.01,\"D\":0.0}},\"9.3\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.54,\"probabilities\":{\"C\":0.18,\"B\":0.05,\"A\":0.11,\"D\":0.66}},\"9.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.97,\"probabilities\":{\"C\":0.0,\"B\":0.98,\"A\":0.02,\"D\":0.0}},\"9.5\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"C\":1.0,\"B\":0.0,\"A\":0.0,\"D\":0.0}},\"9.6\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.57,\"probabilities\":{\"C\":0.02,\"B\":0.68,\"A\":0.17,\"D\":0.13}},\"9.7\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"C\":0.0,\"A\":1.0,\"B\":0.0}}},\"usage\":{\"input_tokens\":1186,\"output_tokens\":311}}" } ], "response": { "model": "jev-1.13.0", "answers": { "9.1": { "type": "choice", "choice": "A", "confidence": 0.32, "probabilities": { "C": 0.32, "B": 0.01, "A": 0.49, "D": 0.18 } }, "9.2": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "C": 0.0, "B": 0.99, "A": 0.01, "D": 0.0 } }, "9.3": { "type": "choice", "choice": "D", "confidence": 0.54, "probabilities": { "C": 0.18, "B": 0.05, "A": 0.11, "D": 0.66 } }, "9.4": { "type": "choice", "choice": "B", "confidence": 0.97, "probabilities": { "C": 0.0, "B": 0.98, "A": 0.02, "D": 0.0 } }, "9.5": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "C": 1.0, "B": 0.0, "A": 0.0, "D": 0.0 } }, "9.6": { "type": "choice", "choice": "B", "confidence": 0.57, "probabilities": { "C": 0.02, "B": 0.68, "A": 0.17, "D": 0.13 } }, "9.7": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "C": 0.0, "A": 1.0, "B": 0.0 } } }, "usage": { "input_tokens": 1186, "output_tokens": 311 } } }, { "exam": "final_su25", "section": 10, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "10.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Functions N->{True,False} form a countable set.", "criteria": { "A": "True", "B": "False" } }, "10.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Finite binary strings form a countable set.", "criteria": { "A": "False", "B": "True" } }, "10.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: It is impossible to write a halting analyzer that correctly handles even some programs.", "criteria": { "A": "False", "B": "True" } }, "10.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: No always-terminating analyzer can correctly decide for all programs whether they ever print a particular fixed string.", "criteria": { "A": "False", "B": "True" } }, "10.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: It is impossible to decide for arbitrary program P and finite N whether P hits a runtime error within its first N execution steps.", "criteria": { "A": "True", "B": "False" } }, "10.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Why can no total analyzer decide whether an arbitrary Python program ever assigns a string to a given variable? Assume unbounded theoretical computation.", "criteria": { "A": "A variable has no value before assignment, so no analyzer can read any program.", "B": "Finite programs contain uncountably many variable names.", "C": "Python never exposes variable types, even during execution.", "D": "Given P,input x, construct a program with a fresh variable initially 0, simulate P(x), then assign a string to that variable if simulation halts. The analyzer would decide halting." } } } }, "attempts": [ { "status": 200, "full_response_ms": 396.4985419988807, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"10.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.92,\"probabilities\":{\"B\":0.96,\"A\":0.04}},\"10.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.98,\"probabilities\":{\"B\":0.99,\"A\":0.01}},\"10.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.63,\"probabilities\":{\"B\":0.19,\"A\":0.81}},\"10.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.95,\"probabilities\":{\"B\":0.97,\"A\":0.03}},\"10.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.67,\"probabilities\":{\"B\":0.83,\"A\":0.17}},\"10.6\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":0.0,\"A\":0.0,\"D\":1.0}}},\"usage\":{\"input_tokens\":910,\"output_tokens\":203}}" } ], "response": { "model": "jev-1.13.0", "answers": { "10.1": { "type": "choice", "choice": "B", "confidence": 0.92, "probabilities": { "B": 0.96, "A": 0.04 } }, "10.2": { "type": "choice", "choice": "B", "confidence": 0.98, "probabilities": { "B": 0.99, "A": 0.01 } }, "10.3": { "type": "choice", "choice": "A", "confidence": 0.63, "probabilities": { "B": 0.19, "A": 0.81 } }, "10.4": { "type": "choice", "choice": "B", "confidence": 0.95, "probabilities": { "B": 0.97, "A": 0.03 } }, "10.5": { "type": "choice", "choice": "B", "confidence": 0.67, "probabilities": { "B": 0.83, "A": 0.17 } }, "10.6": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 0.0, "D": 1.0 } } }, "usage": { "input_tokens": 910, "output_tokens": 203 } } }, { "exam": "final_su25", "section": 11, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "11.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A five-sided die has P(1)=1/2; values 2,3,4,5 share the remaining probability equally. A is an odd result; B is a perfect-square result.\nProblem: P(A)?", "criteria": { "A": "1/2", "B": "3/5", "C": "5/8", "D": "3/4" } }, "11.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A five-sided die has P(1)=1/2; values 2,3,4,5 share the remaining probability equally. A is an odd result; B is a perfect-square result.\nProblem: P(NOT B)?", "criteria": { "A": "2/5", "B": "1/2", "C": "5/8", "D": "3/8" } }, "11.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A five-sided die has P(1)=1/2; values 2,3,4,5 share the remaining probability equally. A is an odd result; B is a perfect-square result.\nProblem: P(A union B)?", "criteria": { "A": "3/4", "B": "1", "C": "5/8", "D": "7/8" } }, "11.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A five-sided die has P(1)=1/2; values 2,3,4,5 share the remaining probability equally. A is an odd result; B is a perfect-square result.\nProblem: P(A given B)?", "criteria": { "A": "4/5", "B": "3/4", "C": "1/2", "D": "5/8" } }, "11.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A five-sided die has P(1)=1/2; values 2,3,4,5 share the remaining probability equally. A is an odd result; B is a perfect-square result.\nProblem: P(A intersection B given A union B)?", "criteria": { "A": "4/7", "B": "4/5", "C": "1/2", "D": "7/8" } }, "11.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A five-sided die has P(1)=1/2; values 2,3,4,5 share the remaining probability equally. A is an odd result; B is a perfect-square result.\nProblem: P(A intersection NOT B given B)?", "criteria": { "A": "0", "B": "3/8", "C": "1", "D": "4/5" } }, "11.7": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A five-sided die has P(1)=1/2; values 2,3,4,5 share the remaining probability equally. A is an odd result; B is a perfect-square result.\nProblem: Now vary P(1)=p<1 and keep the other four outcomes equally likely. Which p makes A and B independent?", "criteria": { "A": "1/5", "B": "1/2", "C": "1/3", "D": "0" } } } }, "attempts": [ { "status": 200, "full_response_ms": 352.26962500019, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"11.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.74,\"probabilities\":{\"A\":0.02,\"C\":0.8,\"B\":0.06,\"D\":0.12}},\"11.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.76,\"probabilities\":{\"A\":0.04,\"C\":0.82,\"B\":0.05,\"D\":0.09}},\"11.3\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.31,\"probabilities\":{\"A\":0.3,\"C\":0.1,\"B\":0.12,\"D\":0.48}},\"11.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.12,\"probabilities\":{\"A\":0.29,\"C\":0.28,\"B\":0.33,\"D\":0.1}},\"11.5\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.51,\"probabilities\":{\"A\":0.63,\"C\":0.13,\"B\":0.21,\"D\":0.03}},\"11.6\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.92,\"probabilities\":{\"A\":0.95,\"C\":0.01,\"B\":0.02,\"D\":0.02}},\"11.7\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.1,\"probabilities\":{\"A\":0.31,\"C\":0.3,\"B\":0.33,\"D\":0.06}}},\"usage\":{\"input_tokens\":1319,\"output_tokens\":318}}" } ], "response": { "model": "jev-1.13.0", "answers": { "11.1": { "type": "choice", "choice": "C", "confidence": 0.74, "probabilities": { "A": 0.02, "C": 0.8, "B": 0.06, "D": 0.12 } }, "11.2": { "type": "choice", "choice": "C", "confidence": 0.76, "probabilities": { "A": 0.04, "C": 0.82, "B": 0.05, "D": 0.09 } }, "11.3": { "type": "choice", "choice": "D", "confidence": 0.31, "probabilities": { "A": 0.3, "C": 0.1, "B": 0.12, "D": 0.48 } }, "11.4": { "type": "choice", "choice": "B", "confidence": 0.12, "probabilities": { "A": 0.29, "C": 0.28, "B": 0.33, "D": 0.1 } }, "11.5": { "type": "choice", "choice": "A", "confidence": 0.51, "probabilities": { "A": 0.63, "C": 0.13, "B": 0.21, "D": 0.03 } }, "11.6": { "type": "choice", "choice": "A", "confidence": 0.92, "probabilities": { "A": 0.95, "C": 0.01, "B": 0.02, "D": 0.02 } }, "11.7": { "type": "choice", "choice": "B", "confidence": 0.1, "probabilities": { "A": 0.31, "C": 0.3, "B": 0.33, "D": 0.06 } } }, "usage": { "input_tokens": 1319, "output_tokens": 318 } } }, { "exam": "final_su25", "section": 12, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "12.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A detector flags plagiarized submissions with probability p1 and original submissions with probability p2. Class CS1 has 10% plagiarized submissions; CS100 has 2%.\nProblem: Must p1+p2 equal one?", "criteria": { "A": "No, it cannot.", "B": "Yes, always.", "C": "Depends; it can but need not." } }, "12.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A detector flags plagiarized submissions with probability p1 and original submissions with probability p2. Class CS1 has 10% plagiarized submissions; CS100 has 2%.\nProblem: p1=0.8,p2=0.2. Given a CS1 submission was flagged, probability it was plagiarized?", "criteria": { "A": "4/5", "B": "4/13", "C": "1/4", "D": "1/10" } }, "12.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A detector flags plagiarized submissions with probability p1 and original submissions with probability p2. Class CS1 has 10% plagiarized submissions; CS100 has 2%.\nProblem: p1=0.8,p2=0.2. Given a CS100 submission was flagged, probability it was plagiarized?", "criteria": { "A": "4/13", "B": "4/5", "C": "1/50", "D": "4/53" } }, "12.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A detector flags plagiarized submissions with probability p1 and original submissions with probability p2. Class CS1 has 10% plagiarized submissions; CS100 has 2%.\nProblem: Choose either class with equal probability, then a uniform submission within it. An auditor establishes plagiarism without the detector. Given plagiarism, probability the class was CS1?", "criteria": { "A": "1/10", "B": "1/2", "C": "5/6", "D": "4/13" } } } }, "attempts": [ { "status": 200, "full_response_ms": 298.2502080012637, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"12.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.38,\"probabilities\":{\"B\":0.01,\"A\":0.4,\"C\":0.59}},\"12.2a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.7,\"probabilities\":{\"B\":0.78,\"D\":0.0,\"C\":0.01,\"A\":0.21}},\"12.2b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.47,\"probabilities\":{\"B\":0.04,\"D\":0.34,\"C\":0.01,\"A\":0.61}},\"12.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.59,\"probabilities\":{\"B\":0.11,\"D\":0.18,\"C\":0.69,\"A\":0.02}}},\"usage\":{\"input_tokens\":960,\"output_tokens\":178}}" } ], "response": { "model": "jev-1.13.0", "answers": { "12.1": { "type": "choice", "choice": "C", "confidence": 0.38, "probabilities": { "B": 0.01, "A": 0.4, "C": 0.59 } }, "12.2a": { "type": "choice", "choice": "B", "confidence": 0.7, "probabilities": { "B": 0.78, "D": 0.0, "C": 0.01, "A": 0.21 } }, "12.2b": { "type": "choice", "choice": "A", "confidence": 0.47, "probabilities": { "B": 0.04, "D": 0.34, "C": 0.01, "A": 0.61 } }, "12.3": { "type": "choice", "choice": "C", "confidence": 0.59, "probabilities": { "B": 0.11, "D": 0.18, "C": 0.69, "A": 0.02 } } }, "usage": { "input_tokens": 960, "output_tokens": 178 } } }, { "exam": "final_su25", "section": 13, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "13.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.\nProblem: Probability the two California senators sit adjacent?", "criteria": { "A": "2/(2n)", "B": "2/(2n−1)", "C": "1/(2n−1)", "D": "1/n" } }, "13.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.\nProblem: Probability the California pair is adjacent given the New Mexico pair is adjacent?", "criteria": { "A": "2/(2n−1)", "B": "2/(2n−2)", "C": "1/(2n−2)", "D": "4/((2n−1)(2n−2))" } }, "13.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.\nProblem: One California senator is happy if adjacent to either of two specified other senators. Probability of happiness?", "criteria": { "A": "4/(2n−1)−2/((2n−1)(2n−2))", "B": "2/(2n−1)", "C": "4/(2n−1)−4/((2n−1)(2n−2))", "D": "4/(2n−1)" } }, "13.4a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.\nProblem: Ni means senator i is adjacent to their state partner. P(Ni)?", "criteria": { "A": "1/(2n−1)", "B": "2/(2n−2)", "C": "1/n", "D": "2/(2n−1)" } }, "13.4b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.\nProblem: Are the events Ni pairwise independent over all senators?", "criteria": { "A": "Yes", "B": "No" } }, "13.4c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.\nProblem: Why are the Ni not pairwise independent?", "criteria": { "A": "Every Ni has probability one.", "B": "For the two senators of one state, their events are identical with probability strictly between zero and one; conditioning on one makes the other certain.", "C": "Events on a finite sample space can never be independent.", "D": "All adjacency events are mutually exclusive." } }, "13.4d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.\nProblem: Expected number of senators adjacent to their state partner?", "criteria": { "A": "4n/(2n−2)", "B": "4n/(2n−1)", "C": "2", "D": "2n/(2n−1)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 396.4044999993348, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"13.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.46,\"probabilities\":{\"C\":0.38,\"B\":0.6,\"A\":0.01,\"D\":0.01}},\"13.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.25,\"probabilities\":{\"C\":0.17,\"A\":0.43,\"B\":0.33,\"D\":0.07}},\"13.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.4,\"probabilities\":{\"C\":0.55,\"B\":0.01,\"A\":0.36,\"D\":0.08}},\"13.4a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.7,\"probabilities\":{\"C\":0.01,\"A\":0.21,\"B\":0.01,\"D\":0.77}},\"13.4b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.63,\"probabilities\":{\"A\":0.18,\"B\":0.82}},\"13.4c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"C\":0.0,\"A\":0.0,\"B\":0.99,\"D\":0.01}},\"13.4d\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.37,\"probabilities\":{\"C\":0.05,\"A\":0.03,\"B\":0.39,\"D\":0.53}}},\"usage\":{\"input_tokens\":1361,\"output_tokens\":308}}" } ], "response": { "model": "jev-1.13.0", "answers": { "13.1": { "type": "choice", "choice": "B", "confidence": 0.46, "probabilities": { "C": 0.38, "B": 0.6, "A": 0.01, "D": 0.01 } }, "13.2": { "type": "choice", "choice": "A", "confidence": 0.25, "probabilities": { "C": 0.17, "A": 0.43, "B": 0.33, "D": 0.07 } }, "13.3": { "type": "choice", "choice": "C", "confidence": 0.4, "probabilities": { "C": 0.55, "B": 0.01, "A": 0.36, "D": 0.08 } }, "13.4a": { "type": "choice", "choice": "D", "confidence": 0.7, "probabilities": { "C": 0.01, "A": 0.21, "B": 0.01, "D": 0.77 } }, "13.4b": { "type": "choice", "choice": "B", "confidence": 0.63, "probabilities": { "A": 0.18, "B": 0.82 } }, "13.4c": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "C": 0.0, "A": 0.0, "B": 0.99, "D": 0.01 } }, "13.4d": { "type": "choice", "choice": "D", "confidence": 0.37, "probabilities": { "C": 0.05, "A": 0.03, "B": 0.39, "D": 0.53 } } }, "usage": { "input_tokens": 1361, "output_tokens": 308 } } }, { "exam": "final_su25", "section": 14, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "14.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Pairwise independent Xi have finite second moments. An attempted induction sets Y=X1+...+X(n−1), then applies a lemma requiring independent Y and Xn to add their variances. What is wrong?", "criteria": { "A": "Pairwise independence of the Xi does not ensure independence of Y and Xn.", "B": "Variance never distributes over independent sums.", "C": "The sum Y is not a random variable.", "D": "Pairwise independence forbids defining a sum." } }, "14.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prove variance of a sum of pairwise independent finite-variance variables equals the sum of variances.", "criteria": { "A": "Pairwise independence implies mutual independence, so group arbitrary sums and use independence recursively.", "B": "Expand Var(sum Xi)=sum Var(Xi)+2sum(i=0. Give (E[X],E[Y],Var(Y),Cov(X,Y)).", "criteria": { "A": "(9/7,9/7,52/49,−52/49)", "B": "(0,9/7,19/7,0)", "C": "(0,3/2,5/4,0)", "D": "(0,9/7,52/49,0)" } }, "15.1e": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Choose uniformly from S={(x,y) in Z²: |x|+|y|=3}, the twelve integer points on the perimeter of the diamond with vertices (0,3),(3,0),(0,−3),(−3,0).\nProblem: Instead choose uniformly from the four points in S with x,y>=0. Give (E[X],E[Y],Var(Y),Cov(X,Y)).", "criteria": { "A": "(1,1,2,−2)", "B": "(3/2,3/2,5/4,−5/4)", "C": "(3/2,3/2,5/4,0)", "D": "(3/2,3/2,5/4,5/4)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 328.6960830009775, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"15.1a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.45,\"probabilities\":{\"D\":0.0,\"C\":0.03,\"B\":0.59,\"A\":0.38}},\"15.1b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"B\":1.0,\"A\":0.0}},\"15.1c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"C\":0.0,\"B\":1.0,\"A\":0.0}},\"15.1d\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.79,\"probabilities\":{\"D\":0.85,\"C\":0.01,\"B\":0.12,\"A\":0.02}},\"15.1e\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.96,\"probabilities\":{\"D\":0.01,\"C\":0.01,\"B\":0.97,\"A\":0.01}}},\"usage\":{\"input_tokens\":1310,\"output_tokens\":219}}" } ], "response": { "model": "jev-1.13.0", "answers": { "15.1a": { "type": "choice", "choice": "B", "confidence": 0.45, "probabilities": { "D": 0.0, "C": 0.03, "B": 0.59, "A": 0.38 } }, "15.1b": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "B": 1.0, "A": 0.0 } }, "15.1c": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "C": 0.0, "B": 1.0, "A": 0.0 } }, "15.1d": { "type": "choice", "choice": "D", "confidence": 0.79, "probabilities": { "D": 0.85, "C": 0.01, "B": 0.12, "A": 0.02 } }, "15.1e": { "type": "choice", "choice": "B", "confidence": 0.96, "probabilities": { "D": 0.01, "C": 0.01, "B": 0.97, "A": 0.01 } } }, "usage": { "input_tokens": 1310, "output_tokens": 219 } } }, { "exam": "final_su25", "section": 16, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "16.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X~Bin(n,p),Y~Bin(m,p).\nProblem: P(X+Y=k)?", "criteria": { "A": "C(n+m,k)p^k(1−p)^(n+m−k)", "B": "C(n,k)C(m,k)p^(2k)(1−p)^(n+m−2k)", "C": "C(n+m,k)p^(n+m−k)(1−p)^k", "D": "p^k(1−p)^(n+m−k)" } }, "16.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X~Bin(n,p),Y~Bin(m,p).\nProblem: Distribution of X+Y?", "criteria": { "A": "Bin(n+m,p)", "B": "Poisson((n+m)p)", "C": "Bin(nm,p)", "D": "Bin(n+m,2p)" } }, "16.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent X~Bin(n,p),Y~Bin(m,p).\nProblem: Explain the distribution of X+Y without a probability formula.", "criteria": { "A": "The sum of any two distributions in the same family stays in that family.", "B": "Combine n and m independent Bernoulli trials with the same success probability p; their total counts successes in n+m such trials.", "C": "Independent random variables always sum to a Poisson variable.", "D": "Adding the trial counts also doubles the success probability." } }, "16.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: X counts heads in 100 independent fair flips. Chebyshev bound for P(X<=10 or X>=90)?", "criteria": { "A": "1/16", "B": "1/8", "C": "1/64", "D": "1/40" } }, "16.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For X counting heads in 100 independent fair flips, P(X<=10)=P(X>=90).", "criteria": { "A": "True", "B": "False" } }, "16.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: X counts heads in 100 independent fair flips. Use a normal approximation without continuity correction for P(X<=10 or X>=90), with Z~N(0,1).", "criteria": { "A": "2P(Z>=4)", "B": "2P(Z>=8)", "C": "2P(Z>=40)", "D": "P(Z>=8)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 379.76145799984806, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"16.1a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"A\":1.0,\"B\":0.0,\"C\":0.0,\"D\":0.0}},\"16.1b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"A\":1.0,\"B\":0.0,\"C\":0.0,\"D\":0.0}},\"16.1c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"A\":0.0,\"B\":1.0,\"C\":0.0,\"D\":0.0}},\"16.2a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.29,\"probabilities\":{\"A\":0.47,\"B\":0.13,\"C\":0.32,\"D\":0.08}},\"16.2b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.94,\"probabilities\":{\"A\":0.97,\"B\":0.03}},\"16.2c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.83,\"probabilities\":{\"A\":0.11,\"B\":0.87,\"C\":0.01,\"D\":0.01}}},\"usage\":{\"input_tokens\":1151,\"output_tokens\":265}}" } ], "response": { "model": "jev-1.13.0", "answers": { "16.1a": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "A": 1.0, "B": 0.0, "C": 0.0, "D": 0.0 } }, "16.1b": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "A": 1.0, "B": 0.0, "C": 0.0, "D": 0.0 } }, "16.1c": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "A": 0.0, "B": 1.0, "C": 0.0, "D": 0.0 } }, "16.2a": { "type": "choice", "choice": "A", "confidence": 0.29, "probabilities": { "A": 0.47, "B": 0.13, "C": 0.32, "D": 0.08 } }, "16.2b": { "type": "choice", "choice": "A", "confidence": 0.94, "probabilities": { "A": 0.97, "B": 0.03 } }, "16.2c": { "type": "choice", "choice": "B", "confidence": 0.83, "probabilities": { "A": 0.11, "B": 0.87, "C": 0.01, "D": 0.01 } } }, "usage": { "input_tokens": 1151, "output_tokens": 265 } } }, { "exam": "final_su25", "section": 17, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "17.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Pay $8 to roll two independent fair six-sided dice. Receive their final sum. A reroll, if allowed, is independent and replaces that die; each die can be rerolled at most once. Net winnings subtract entry and reroll costs.\nProblem: Expected net winnings with no rerolls?", "criteria": { "A": "7", "B": "−1", "C": "−2", "D": "1" } }, "17.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Pay $8 to roll two independent fair six-sided dice. Receive their final sum. A reroll, if allowed, is independent and replaces that die; each die can be rerolled at most once. Net winnings subtract entry and reroll costs.\nProblem: Reroll each initial 1 for free. Expected net winnings?", "criteria": { "A": "−1/6", "B": "−1", "C": "1/6", "D": "5/6" } }, "17.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Pay $8 to roll two independent fair six-sided dice. Receive their final sum. A reroll, if allowed, is independent and replaces that die; each die can be rerolled at most once. Net winnings subtract entry and reroll costs.\nProblem: Choose integer threshold c in advance and reroll each die <=c for free. Give optimal c and expected net winnings.", "criteria": { "A": "(2,1/3)", "B": "(4,2/3)", "C": "(3,3/2)", "D": "(3,1/2)" } }, "17.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Pay $8 to roll two independent fair six-sided dice. Receive their final sum. A reroll, if allowed, is independent and replaces that die; each die can be rerolled at most once. Net winnings subtract entry and reroll costs.\nProblem: Each reroll costs $1. Give optimal integer threshold c and expected net winnings.", "criteria": { "A": "(1,−1/6)", "B": "(3,−1/2)", "C": "(2,−1/3)", "D": "(2,1/3)" } }, "17.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Pay $8 to roll two independent fair six-sided dice. Receive their final sum. A reroll, if allowed, is independent and replaces that die; each die can be rerolled at most once. Net winnings subtract entry and reroll costs.\nProblem: Smallest integer reroll fee d such that any reroll strictly decreases conditional expected net winnings?", "criteria": { "A": "4", "B": "2", "C": "3", "D": "6" } } } }, "attempts": [ { "status": 200, "full_response_ms": 332.8514999993786, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"17.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.64,\"probabilities\":{\"D\":0.12,\"A\":0.02,\"C\":0.13,\"B\":0.73}},\"17.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.19,\"probabilities\":{\"D\":0.29,\"A\":0.21,\"C\":0.4,\"B\":0.1}},\"17.3\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.12,\"probabilities\":{\"D\":0.34,\"C\":0.33,\"A\":0.1,\"B\":0.23}},\"17.4\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.34,\"probabilities\":{\"D\":0.17,\"C\":0.51,\"A\":0.18,\"B\":0.14}},\"17.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.18,\"probabilities\":{\"D\":0.08,\"A\":0.24,\"C\":0.29,\"B\":0.39}}},\"usage\":{\"input_tokens\":1152,\"output_tokens\":228}}" } ], "response": { "model": "jev-1.13.0", "answers": { "17.1": { "type": "choice", "choice": "B", "confidence": 0.64, "probabilities": { "D": 0.12, "A": 0.02, "C": 0.13, "B": 0.73 } }, "17.2": { "type": "choice", "choice": "C", "confidence": 0.19, "probabilities": { "D": 0.29, "A": 0.21, "C": 0.4, "B": 0.1 } }, "17.3": { "type": "choice", "choice": "D", "confidence": 0.12, "probabilities": { "D": 0.34, "C": 0.33, "A": 0.1, "B": 0.23 } }, "17.4": { "type": "choice", "choice": "C", "confidence": 0.34, "probabilities": { "D": 0.17, "C": 0.51, "A": 0.18, "B": 0.14 } }, "17.5": { "type": "choice", "choice": "B", "confidence": 0.18, "probabilities": { "D": 0.08, "A": 0.24, "C": 0.29, "B": 0.39 } } }, "usage": { "input_tokens": 1152, "output_tokens": 228 } } }, { "exam": "final_su25", "section": 18, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "18.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent T1~Exp(rate lambda1),T2~Exp(rate lambda2), both rates positive. R=min(T1,T2).\nProblem: PDF fR(x) for x>=0?", "criteria": { "A": "exp(−(lambda1+lambda2)x)", "B": "lambda1*lambda2 exp(−(lambda1+lambda2)x)", "C": "(lambda1+lambda2)exp(−(lambda1+lambda2)x)", "D": "lambda1 exp(−lambda1*x)+lambda2 exp(−lambda2*x)" } }, "18.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent T1~Exp(rate lambda1),T2~Exp(rate lambda2), both rates positive. R=min(T1,T2).\nProblem: E[R]?", "criteria": { "A": "1/(lambda1+lambda2)", "B": "1/lambda1+1/lambda2", "C": "min(1/lambda1,1/lambda2)", "D": "1/(lambda1*lambda2)" } }, "18.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent T1~Exp(rate lambda1),T2~Exp(rate lambda2), both rates positive. R=min(T1,T2).\nProblem: Choose an integral for P(T1<=T2).", "criteria": { "A": "integral(y=0..infinity) integral(x=0..y) lambda1*lambda2*exp(−lambda1*x−lambda2*y) dx dy", "B": "integral(x=0..infinity) exp(−(lambda1+lambda2)*x) dx", "C": "integral(x=0..infinity) lambda1*lambda2*exp(−(lambda1+lambda2)*x) dx", "D": "integral(y=0..infinity) integral(x=y..infinity) lambda1*lambda2*exp(−lambda1*x−lambda2*y) dx dy" } }, "18.1d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent T1~Exp(rate lambda1),T2~Exp(rate lambda2), both rates positive. R=min(T1,T2).\nProblem: Evaluate P(T1<=T2).", "criteria": { "A": "1/2", "B": "lambda1*lambda2/(lambda1+lambda2)", "C": "lambda2/(lambda1+lambda2)", "D": "lambda1/(lambda1+lambda2)" } }, "18.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent T1,T2~Uniform[0,10], S=min(T1,T2).\nProblem: PDF of S for 0<=x<=10?", "criteria": { "A": "(10−x)/50", "B": "(10−x)²/100", "C": "x/50", "D": "1/10" } }, "18.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent T1,T2~Uniform[0,10], S=min(T1,T2).\nProblem: E[S]?", "criteria": { "A": "20/3", "B": "5", "C": "5/2", "D": "10/3" } }, "18.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Independent T1,T2~Uniform[0,10], S=min(T1,T2).\nProblem: P(|T1−T2|<=1)?", "criteria": { "A": "1/10", "B": "9/100", "C": "1/5", "D": "19/100" } } } }, "attempts": [ { "status": 200, "full_response_ms": 309.09583299944643, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"18.1a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"A\":0.0,\"D\":0.0,\"C\":1.0}},\"18.1b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"A\":1.0,\"D\":0.0,\"C\":0.0}},\"18.1c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.87,\"probabilities\":{\"B\":0.01,\"A\":0.91,\"D\":0.06,\"C\":0.02}},\"18.1d\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.95,\"probabilities\":{\"B\":0.0,\"A\":0.0,\"D\":0.96,\"C\":0.04}},\"18.2a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"B\":0.0,\"A\":0.99,\"D\":0.0,\"C\":0.01}},\"18.2b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.96,\"probabilities\":{\"B\":0.0,\"A\":0.03,\"D\":0.97,\"C\":0.0}},\"18.2c\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.86,\"probabilities\":{\"B\":0.01,\"A\":0.0,\"D\":0.9,\"C\":0.09}}},\"usage\":{\"input_tokens\":1473,\"output_tokens\":325}}" } ], "response": { "model": "jev-1.13.0", "answers": { "18.1a": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "A": 0.0, "D": 0.0, "C": 1.0 } }, "18.1b": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "A": 1.0, "D": 0.0, "C": 0.0 } }, "18.1c": { "type": "choice", "choice": "A", "confidence": 0.87, "probabilities": { "B": 0.01, "A": 0.91, "D": 0.06, "C": 0.02 } }, "18.1d": { "type": "choice", "choice": "D", "confidence": 0.95, "probabilities": { "B": 0.0, "A": 0.0, "D": 0.96, "C": 0.04 } }, "18.2a": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "B": 0.0, "A": 0.99, "D": 0.0, "C": 0.01 } }, "18.2b": { "type": "choice", "choice": "D", "confidence": 0.96, "probabilities": { "B": 0.0, "A": 0.03, "D": 0.97, "C": 0.0 } }, "18.2c": { "type": "choice", "choice": "D", "confidence": 0.86, "probabilities": { "B": 0.01, "A": 0.0, "D": 0.9, "C": 0.09 } } }, "usage": { "input_tokens": 1473, "output_tokens": 325 } } }, { "exam": "final_su25", "section": 19, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "19.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: X~Exp(rate lambda>0), Y=min(X,2/lambda). Choose a derivation of E[Y].", "criteria": { "A": "Integrate x*lambda*exp(−lambda*x) from 0 to 2/lambda and add (2/lambda)P(X>2/lambda); the result is (1−exp(−2))/lambda.", "B": "Discard X values above 2/lambda without assigning their clamped mass; the result is (1−3exp(−2))/lambda.", "C": "Clamping at twice the mean leaves the mean unchanged at 1/lambda.", "D": "The clamp makes Y uniform on [0,2/lambda], so E[Y]=1/lambda." } } } }, "attempts": [ { "status": 200, "full_response_ms": 398.3535000006668, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"19.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.95,\"probabilities\":{\"B\":0.04,\"C\":0.0,\"D\":0.0,\"A\":0.96}}},\"usage\":{\"input_tokens\":560,\"output_tokens\":48}}" } ], "response": { "model": "jev-1.13.0", "answers": { "19.1": { "type": "choice", "choice": "A", "confidence": 0.95, "probabilities": { "B": 0.04, "C": 0.0, "D": 0.0, "A": 0.96 } } }, "usage": { "input_tokens": 560, "output_tokens": 48 } } }, { "exam": "final_su25", "section": 20, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "20.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Markov states 0,1,2,3. Transition rows (destination order 0,1,2,3) are [1/2,1/4,1/4,0], [1/2,1/4,1/4,0], [1/2,0,1/4,1/4], [0,1/4,0,3/4]. State 0 is studying, 3 sleeping.\nProblem: This chain is irreducible.", "criteria": { "A": "True", "B": "False" } }, "20.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Markov states 0,1,2,3. Transition rows (destination order 0,1,2,3) are [1/2,1/4,1/4,0], [1/2,1/4,1/4,0], [1/2,0,1/4,1/4], [0,1/4,0,3/4]. State 0 is studying, 3 sleeping.\nProblem: Period of the chain?", "criteria": { "A": "3", "B": "2", "C": "4", "D": "1" } }, "20.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Markov states 0,1,2,3. Transition rows (destination order 0,1,2,3) are [1/2,1/4,1/4,0], [1/2,1/4,1/4,0], [1/2,0,1/4,1/4], [0,1/4,0,3/4]. State 0 is studying, 3 sleeping.\nProblem: Why does the distribution converge to a unique stationary distribution from every initial state?", "criteria": { "A": "The rows sum to one, which alone guarantees convergence.", "B": "State 3 is absorbing and hence gives the unique stationary mass.", "C": "The finite chain is irreducible and aperiodic; self-loops give period one.", "D": "All finite chains converge uniquely, even if reducible or periodic." } }, "20.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Markov states 0,1,2,3. Transition rows (destination order 0,1,2,3) are [1/2,1/4,1/4,0], [1/2,1/4,1/4,0], [1/2,0,1/4,1/4], [0,1/4,0,3/4]. State 0 is studying, 3 sleeping.\nProblem: Choose stationary equations, plus pi0+pi1+pi2+pi3=1.", "criteria": { "A": "pi0=pi1=pi2=pi3=1/4", "B": "pi0=(pi0+pi1+pi2)/2; pi1=(pi0+pi1+pi3)/4; pi2=(pi0+pi1+pi2)/4; pi3=pi2/4+3pi3/4", "C": "pi0=(pi0+pi1)/2; pi1=(pi0+pi1)/4; pi2=pi2/4; pi3=3pi3/4", "D": "pi0=pi0/2+pi1/4+pi2/4; pi1=pi0/2+pi1/4+pi2/4; pi2=pi0/2+pi2/4+pi3/4; pi3=pi1/4+3pi3/4" } }, "20.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Markov states 0,1,2,3. Transition rows (destination order 0,1,2,3) are [1/2,1/4,1/4,0], [1/2,1/4,1/4,0], [1/2,0,1/4,1/4], [0,1/4,0,3/4]. State 0 is studying, 3 sleeping.\nProblem: Expected steps to first reach state 3, starting at 0?", "criteria": { "A": "4", "B": "8", "C": "16", "D": "12" } } } }, "attempts": [ { "status": 200, "full_response_ms": 368.52491599711357, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"20.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.01,\"probabilities\":{\"A\":0.5,\"B\":0.5}},\"20.2\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.81,\"probabilities\":{\"C\":0.01,\"D\":0.86,\"A\":0.01,\"B\":0.12}},\"20.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"C\":1.0,\"D\":0.0,\"A\":0.0,\"B\":0.0}},\"20.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.96,\"probabilities\":{\"C\":0.0,\"A\":0.01,\"D\":0.02,\"B\":0.97}},\"20.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.29,\"probabilities\":{\"C\":0.2,\"A\":0.21,\"D\":0.13,\"B\":0.45999999999999996}}},\"usage\":{\"input_tokens\":1490,\"output_tokens\":214}}" } ], "response": { "model": "jev-1.13.0", "answers": { "20.1": { "type": "choice", "choice": "B", "confidence": 0.01, "probabilities": { "A": 0.5, "B": 0.5 } }, "20.2": { "type": "choice", "choice": "D", "confidence": 0.81, "probabilities": { "C": 0.01, "D": 0.86, "A": 0.01, "B": 0.12 } }, "20.3": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "C": 1.0, "D": 0.0, "A": 0.0, "B": 0.0 } }, "20.4": { "type": "choice", "choice": "B", "confidence": 0.96, "probabilities": { "C": 0.0, "A": 0.01, "D": 0.02, "B": 0.97 } }, "20.5": { "type": "choice", "choice": "B", "confidence": 0.29, "probabilities": { "C": 0.2, "A": 0.21, "D": 0.13, "B": 0.45999999999999996 } } }, "usage": { "input_tokens": 1490, "output_tokens": 214 } } }, { "exam": "midterm_fa23", "section": 3, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "3.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: P implies Q is equivalent to NOT P implies NOT Q.", "criteria": { "A": "True", "B": "False" } }, "3.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Complete the contrapositive: (P AND Q) implies R is equivalent to NOT R implies __.", "criteria": { "A": "NOT P OR NOT Q", "B": "NOT P AND NOT Q", "C": "P OR Q", "D": "P AND Q" } }, "3.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: NOT exists integer x [P(x) OR Q(x)] is equivalent to forall integer x [NOT P(x) AND NOT Q(x)].", "criteria": { "A": "False", "B": "True" } }, "3.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: NOT forall natural x exists natural y [P(x) AND Q(x,y)] is equivalent to exists natural x forall natural y [NOT P(x) AND NOT Q(x,y)].", "criteria": { "A": "True", "B": "False" } }, "3.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For all natural a and positive integer b, a/b is not sqrt(2).", "criteria": { "A": "True", "B": "False" } }, "3.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Every natural n is either a perfect square or has irrational square root.", "criteria": { "A": "True", "B": "False" } } } }, "attempts": [ { "status": 200, "full_response_ms": 356.62395800318336, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"3.1a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.97,\"probabilities\":{\"B\":0.98,\"A\":0.02}},\"3.1b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"C\":0.0,\"D\":0.0,\"A\":1.0}},\"3.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"B\":0.99,\"A\":0.01}},\"3.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.98,\"probabilities\":{\"B\":0.99,\"A\":0.01}},\"3.4\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.97,\"probabilities\":{\"B\":0.01,\"A\":0.99}},\"3.5\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.06,\"probabilities\":{\"B\":0.47,\"A\":0.53}}},\"usage\":{\"input_tokens\":864,\"output_tokens\":199}}" } ], "response": { "model": "jev-1.13.0", "answers": { "3.1a": { "type": "choice", "choice": "B", "confidence": 0.97, "probabilities": { "B": 0.98, "A": 0.02 } }, "3.1b": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "D": 0.0, "A": 1.0 } }, "3.2": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "B": 0.99, "A": 0.01 } }, "3.3": { "type": "choice", "choice": "B", "confidence": 0.98, "probabilities": { "B": 0.99, "A": 0.01 } }, "3.4": { "type": "choice", "choice": "A", "confidence": 0.97, "probabilities": { "B": 0.01, "A": 0.99 } }, "3.5": { "type": "choice", "choice": "A", "confidence": 0.06, "probabilities": { "B": 0.47, "A": 0.53 } } }, "usage": { "input_tokens": 864, "output_tokens": 199 } } }, { "exam": "midterm_fa23", "section": 4, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "4.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For integers a,b,c, does a|b and a|c imply a|(b+5c)? Justify.", "criteria": { "A": "No: choose a=2,b=4,c=6.", "B": "No: multiplying c by 5 destroys divisibility.", "C": "Yes: b=ak,c=al imply b+5c=a(k+5l).", "D": "Yes: b+5c equals a whenever both are divisible by a." } }, "4.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Does b²=n imply n divides b for all positive integers b,n?", "criteria": { "A": "Yes: any square divides its square root.", "B": "Yes: take square roots of the divisibility n|b².", "C": "No: b=1,n=1 is a counterexample.", "D": "No: b=2,n=4 is a counterexample." } }, "4.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Why must a positive nonsquare integer have an odd exponent in its prime factorization?", "criteria": { "A": "Every nonsquare integer is prime and has exponent one.", "B": "An even exponent would force the integer itself to be even.", "C": "All prime factors of a nonsquare must have odd exponents.", "D": "If all exponents were even, halving them gives an integer whose square is the original number, a contradiction." } } } }, "attempts": [ { "status": 200, "full_response_ms": 363.1284589973802, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"4.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":0.0,\"C\":1.0}},\"4.2\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.87,\"probabilities\":{\"D\":0.9,\"B\":0.02,\"A\":0.07,\"C\":0.01}},\"4.3\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"B\":0.0,\"A\":0.0,\"C\":0.0}}},\"usage\":{\"input_tokens\":814,\"output_tokens\":135}}" } ], "response": { "model": "jev-1.13.0", "answers": { "4.1": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.0, "C": 1.0 } }, "4.2": { "type": "choice", "choice": "D", "confidence": 0.87, "probabilities": { "D": 0.9, "B": 0.02, "A": 0.07, "C": 0.01 } }, "4.3": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "A": 0.0, "C": 0.0 } } }, "usage": { "input_tokens": 814, "output_tokens": 135 } } }, { "exam": "midterm_fa23", "section": 5, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "5.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: x1=1; xn=sqrt(1+x(n−1)) for n>=2. Prove xn<2.", "criteria": { "A": "Every term equals 1, so the result is immediate.", "B": "Base x1=1<2. If 0<=xk<2, then x(k+1)=sqrt(1+xk)=1.", "criteria": { "A": "A1=35; A(k+1)=2Ak+7*3^(2k+1), so the induction preserves divisibility.", "B": "A1=35 and A2=259, so all remaining cases follow without an induction step.", "C": "A1=35; A(k+1)=9Ak, proving the induction.", "D": "Both summands are separately multiples of seven." } } } }, "attempts": [ { "status": 200, "full_response_ms": 349.32162500263075, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"5.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":1.0,\"A\":0.0,\"C\":0.0}},\"5.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":0.99,\"C\":0.01}}},\"usage\":{\"input_tokens\":719,\"output_tokens\":91}}" } ], "response": { "model": "jev-1.13.0", "answers": { "5.1": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 1.0, "A": 0.0, "C": 0.0 } }, "5.2": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.99, "C": 0.01 } } }, "usage": { "input_tokens": 719, "output_tokens": 91 } } }, { "exam": "midterm_fa23", "section": 6, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "6.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A pair present in both proposer-optimal matchings is present in every stable matching.", "criteria": { "A": "False", "B": "True" } }, "6.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If some pair appears in both proposer-optimal matchings, then the entire stable matching is unique.", "criteria": { "A": "False", "B": "True" } }, "6.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: With two jobs and two candidates, every stable matching is optimal for one proposer side or both.", "criteria": { "A": "True", "B": "False" } }, "6.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: There exist instances where a candidate rejecting the less-preferred-of-two rule in job-proposing deferred acceptance can obtain a better final partner.", "criteria": { "A": "True", "B": "False" } } } }, "attempts": [ { "status": 200, "full_response_ms": 315.8796669995354, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"6.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.55,\"probabilities\":{\"B\":0.78,\"A\":0.22}},\"6.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.44,\"probabilities\":{\"B\":0.28,\"A\":0.72}},\"6.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.9,\"probabilities\":{\"B\":0.05,\"A\":0.95}},\"6.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.06,\"probabilities\":{\"B\":0.53,\"A\":0.47}}},\"usage\":{\"input_tokens\":663,\"output_tokens\":123}}" } ], "response": { "model": "jev-1.13.0", "answers": { "6.1": { "type": "choice", "choice": "B", "confidence": 0.55, "probabilities": { "B": 0.78, "A": 0.22 } }, "6.2": { "type": "choice", "choice": "A", "confidence": 0.44, "probabilities": { "B": 0.28, "A": 0.72 } }, "6.3": { "type": "choice", "choice": "A", "confidence": 0.9, "probabilities": { "B": 0.05, "A": 0.95 } }, "6.4": { "type": "choice", "choice": "B", "confidence": 0.06, "probabilities": { "B": 0.53, "A": 0.47 } } }, "usage": { "input_tokens": 663, "output_tokens": 123 } } }, { "exam": "midterm_fa23", "section": 7, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "7.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Smallest N such that every n-dimensional hypercube with n>=N has an even edge count?", "criteria": { "A": "1", "B": "3", "C": "2", "D": "4" } }, "7.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Kn has an even number of edges for every n>3.", "criteria": { "A": "True", "B": "False" } }, "7.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: In Kn, how many edges cross a cut with k vertices on one side?", "criteria": { "A": "n−1", "B": "C(k,2)", "C": "k(n−k)", "D": "k*n" } }, "7.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Minimum edges crossing a nonempty proper cut of an n-vertex tree, n>=2?", "criteria": { "A": "n−1", "B": "0", "C": "2", "D": "1" } }, "7.3c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Maximum edges crossing a cut of an n-vertex tree?", "criteria": { "A": "n", "B": "1", "C": "n−1", "D": "floor(n/2)" } }, "7.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Adding a new edge either reduces the component count or creates a cycle containing that edge.", "criteria": { "A": "True", "B": "False" } }, "7.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Minimum possible component count with n>=1 vertices and e edges?", "criteria": { "A": "n−1", "B": "e−n+1", "C": "n−e", "D": "max(n−e,1)" } }, "7.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Smallest edge-count threshold guaranteeing a cycle in a simple n-vertex graph?", "criteria": { "A": "n", "B": "2n", "C": "n+1", "D": "n−1" } } } }, "attempts": [ { "status": 200, "full_response_ms": 394.22216700040735, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"7.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.12,\"probabilities\":{\"D\":0.11,\"A\":0.28,\"C\":0.34,\"B\":0.27}},\"7.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.09,\"probabilities\":{\"A\":0.55,\"B\":0.45}},\"7.3a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"C\":1.0,\"B\":0.0}},\"7.3b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"A\":0.0,\"C\":0.0,\"B\":0.0}},\"7.3c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.27,\"probabilities\":{\"D\":0.4,\"A\":0.01,\"C\":0.45,\"B\":0.14}},\"7.4\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"A\":0.99,\"B\":0.01}},\"7.5\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.93,\"probabilities\":{\"D\":0.95,\"A\":0.0,\"C\":0.05,\"B\":0.0}},\"7.6\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.94,\"probabilities\":{\"D\":0.01,\"A\":0.95,\"C\":0.04,\"B\":0.0}}},\"usage\":{\"input_tokens\":1141,\"output_tokens\":330}}" } ], "response": { "model": "jev-1.13.0", "answers": { "7.1": { "type": "choice", "choice": "C", "confidence": 0.12, "probabilities": { "D": 0.11, "A": 0.28, "C": 0.34, "B": 0.27 } }, "7.2": { "type": "choice", "choice": "A", "confidence": 0.09, "probabilities": { "A": 0.55, "B": 0.45 } }, "7.3a": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "D": 0.0, "A": 0.0, "C": 1.0, "B": 0.0 } }, "7.3b": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "A": 0.0, "C": 0.0, "B": 0.0 } }, "7.3c": { "type": "choice", "choice": "C", "confidence": 0.27, "probabilities": { "D": 0.4, "A": 0.01, "C": 0.45, "B": 0.14 } }, "7.4": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "A": 0.99, "B": 0.01 } }, "7.5": { "type": "choice", "choice": "D", "confidence": 0.93, "probabilities": { "D": 0.95, "A": 0.0, "C": 0.05, "B": 0.0 } }, "7.6": { "type": "choice", "choice": "A", "confidence": 0.94, "probabilities": { "D": 0.01, "A": 0.95, "C": 0.04, "B": 0.0 } } }, "usage": { "input_tokens": 1141, "output_tokens": 330 } } }, { "exam": "midterm_fa23", "section": 8, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "8.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: The n-dimensional hypercube is edge-colorable with n colors.", "criteria": { "A": "True", "B": "False" } }, "8.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Every graph of maximum degree d is edge-colorable with d colors.", "criteria": { "A": "True", "B": "False" } }, "8.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Every bipartite graph is edge-colorable with two colors.", "criteria": { "A": "True", "B": "False" } }, "8.1di": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A bipartite graph on n>2 vertices is d-regular, and its edges partition into Hamiltonian cycles.\nProblem: Number of edges?", "criteria": { "A": "nd", "B": "d/2", "C": "nd/2", "D": "n+d" } }, "8.1dii": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A bipartite graph on n>2 vertices is d-regular, and its edges partition into Hamiltonian cycles.\nProblem: Number of cycles in this edge partition?", "criteria": { "A": "nd/2", "B": "d/2", "C": "d", "D": "n/2" } }, "8.1diii": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A bipartite graph on n>2 vertices is d-regular, and its edges partition into Hamiltonian cycles.\nProblem: Why can the graph be edge-colored with d colors?", "criteria": { "A": "Every d-regular graph automatically has a d-edge-coloring regardless of bipartiteness.", "B": "Each cycle gets one color, since its edges do not share endpoints.", "C": "Use the two vertex colors as edge colors for all cycles.", "D": "Each Hamiltonian cycle is even because the graph is bipartite, so alternate two colors on it; use a disjoint color pair for each of the d/2 cycles." } }, "8.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: In a d-regular bipartite graph, d>0, prove every S on the left satisfies |N(S)|>=|S|.", "criteria": { "A": "Exactly d|S| edges leave S and at most d|N(S)| can meet its neighbors, so d|S|<=d|N(S)|; divide by d.", "B": "All neighbors of different left vertices are distinct, so each adds d new vertices.", "C": "Each left vertex has at least one neighbor, which alone ensures Hall’s condition.", "D": "The graph is bipartite, so |N(S)|=|S| for every subset regardless of degree." } } } }, "attempts": [ { "status": 200, "full_response_ms": 292.3393749988463, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"8.1a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.97,\"probabilities\":{\"B\":0.02,\"A\":0.98}},\"8.1b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.92,\"probabilities\":{\"B\":0.96,\"A\":0.04}},\"8.1c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.13,\"probabilities\":{\"B\":0.57,\"A\":0.43}},\"8.1di\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.42,\"probabilities\":{\"B\":0.0,\"C\":0.57,\"D\":0.0,\"A\":0.43}},\"8.1dii\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.92,\"probabilities\":{\"B\":0.94,\"C\":0.05,\"A\":0.01,\"D\":0.0}},\"8.1diii\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.95,\"probabilities\":{\"B\":0.03,\"C\":0.0,\"A\":0.0,\"D\":0.97}},\"8.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"C\":0.0,\"A\":1.0,\"D\":0.0}}},\"usage\":{\"input_tokens\":1210,\"output_tokens\":277}}" } ], "response": { "model": "jev-1.13.0", "answers": { "8.1a": { "type": "choice", "choice": "A", "confidence": 0.97, "probabilities": { "B": 0.02, "A": 0.98 } }, "8.1b": { "type": "choice", "choice": "B", "confidence": 0.92, "probabilities": { "B": 0.96, "A": 0.04 } }, "8.1c": { "type": "choice", "choice": "B", "confidence": 0.13, "probabilities": { "B": 0.57, "A": 0.43 } }, "8.1di": { "type": "choice", "choice": "C", "confidence": 0.42, "probabilities": { "B": 0.0, "C": 0.57, "D": 0.0, "A": 0.43 } }, "8.1dii": { "type": "choice", "choice": "B", "confidence": 0.92, "probabilities": { "B": 0.94, "C": 0.05, "A": 0.01, "D": 0.0 } }, "8.1diii": { "type": "choice", "choice": "D", "confidence": 0.95, "probabilities": { "B": 0.03, "C": 0.0, "A": 0.0, "D": 0.97 } }, "8.2": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "A": 1.0, "D": 0.0 } } }, "usage": { "input_tokens": 1210, "output_tokens": 277 } } }, { "exam": "midterm_fa23", "section": 9, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "9.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Compute 7^50 modulo 35.", "criteria": { "A": "14", "B": "1", "C": "7", "D": "0" } }, "9.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prime q, N=q². How many residues modulo N are coprime to N?", "criteria": { "A": "q", "B": "q²−q", "C": "q²−1", "D": "q−1" } }, "9.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If q is prime and gcd(a,q)=1, then a has a multiplicative inverse modulo q².", "criteria": { "A": "False", "B": "True" } }, "9.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prime q, N=q², gcd(a,N)=1. Besides exponent 1, choose an exponent x>1 guaranteed to satisfy a^x=a mod N for every such a.", "criteria": { "A": "q", "B": "q−1", "C": "q(q−1)+1", "D": "q²" } }, "9.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For all moduli n, ab=0 mod n implies a=0 mod n or b=0 mod n.", "criteria": { "A": "False", "B": "True" } }, "9.4a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If m and n are integer linear combinations of x,y, every integer linear combination of m,n is also an integer linear combination of x,y.", "criteria": { "A": "False", "B": "True" } }, "9.4b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For positive x,y, the least positive integer linear combination is min(x,y).", "criteria": { "A": "False", "B": "True" } }, "9.4c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For positive x,y, the least positive integer linear combination is gcd(x,y).", "criteria": { "A": "False", "B": "True" } }, "9.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If x=a mod m and x=a mod n, with coprime positive m,n, then x=a mod mn.", "criteria": { "A": "False", "B": "True" } }, "9.6a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Positive m,n have gcd q. Smallest positive k with a+kn=a mod m?", "criteria": { "A": "n/q", "B": "q", "C": "m/q", "D": "mn/q" } }, "9.6b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Positive m,n have gcd q. How many x in 0..mn−1 satisfy x=a mod m and x=a mod n?", "criteria": { "A": "q", "B": "1", "C": "mn/q", "D": "m+n" } } } }, "attempts": [ { "status": 200, "full_response_ms": 499.06333300168626, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"9.1\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.72,\"probabilities\":{\"D\":0.79,\"A\":0.11,\"C\":0.1,\"B\":0.0}},\"9.2a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"C\":0.0,\"A\":0.0,\"D\":0.0,\"B\":1.0}},\"9.2b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.91,\"probabilities\":{\"A\":0.04,\"B\":0.96}},\"9.2c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.93,\"probabilities\":{\"C\":0.94,\"A\":0.04,\"D\":0.02,\"B\":0.0}},\"9.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"A\":1.0,\"B\":0.0}},\"9.4a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"A\":0.0,\"B\":1.0}},\"9.4b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.7,\"probabilities\":{\"A\":0.85,\"B\":0.15}},\"9.4c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.75,\"probabilities\":{\"A\":0.13,\"B\":0.87}},\"9.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.94,\"probabilities\":{\"A\":0.03,\"B\":0.97}},\"9.6a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.93,\"probabilities\":{\"C\":0.94,\"A\":0.02,\"D\":0.03,\"B\":0.01}},\"9.6b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.65,\"probabilities\":{\"C\":0.01,\"A\":0.74,\"D\":0.0,\"B\":0.25}}},\"usage\":{\"input_tokens\":1402,\"output_tokens\":411}}" } ], "response": { "model": "jev-1.13.0", "answers": { "9.1": { "type": "choice", "choice": "D", "confidence": 0.72, "probabilities": { "D": 0.79, "A": 0.11, "C": 0.1, "B": 0.0 } }, "9.2a": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "C": 0.0, "A": 0.0, "D": 0.0, "B": 1.0 } }, "9.2b": { "type": "choice", "choice": "B", "confidence": 0.91, "probabilities": { "A": 0.04, "B": 0.96 } }, "9.2c": { "type": "choice", "choice": "C", "confidence": 0.93, "probabilities": { "C": 0.94, "A": 0.04, "D": 0.02, "B": 0.0 } }, "9.3": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "A": 1.0, "B": 0.0 } }, "9.4a": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "A": 0.0, "B": 1.0 } }, "9.4b": { "type": "choice", "choice": "A", "confidence": 0.7, "probabilities": { "A": 0.85, "B": 0.15 } }, "9.4c": { "type": "choice", "choice": "B", "confidence": 0.75, "probabilities": { "A": 0.13, "B": 0.87 } }, "9.5": { "type": "choice", "choice": "B", "confidence": 0.94, "probabilities": { "A": 0.03, "B": 0.97 } }, "9.6a": { "type": "choice", "choice": "C", "confidence": 0.93, "probabilities": { "C": 0.94, "A": 0.02, "D": 0.03, "B": 0.01 } }, "9.6b": { "type": "choice", "choice": "A", "confidence": 0.65, "probabilities": { "C": 0.01, "A": 0.74, "D": 0.0, "B": 0.25 } } }, "usage": { "input_tokens": 1402, "output_tokens": 411 } } }, { "exam": "midterm_fa23", "section": 10, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "10.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Prime p>2 has p−1=q1*...*qk, a product of distinct primes.\nProblem: For 1<=n2 has p−1=q1*...*qk, a product of distinct primes.\nProblem: For each fixed i, prove there exists a≠0,1 modulo p with a^qi=1.", "criteria": { "A": "Set a=1, which satisfies both required inequalities.", "B": "Choose one a with order divisible by all qi; then each qi separately annihilates it.", "C": "Let h=(p−1)/qi. The polynomial x^h−1 has fewer than p−1 roots, so choose nonzero b with b^h≠1. Put a=b^h; then a^qi=b^(p−1)=1 by Fermat.", "D": "Choose any a≠0,1; Fermat says its qith power is one." } }, "10.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If x has order d modulo prime p and x^q=1 mod p, prove d divides q.", "criteria": { "A": "The order d is always p−1 and q is always p.", "B": "Every divisor of x also divides q.", "C": "Write q=dm+r, 0<=r=dQ>=1. Euclidean division gives P=DQ+R, with R zero or deg R=dQ>=1. Euclidean division gives P=DQ+R, with R zero or deg R=dQ>=1. Euclidean division gives P=DQ+R, with R zero or deg R4. Hint: separate square and nonsquare factorizations.", "criteria": { "A": "Write n=ab with 12 and a,2a=1.", "criteria": { "A": "Checking the first seven n proves the rest by periodicity over the integers.", "B": "A_1=28. Both exponential terms are independently divisible by 7.", "C": "A_1=28. If 7 divides A_n, then A_(n+1)=7*3^(2n+1)+2A_n is divisible by 7.", "D": "A_1=28. The recurrence is A_(n+1)=9A_n, so the result follows." } } } }, "attempts": [ { "status": 200, "full_response_ms": 371.74975000016275, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"5.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.98,\"probabilities\":{\"B\":0.0,\"C\":0.99,\"A\":0.0,\"D\":0.01}}},\"usage\":{\"input_tokens\":535,\"output_tokens\":47}}" } ], "response": { "model": "jev-1.13.0", "answers": { "5.1": { "type": "choice", "choice": "C", "confidence": 0.98, "probabilities": { "B": 0.0, "C": 0.99, "A": 0.0, "D": 0.01 } } }, "usage": { "input_tokens": 535, "output_tokens": 47 } } }, { "exam": "midterm_fa24", "section": 6, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "6.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Jobs: A:1>2>3, B:2>3>1, C:2>3>1. Candidates: 1:B>C>A, 2:C>A>B, 3:C>A>B.\nProblem: Job-optimal matching?", "criteria": { "A": "A–3, B–1, C–2", "B": "A–1, B–2, C–3", "C": "A–1, B–3, C–2", "D": "A–2, B–3, C–1" } }, "6.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Jobs: A:1>2>3, B:2>3>1, C:2>3>1. Candidates: 1:B>C>A, 2:C>A>B, 3:C>A>B.\nProblem: Candidate-optimal matching?", "criteria": { "A": "A–1, B–3, C–2", "B": "A–3, B–1, C–2", "C": "A–1, B–2, C–3", "D": "A–2, B–3, C–1" } }, "6.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Jobs: A:1>2>3, B:2>3>1, C:2>3>1. Candidates: 1:B>C>A, 2:C>A>B, 3:C>A>B.\nProblem: Another stable matching beyond the two proposer-optimal ones?", "criteria": { "A": "A–2, B–1, C–3", "B": "A–3, B–2, C–1", "C": "A–1, B–2, C–3", "D": "None" } }, "6.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A job receiving its least-preferred candidate in the job-optimal matching must be every candidate’s least-preferred job.", "criteria": { "A": "False", "B": "True" } }, "6.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If a job has the same candidate in every stable matching, that candidate must be its first choice.", "criteria": { "A": "True", "B": "False" } }, "6.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If a candidate rejects all offers one day instead of retaining its favorite in job-proposing deferred acceptance, any resulting complete matching must be worse for that candidate.", "criteria": { "A": "True", "B": "False" } }, "6.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A job–candidate pair present in both proposer-optimal matchings is present in every stable matching.", "criteria": { "A": "True", "B": "False" } } } }, "attempts": [ { "status": 200, "full_response_ms": 337.23529200142366, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"6.1a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.25,\"probabilities\":{\"C\":0.23,\"A\":0.44,\"D\":0.09,\"B\":0.24}},\"6.1b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.4,\"probabilities\":{\"C\":0.13,\"A\":0.23,\"D\":0.09,\"B\":0.55}},\"6.1c\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.2,\"probabilities\":{\"C\":0.06,\"A\":0.32,\"D\":0.4,\"B\":0.22}},\"6.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.32,\"probabilities\":{\"A\":0.66,\"B\":0.34}},\"6.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.02,\"probabilities\":{\"B\":0.51,\"A\":0.49}},\"6.4\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.3,\"probabilities\":{\"A\":0.65,\"B\":0.35}},\"6.5\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.55,\"probabilities\":{\"A\":0.77,\"B\":0.23}}},\"usage\":{\"input_tokens\":1201,\"output_tokens\":258}}" } ], "response": { "model": "jev-1.13.0", "answers": { "6.1a": { "type": "choice", "choice": "A", "confidence": 0.25, "probabilities": { "C": 0.23, "A": 0.44, "D": 0.09, "B": 0.24 } }, "6.1b": { "type": "choice", "choice": "B", "confidence": 0.4, "probabilities": { "C": 0.13, "A": 0.23, "D": 0.09, "B": 0.55 } }, "6.1c": { "type": "choice", "choice": "D", "confidence": 0.2, "probabilities": { "C": 0.06, "A": 0.32, "D": 0.4, "B": 0.22 } }, "6.2": { "type": "choice", "choice": "A", "confidence": 0.32, "probabilities": { "A": 0.66, "B": 0.34 } }, "6.3": { "type": "choice", "choice": "B", "confidence": 0.02, "probabilities": { "B": 0.51, "A": 0.49 } }, "6.4": { "type": "choice", "choice": "A", "confidence": 0.3, "probabilities": { "A": 0.65, "B": 0.35 } }, "6.5": { "type": "choice", "choice": "A", "confidence": 0.55, "probabilities": { "A": 0.77, "B": 0.23 } } }, "usage": { "input_tokens": 1201, "output_tokens": 258 } } }, { "exam": "midterm_fa24", "section": 7, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "7.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A graph with n vertices and m>=n−1 edges must be connected.", "criteria": { "A": "False", "B": "True" } }, "7.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A graph with n vertices and m<=n−1 edges must be acyclic.", "criteria": { "A": "True", "B": "False" } }, "7.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of edges in Kn?", "criteria": { "A": "C(n,2)", "B": "n²", "C": "2n", "D": "n−1" } }, "7.1d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A hypercube has n vertices. Number of edges?", "criteria": { "A": "n log2(n)", "B": "n log2(n)/2", "C": "n/2", "D": "2^n" } }, "7.1e": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A graph with an Eulerian tour must have an even number of edges.", "criteria": { "A": "True", "B": "False" } }, "7.1f": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Minimum possible connected components for a simple graph with n>=1 vertices and feasible edge count m?", "criteria": { "A": "max(1,n−m)", "B": "n−m", "C": "n−1", "D": "max(1,m−n)" } }, "7.1g": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A connected acyclic graph with at least two vertices has at least how many degree-one vertices? Give the sharp universal bound.", "criteria": { "A": "3", "B": "1", "C": "2", "D": "n−1" } }, "7.1h": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A forest has n vertices and c components. Number of edges?", "criteria": { "A": "c−1", "B": "n−1", "C": "n+c", "D": "n−c" } }, "7.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Every graph has an even number of odd-degree vertices.", "criteria": { "A": "True", "B": "False" } }, "7.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Minimum edge colors for a d-dimensional hypercube?", "criteria": { "A": "d+1", "B": "d", "C": "2^d", "D": "2d" } }, "7.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Minimum edge colors for an odd cycle?", "criteria": { "A": "4", "B": "2", "C": "1", "D": "3" } }, "7.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A simple graph has n vertices and n+1 edges. Largest constant degree guaranteed for at least one vertex?", "criteria": { "A": "3", "B": "2", "C": "4", "D": "n−1" } } } }, "attempts": [ { "status": 200, "full_response_ms": 402.2258749973844, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"7.1a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"A\":0.99,\"B\":0.01}},\"7.1b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.88,\"probabilities\":{\"A\":0.06,\"B\":0.94}},\"7.1c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"A\":1.0,\"B\":0.0,\"C\":0.0,\"D\":0.0}},\"7.1d\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.89,\"probabilities\":{\"A\":0.08,\"B\":0.91,\"C\":0.01,\"D\":0.0}},\"7.1e\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.6,\"probabilities\":{\"A\":0.2,\"B\":0.8}},\"7.1f\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.96,\"probabilities\":{\"A\":0.97,\"B\":0.03,\"C\":0.0,\"D\":0.0}},\"7.1g\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"A\":0.0,\"B\":0.0,\"C\":1.0,\"D\":0.0}},\"7.1h\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"A\":0.0,\"B\":0.0,\"C\":0.0,\"D\":1.0}},\"7.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"A\":0.99,\"B\":0.01}},\"7.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.92,\"probabilities\":{\"A\":0.05,\"B\":0.94,\"C\":0.0,\"D\":0.01}},\"7.4\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.96,\"probabilities\":{\"A\":0.01,\"B\":0.02,\"C\":0.0,\"D\":0.97}},\"7.5\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.6,\"probabilities\":{\"A\":0.71,\"B\":0.21,\"C\":0.08,\"D\":0.0}}},\"usage\":{\"input_tokens\":1495,\"output_tokens\":483}}" } ], "response": { "model": "jev-1.13.0", "answers": { "7.1a": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "A": 0.99, "B": 0.01 } }, "7.1b": { "type": "choice", "choice": "B", "confidence": 0.88, "probabilities": { "A": 0.06, "B": 0.94 } }, "7.1c": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "A": 1.0, "B": 0.0, "C": 0.0, "D": 0.0 } }, "7.1d": { "type": "choice", "choice": "B", "confidence": 0.89, "probabilities": { "A": 0.08, "B": 0.91, "C": 0.01, "D": 0.0 } }, "7.1e": { "type": "choice", "choice": "B", "confidence": 0.6, "probabilities": { "A": 0.2, "B": 0.8 } }, "7.1f": { "type": "choice", "choice": "A", "confidence": 0.96, "probabilities": { "A": 0.97, "B": 0.03, "C": 0.0, "D": 0.0 } }, "7.1g": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "A": 0.0, "B": 0.0, "C": 1.0, "D": 0.0 } }, "7.1h": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "A": 0.0, "B": 0.0, "C": 0.0, "D": 1.0 } }, "7.2": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "A": 0.99, "B": 0.01 } }, "7.3": { "type": "choice", "choice": "B", "confidence": 0.92, "probabilities": { "A": 0.05, "B": 0.94, "C": 0.0, "D": 0.01 } }, "7.4": { "type": "choice", "choice": "D", "confidence": 0.96, "probabilities": { "A": 0.01, "B": 0.02, "C": 0.0, "D": 0.97 } }, "7.5": { "type": "choice", "choice": "A", "confidence": 0.6, "probabilities": { "A": 0.71, "B": 0.21, "C": 0.08, "D": 0.0 } } }, "usage": { "input_tokens": 1495, "output_tokens": 483 } } }, { "exam": "midterm_fa24", "section": 8, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "8.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A connected simple planar drawing has v>2 vertices and e<3v−6. Prove some face has boundary length at least 4, counting repeated edge sides.", "criteria": { "A": "If every face had length 3, then 3f=2e. Euler’s v−e+f=2 would force e=3v−6, a contradiction; each face has length at least 3.", "B": "Since e<3v−6, every vertex has degree at most two, and every face is a quadrilateral.", "C": "A planar graph can have only triangular or quadrilateral faces; fewer edges forbid triangles.", "D": "Euler says f=3 for every planar graph, so one face has four sides." } }, "8.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Partition the edges of a bipartite Eulerian graph into two spanning subgraphs with half of each vertex’s degree.", "criteria": { "A": "Color a spanning tree one color and all remaining edges the other; this halves each degree.", "B": "Alternate the edges of an Euler tour between the two sets. The tour has even length because the graph is bipartite; paired arrival/departure edges, including the wraparound, receive opposite colors at every vertex.", "C": "Assign every edge incident to the left bipartition to the first set and all others to the second.", "D": "Take any half of the edges; equal edge counts imply equal degrees at every vertex." } } } }, "attempts": [ { "status": 200, "full_response_ms": 370.90579199866625, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"8.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":1.0,\"C\":0.0,\"B\":0.0}},\"8.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"C\":0.0,\"A\":0.0,\"B\":1.0}}},\"usage\":{\"input_tokens\":767,\"output_tokens\":91}}" } ], "response": { "model": "jev-1.13.0", "answers": { "8.1": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 1.0, "C": 0.0, "B": 0.0 } }, "8.2": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "C": 0.0, "A": 0.0, "B": 1.0 } } }, "usage": { "input_tokens": 767, "output_tokens": 91 } } }, { "exam": "midterm_fa24", "section": 9, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "9.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For integers a,x,y, gcd(x,y)=1 and a divides xy imply a divides x or a divides y.", "criteria": { "A": "True", "B": "False" } }, "9.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Choose a counterexample to: gcd(x,y)=1 and a|xy imply a|x or a|y.", "criteria": { "A": "a=3,x=9,y=4", "B": "a=6,x=9,y=4", "C": "a=6,x=6,y=5", "D": "a=5,x=9,y=4" } }, "9.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: N=pqr for three distinct primes p,q,r.\nProblem: Number of residues x mod N satisfying qx=0 mod N?", "criteria": { "A": "1", "B": "pqr", "C": "q", "D": "pr" } }, "9.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: N=pqr for three distinct primes p,q,r.\nProblem: If gcd(x,N)=1, what is x^((p−1)(q−1)(r−1)) mod N?", "criteria": { "A": "N−1", "B": "x", "C": "0", "D": "1" } }, "9.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: N=pqr for three distinct primes p,q,r.\nProblem: Why is x^((p−1)(q−1)(r−1))=1 mod N when gcd(x,N)=1?", "criteria": { "A": "Being coprime to N means x=1 modulo N.", "B": "Fermat applies directly to the composite modulus pqr with exponent pqr−1.", "C": "Fermat gives value 1 modulo each prime because the exponent is a multiple of each prime minus one. CRT makes their common solution 1 modulo pqr.", "D": "The exponent is zero modulo every prime, so the power is one." } } } }, "attempts": [ { "status": 200, "full_response_ms": 338.06458300023223, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"9.1a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.12,\"probabilities\":{\"B\":0.56,\"A\":0.44}},\"9.1b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.96,\"probabilities\":{\"B\":0.97,\"D\":0.02,\"C\":0.0,\"A\":0.01}},\"9.2a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.82,\"probabilities\":{\"B\":0.01,\"D\":0.86,\"C\":0.11,\"A\":0.02}},\"9.2b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.98,\"probabilities\":{\"B\":0.01,\"D\":0.99,\"C\":0.0,\"A\":0.0}},\"9.2c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"C\":1.0,\"A\":0.0}}},\"usage\":{\"input_tokens\":993,\"output_tokens\":214}}" } ], "response": { "model": "jev-1.13.0", "answers": { "9.1a": { "type": "choice", "choice": "B", "confidence": 0.12, "probabilities": { "B": 0.56, "A": 0.44 } }, "9.1b": { "type": "choice", "choice": "B", "confidence": 0.96, "probabilities": { "B": 0.97, "D": 0.02, "C": 0.0, "A": 0.01 } }, "9.2a": { "type": "choice", "choice": "D", "confidence": 0.82, "probabilities": { "B": 0.01, "D": 0.86, "C": 0.11, "A": 0.02 } }, "9.2b": { "type": "choice", "choice": "D", "confidence": 0.98, "probabilities": { "B": 0.01, "D": 0.99, "C": 0.0, "A": 0.0 } }, "9.2c": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "C": 1.0, "A": 0.0 } } }, "usage": { "input_tokens": 993, "output_tokens": 214 } } }, { "exam": "midterm_fa24", "section": 10, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "10.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Multiplicative order of 3 modulo 5?", "criteria": { "A": "3", "B": "5", "C": "2", "D": "4" } }, "10.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Let t be the order of a modulo n, gcd(a,n)=1. Prove a^m=1 mod n iff t divides m.", "criteria": { "A": "Multiples of t give powers of a^t=1. Conversely write m=qt+r, 0<=r1 with n dividing 2^n−1.", "criteria": { "A": "Since 2^n−1 is odd, no integer n can divide it.", "B": "If n divides 2^n−1, then n divides 2 and n divides 1 separately.", "C": "Such n is odd. Let p be its smallest prime factor and t=ord_p(2)>1. Then t divides n and p−1, so a prime factor of t is a prime factor of n smaller than p, a contradiction.", "D": "Fermat says 2^n=2 mod n even for composite n, so the remainder is always one." } } } }, "attempts": [ { "status": 200, "full_response_ms": 351.1053330003051, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"10.1\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.91,\"probabilities\":{\"C\":0.06,\"B\":0.0,\"A\":0.0,\"D\":0.9400000000000001}},\"10.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"C\":0.0,\"A\":1.0,\"D\":0.0}},\"10.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":1.0,\"A\":0.0,\"D\":0.0}},\"10.4\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"C\":1.0,\"B\":0.0,\"A\":0.0,\"D\":0.0}}},\"usage\":{\"input_tokens\":1008,\"output_tokens\":183}}" } ], "response": { "model": "jev-1.13.0", "answers": { "10.1": { "type": "choice", "choice": "D", "confidence": 0.91, "probabilities": { "C": 0.06, "B": 0.0, "A": 0.0, "D": 0.9400000000000001 } }, "10.2": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "A": 1.0, "D": 0.0 } }, "10.3": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 1.0, "A": 0.0, "D": 0.0 } }, "10.4": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "C": 1.0, "B": 0.0, "A": 0.0, "D": 0.0 } } }, "usage": { "input_tokens": 1008, "output_tokens": 183 } } }, { "exam": "midterm_fa24", "section": 11, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "11.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: 7^26 modulo 10?", "criteria": { "A": "1", "B": "9", "C": "3", "D": "7" } }, "11.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Inverse of 7 modulo 68 in 0..67?", "criteria": { "A": "29", "B": "39", "C": "10", "D": "61" } }, "11.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct integers x,y are congruent modulo positive m and n, gcd(m,n)=d. Sharp lower bound on |x−y|?", "criteria": { "A": "mn", "B": "m+n", "C": "mn/d", "D": "d" } }, "11.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of x in 0..504 solving 10x=5 mod 505?", "criteria": { "A": "1", "B": "5", "C": "10", "D": "0" } }, "11.5a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For valid textbook RSA encryption E and decryption D, D(E(x))=E(D(x)) modulo N for every message x.", "criteria": { "A": "True", "B": "False" } }, "11.5b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Given y=E(E(x)) in valid RSA, which expression recovers x?", "criteria": { "A": "D(D(y))", "B": "E(E(y))", "C": "E(D(y))", "D": "D(y)" } }, "11.5c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: In valid textbook RSA, a=E(x), b=E(y). What is D(ab) modulo N?", "criteria": { "A": "xy", "B": "x^y", "C": "x+y", "D": "x/y" } } } }, "attempts": [ { "status": 200, "full_response_ms": 466.0425000001851, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"11.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.44,\"probabilities\":{\"D\":0.18,\"A\":0.08,\"B\":0.59,\"C\":0.15}},\"11.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.66,\"probabilities\":{\"D\":0.03,\"A\":0.21,\"B\":0.75,\"C\":0.01}},\"11.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.85,\"probabilities\":{\"D\":0.1,\"A\":0.01,\"B\":0.0,\"C\":0.89}},\"11.4\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.51,\"probabilities\":{\"D\":0.13,\"A\":0.64,\"B\":0.18,\"C\":0.05}},\"11.5a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.3,\"probabilities\":{\"A\":0.35,\"B\":0.65}},\"11.5b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.95,\"probabilities\":{\"D\":0.02,\"A\":0.96,\"B\":0.0,\"C\":0.02}},\"11.5c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"D\":0.0,\"A\":0.99,\"B\":0.01,\"C\":0.0}}},\"usage\":{\"input_tokens\":1076,\"output_tokens\":307}}" } ], "response": { "model": "jev-1.13.0", "answers": { "11.1": { "type": "choice", "choice": "B", "confidence": 0.44, "probabilities": { "D": 0.18, "A": 0.08, "B": 0.59, "C": 0.15 } }, "11.2": { "type": "choice", "choice": "B", "confidence": 0.66, "probabilities": { "D": 0.03, "A": 0.21, "B": 0.75, "C": 0.01 } }, "11.3": { "type": "choice", "choice": "C", "confidence": 0.85, "probabilities": { "D": 0.1, "A": 0.01, "B": 0.0, "C": 0.89 } }, "11.4": { "type": "choice", "choice": "A", "confidence": 0.51, "probabilities": { "D": 0.13, "A": 0.64, "B": 0.18, "C": 0.05 } }, "11.5a": { "type": "choice", "choice": "B", "confidence": 0.3, "probabilities": { "A": 0.35, "B": 0.65 } }, "11.5b": { "type": "choice", "choice": "A", "confidence": 0.95, "probabilities": { "D": 0.02, "A": 0.96, "B": 0.0, "C": 0.02 } }, "11.5c": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "D": 0.0, "A": 0.99, "B": 0.01, "C": 0.0 } } }, "usage": { "input_tokens": 1076, "output_tokens": 307 } } }, { "exam": "midterm_fa24", "section": 12, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "12.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Over GF(5), give a degree-at-most-2 polynomial through (0,1),(1,0),(3,0).", "criteria": { "A": "2(x−1)(x−3)", "B": "3(x−1)(x−3)", "C": "(x−1)(x−3)", "D": "2(x+1)(x+3)" } }, "12.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Over GF(5), ax²+bx+c passes through (0,0),(1,1),(2,4). Give (a,b,c).", "criteria": { "A": "(1,1,0)", "B": "(2,4,0)", "C": "(1,0,0)", "D": "(0,1,0)" } }, "12.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Over GF(p), p>=5, four points have distinct x-coordinates. Number of degree-at-most-4 polynomials through them?", "criteria": { "A": "p²", "B": "1", "C": "p", "D": "p−1" } }, "12.4a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Polynomial division over a field: P=QD+R, deg P=d, deg D=d′, with 1<=d′=max(s+1,6)", "B": "p>=s", "C": "p>=3", "D": "p>=max(s,5)" } }, "13.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P(x)=2x²+3x+3 over GF(5); evaluations at 0,1,2,3,4 are sent and only the value at x=1 is corrupted. Monic E(x)=x+b0 and Q=PE.\nProblem: What is b0 in 0..4?", "criteria": { "A": "1", "B": "0", "C": "4", "D": "2" } }, "13.2bi": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P(x)=2x²+3x+3 over GF(5); evaluations at 0,1,2,3,4 are sent and only the value at x=1 is corrupted. Monic E(x)=x+b0 and Q=PE.\nProblem: Degree d of Q?", "criteria": { "A": "3", "B": "4", "C": "2", "D": "1" } }, "13.2bii": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P(x)=2x²+3x+3 over GF(5); evaluations at 0,1,2,3,4 are sent and only the value at x=1 is corrupted. Monic E(x)=x+b0 and Q=PE.\nProblem: Leading coefficient qd of Q?", "criteria": { "A": "1", "B": "4", "C": "3", "D": "2" } }, "13.2biii": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P(x)=2x²+3x+3 over GF(5); evaluations at 0,1,2,3,4 are sent and only the value at x=1 is corrupted. Monic E(x)=x+b0 and Q=PE.\nProblem: Constant coefficient q0 in terms of b0?", "criteria": { "A": "2b0", "B": "3b0", "C": "b0", "D": "3+b0" } } } }, "attempts": [ { "status": 200, "full_response_ms": 394.3962499979534, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"13.1a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.3,\"probabilities\":{\"B\":0.47,\"D\":0.1,\"A\":0.42,\"C\":0.01}},\"13.1b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.48,\"probabilities\":{\"B\":0.04,\"D\":0.34,\"A\":0.61,\"C\":0.01}},\"13.1c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.79,\"probabilities\":{\"B\":0.02,\"D\":0.11,\"A\":0.85,\"C\":0.02}},\"13.2a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.11,\"probabilities\":{\"B\":0.14,\"D\":0.28,\"A\":0.25,\"C\":0.33}},\"13.2bi\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"B\":0.02,\"D\":0.0,\"C\":0.0,\"A\":0.98}},\"13.2bii\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.76,\"probabilities\":{\"B\":0.1,\"D\":0.82,\"C\":0.02,\"A\":0.06}},\"13.2biii\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.86,\"probabilities\":{\"B\":0.9,\"D\":0.05,\"C\":0.02,\"A\":0.03}}},\"usage\":{\"input_tokens\":1370,\"output_tokens\":327}}" } ], "response": { "model": "jev-1.13.0", "answers": { "13.1a": { "type": "choice", "choice": "B", "confidence": 0.3, "probabilities": { "B": 0.47, "D": 0.1, "A": 0.42, "C": 0.01 } }, "13.1b": { "type": "choice", "choice": "A", "confidence": 0.48, "probabilities": { "B": 0.04, "D": 0.34, "A": 0.61, "C": 0.01 } }, "13.1c": { "type": "choice", "choice": "A", "confidence": 0.79, "probabilities": { "B": 0.02, "D": 0.11, "A": 0.85, "C": 0.02 } }, "13.2a": { "type": "choice", "choice": "C", "confidence": 0.11, "probabilities": { "B": 0.14, "D": 0.28, "A": 0.25, "C": 0.33 } }, "13.2bi": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "B": 0.02, "D": 0.0, "C": 0.0, "A": 0.98 } }, "13.2bii": { "type": "choice", "choice": "D", "confidence": 0.76, "probabilities": { "B": 0.1, "D": 0.82, "C": 0.02, "A": 0.06 } }, "13.2biii": { "type": "choice", "choice": "B", "confidence": 0.86, "probabilities": { "B": 0.9, "D": 0.05, "C": 0.02, "A": 0.03 } } }, "usage": { "input_tokens": 1370, "output_tokens": 327 } } }, { "exam": "midterm_fa24", "section": 14, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "14.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of formal polynomials of degree exactly d over GF(p)?", "criteria": { "A": "(p−1)p^d", "B": "(p−1)^(d+1)", "C": "p^d", "D": "p^(d+1)" } }, "14.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Every degree-exactly-one polynomial over a field has exactly one root.", "criteria": { "A": "False", "B": "True" } }, "14.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Choose a quadratic over GF(p) having x=1 as its only distinct root, for every prime p.", "criteria": { "A": "x²−1", "B": "(x−1)²", "C": "x²+1", "D": "x(x−1)" } }, "14.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of distinct formal polynomials a(x−r)² over GF(p), a nonzero?", "criteria": { "A": "(p−1)C(p,2)", "B": "(p−1)p", "C": "p−1", "D": "p²" } }, "14.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of a(x−r)(x−s), a nonzero and r≠s, over GF(p)?", "criteria": { "A": "C(p,2)", "B": "p²", "C": "(p−1)p(p−1)", "D": "(p−1)C(p,2)" } }, "14.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of degree-exactly-two polynomials over GF(p) with no field roots?", "criteria": { "A": "(p−1)(p²−C(p,2)−p)", "B": "(p−1)C(p,2)−p", "C": "(p−1)(p²−C(p,2))", "D": "(p−1)p²" } }, "14.7": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For fixed distinct field elements r1,...,rk, count polynomials product (x−ri)^bi with each bi>=1 and sum bi=d>=k.", "criteria": { "A": "C(d,k)", "B": "k^d", "C": "C(d+k−1,k−1)", "D": "C(d−1,k−1)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 403.18283399756183, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"14.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"C\":0.01,\"A\":0.99,\"D\":0.0}},\"14.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.73,\"probabilities\":{\"B\":0.86,\"A\":0.14}},\"14.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"B\":0.99,\"C\":0.0,\"A\":0.01,\"D\":0.0}},\"14.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.7,\"probabilities\":{\"B\":0.78,\"C\":0.04,\"A\":0.17,\"D\":0.01}},\"14.5\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.79,\"probabilities\":{\"B\":0.01,\"C\":0.14,\"A\":0.01,\"D\":0.84}},\"14.6\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.25,\"probabilities\":{\"B\":0.2,\"C\":0.31,\"A\":0.45,\"D\":0.04}},\"14.7\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.95,\"probabilities\":{\"B\":0.0,\"C\":0.01,\"A\":0.03,\"D\":0.96}}},\"usage\":{\"input_tokens\":1182,\"output_tokens\":304}}" } ], "response": { "model": "jev-1.13.0", "answers": { "14.1": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "B": 0.0, "C": 0.01, "A": 0.99, "D": 0.0 } }, "14.2": { "type": "choice", "choice": "B", "confidence": 0.73, "probabilities": { "B": 0.86, "A": 0.14 } }, "14.3": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "B": 0.99, "C": 0.0, "A": 0.01, "D": 0.0 } }, "14.4": { "type": "choice", "choice": "B", "confidence": 0.7, "probabilities": { "B": 0.78, "C": 0.04, "A": 0.17, "D": 0.01 } }, "14.5": { "type": "choice", "choice": "D", "confidence": 0.79, "probabilities": { "B": 0.01, "C": 0.14, "A": 0.01, "D": 0.84 } }, "14.6": { "type": "choice", "choice": "A", "confidence": 0.25, "probabilities": { "B": 0.2, "C": 0.31, "A": 0.45, "D": 0.04 } }, "14.7": { "type": "choice", "choice": "D", "confidence": 0.95, "probabilities": { "B": 0.0, "C": 0.01, "A": 0.03, "D": 0.96 } } }, "usage": { "input_tokens": 1182, "output_tokens": 304 } } }, { "exam": "midterm_fa24", "section": 15, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "15.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of unordered 13-card hands from 52 cards?", "criteria": { "A": "C(64,13)", "B": "C(52,13)", "C": "52!/39!", "D": "52^13" } }, "15.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Deal all 52 distinct cards to four named players, 13 each, unordered within each hand. Number of deals?", "criteria": { "A": "C(52,13)", "B": "52!/(4!*(13!)^4)", "C": "52!/(13!)^4", "D": "4^52" } }, "15.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Count integer k-tuples with xi>=−1 and sum xi=n, n>=0, k>=1.", "criteria": { "A": "C(n+2k−1,k−1)", "B": "C(n+k−1,k−1)", "C": "k^n", "D": "C(n−1,k−1)" } }, "15.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of anagrams of CANTSTANFURD?", "criteria": { "A": "12!/2!", "B": "11!/(2!2!2!)", "C": "12!/(3!3!)", "D": "12!/(2!2!2!)" } }, "15.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Let x be the number of anagrams of CANTSTANFURD. How many put C before S before D, without an adjacency requirement?", "criteria": { "A": "x/2", "B": "x/12", "C": "x/6", "D": "x/3" } }, "15.3c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Let x be the number of anagrams of CANTSTANFURD. How many put both As before S, without an adjacency requirement?", "criteria": { "A": "x/2", "B": "2x/3", "C": "x/6", "D": "x/3" } }, "15.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of length-n bitstrings with k ones?", "criteria": { "A": "C(n,k)", "B": "2^k", "C": "n!/k!", "D": "n^k" } }, "15.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distribute five identical dollar bills among four named people, allowing zero. Number of allocations?", "criteria": { "A": "C(5,4)", "B": "C(8,3)", "C": "5^4", "D": "4^5" } } } }, "attempts": [ { "status": 200, "full_response_ms": 391.0184999986086, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"15.1a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"A\":0.0,\"B\":1.0,\"C\":0.0,\"D\":0.0}},\"15.1b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.85,\"probabilities\":{\"A\":0.0,\"B\":0.11,\"C\":0.89,\"D\":0.0}},\"15.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.86,\"probabilities\":{\"A\":0.9,\"B\":0.1,\"C\":0.0,\"D\":0.0}},\"15.3a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.69,\"probabilities\":{\"A\":0.16,\"B\":0.06,\"C\":0.01,\"D\":0.77}},\"15.3b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.92,\"probabilities\":{\"A\":0.0,\"B\":0.04,\"C\":0.94,\"D\":0.02}},\"15.3c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.31,\"probabilities\":{\"A\":0.48,\"B\":0.06,\"C\":0.14,\"D\":0.32}},\"15.4\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"A\":1.0,\"B\":0.0,\"C\":0.0,\"D\":0.0}},\"15.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"A\":0.0,\"B\":1.0,\"C\":0.0,\"D\":0.0}}},\"usage\":{\"input_tokens\":1334,\"output_tokens\":368}}" } ], "response": { "model": "jev-1.13.0", "answers": { "15.1a": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "A": 0.0, "B": 1.0, "C": 0.0, "D": 0.0 } }, "15.1b": { "type": "choice", "choice": "C", "confidence": 0.85, "probabilities": { "A": 0.0, "B": 0.11, "C": 0.89, "D": 0.0 } }, "15.2": { "type": "choice", "choice": "A", "confidence": 0.86, "probabilities": { "A": 0.9, "B": 0.1, "C": 0.0, "D": 0.0 } }, "15.3a": { "type": "choice", "choice": "D", "confidence": 0.69, "probabilities": { "A": 0.16, "B": 0.06, "C": 0.01, "D": 0.77 } }, "15.3b": { "type": "choice", "choice": "C", "confidence": 0.92, "probabilities": { "A": 0.0, "B": 0.04, "C": 0.94, "D": 0.02 } }, "15.3c": { "type": "choice", "choice": "A", "confidence": 0.31, "probabilities": { "A": 0.48, "B": 0.06, "C": 0.14, "D": 0.32 } }, "15.4": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "A": 1.0, "B": 0.0, "C": 0.0, "D": 0.0 } }, "15.5": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "A": 0.0, "B": 1.0, "C": 0.0, "D": 0.0 } } }, "usage": { "input_tokens": 1334, "output_tokens": 368 } } }, { "exam": "midterm_fa25", "section": 1, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "1.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If 6 is prime, then trees can walk.", "criteria": { "A": "True", "B": "False" } }, "1.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Is NOT(P implies Q) equivalent to (NOT P) AND (NOT Q)?", "criteria": { "A": "False", "B": "True" } }, "1.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For integers a,m,n, prove: if a does not divide mn, then a divides neither m nor n. Which argument is valid?", "criteria": { "A": "Contrapositive: if m=ak or n=ak for an integer k, then respectively mn=a(kn) or mn=a(km), so a divides mn.", "B": "If a does not divide m or n, their product cannot be divisible by a.", "C": "If a divides mn, it must divide both m and n; take the converse.", "D": "Since a does not divide mn, neither m nor n can have any divisor in common with a." } } } }, "attempts": [ { "status": 200, "full_response_ms": 370.6732499995269, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"1.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.51,\"probabilities\":{\"A\":0.76,\"B\":0.24}},\"1.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"A\":0.99,\"B\":0.01}},\"1.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":1.0,\"B\":0.0,\"C\":0.0}}},\"usage\":{\"input_tokens\":678,\"output_tokens\":107}}" } ], "response": { "model": "jev-1.13.0", "answers": { "1.1": { "type": "choice", "choice": "A", "confidence": 0.51, "probabilities": { "A": 0.76, "B": 0.24 } }, "1.2": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "A": 0.99, "B": 0.01 } }, "1.3": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 1.0, "B": 0.0, "C": 0.0 } } }, "usage": { "input_tokens": 678, "output_tokens": 107 } } }, { "exam": "midterm_fa25", "section": 2, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "2.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: J means jumping, G on ground, M in motion, H healing, W touching a wall.\nProblem: Express: Bloof is not healing unless it is not in motion.", "criteria": { "A": "NOT M implies NOT H", "B": "M implies NOT H", "C": "NOT H implies M", "D": "H implies M" } }, "2.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: J means jumping, G on ground, M in motion, H healing, W touching a wall.\nProblem: Give the contrapositive of J implies (NOT G AND M).", "criteria": { "A": "(G OR NOT M) implies NOT J", "B": "(NOT G AND M) implies J", "C": "(G AND NOT M) implies NOT J", "D": "NOT J implies (G OR NOT M)" } }, "2.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: J means jumping, G on ground, M in motion, H healing, W touching a wall. Rules: H implies NOT M; J implies (NOT G AND M); NOT(G AND W); NOT G implies (M OR J).\nProblem: H is true. What can be concluded about W?", "criteria": { "A": "True", "B": "Could be either", "C": "False" } } } }, "attempts": [ { "status": 200, "full_response_ms": 385.4925830019056, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"2.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.98,\"probabilities\":{\"B\":0.98,\"D\":0.01,\"A\":0.0,\"C\":0.01}},\"2.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":1.0,\"C\":0.0}},\"2.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.88,\"probabilities\":{\"B\":0.92,\"A\":0.02,\"C\":0.06}}},\"usage\":{\"input_tokens\":741,\"output_tokens\":128}}" } ], "response": { "model": "jev-1.13.0", "answers": { "2.1": { "type": "choice", "choice": "B", "confidence": 0.98, "probabilities": { "B": 0.98, "D": 0.01, "A": 0.0, "C": 0.01 } }, "2.2": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 1.0, "C": 0.0 } }, "2.3": { "type": "choice", "choice": "B", "confidence": 0.88, "probabilities": { "B": 0.92, "A": 0.02, "C": 0.06 } } }, "usage": { "input_tokens": 741, "output_tokens": 128 } } }, { "exam": "midterm_fa25", "section": 3, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "3.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Which proves that an integer n is odd iff 5n+3 is even?", "criteria": { "A": "If n=2k+1, then 5n+3 is even; this alone proves both implications.", "B": "For n=2k+1, 5n+3=2(5k+4). Conversely, if n=2k then 5n+3=2(5k+1)+1 is odd, proving the reverse implication by contrapositive.", "C": "An odd integer plus 3 is always even, regardless of the factor 5; therefore every integer n is odd.", "D": "If 5n+3=2k, then n=2(k-3)/5 is even, so n is odd." } }, "3.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prove 7 divides A_n=2^(n+2)+3^(2n+1), for integers n>=0. Which induction works?", "criteria": { "A": "A_0=7. If 7 divides A_k, then A_(k+1)=2A_k+7*3^(2k+1), so 7 also divides A_(k+1).", "B": "A_0=7 and A_(k+1)=9A_k+7, so divisibility follows.", "C": "Check n=0 and n=1; two successful cases establish every n.", "D": "A_0=7 and A_(k+1)=2A_k, so divisibility follows." } } } }, "attempts": [ { "status": 200, "full_response_ms": 341.8095839988382, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"3.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"B\":1.0,\"C\":0.0,\"D\":0.0,\"A\":0.0}},\"3.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.93,\"probabilities\":{\"B\":0.04,\"C\":0.0,\"D\":0.0,\"A\":0.96}}},\"usage\":{\"input_tokens\":777,\"output_tokens\":91}}" } ], "response": { "model": "jev-1.13.0", "answers": { "3.1": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "B": 1.0, "C": 0.0, "D": 0.0, "A": 0.0 } }, "3.2": { "type": "choice", "choice": "A", "confidence": 0.93, "probabilities": { "B": 0.04, "C": 0.0, "D": 0.0, "A": 0.96 } } }, "usage": { "input_tokens": 777, "output_tokens": 91 } } }, { "exam": "midterm_fa25", "section": 4, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "4.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Jobs prefer A:2>3>1, B:2>3>1, C:1>3>2. Candidates prefer 1:A>B>C, 2:A>C>B, 3:C>A>B.\nProblem: Is matching A–1, B–2, C–3 stable? Include a valid reason.", "criteria": { "A": "No; C and 2 form a blocking pair.", "B": "No; A and 2 form a blocking pair.", "C": "Yes; every candidate has a different job.", "D": "Yes; candidate 3 has its top choice." } }, "4.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Jobs prefer A:2>3>1, B:2>3>1, C:1>3>2. Candidates prefer 1:A>B>C, 2:A>C>B, 3:C>A>B.\nProblem: Which matching is stable?", "criteria": { "A": "A–1, B–3, C–2", "B": "A–3, B–2, C–1", "C": "A–1, B–2, C–3", "D": "A–2, B–1, C–3" } }, "4.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: In a stable matching, it is impossible for two partners to be each other’s least preferred option.", "criteria": { "A": "True", "B": "False" } }, "4.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If a job has the same partner in every stable matching, that partner must be its first choice.", "criteria": { "A": "True", "B": "False" } }, "4.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Job-proposing and candidate-proposing deferred acceptance produce the same matching. How many stable matchings exist?", "criteria": { "A": "Exactly one", "B": "None", "C": "Cannot be determined", "D": "More than one" } } } }, "attempts": [ { "status": 200, "full_response_ms": 326.9495829990774, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"4.1a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.53,\"probabilities\":{\"C\":0.02,\"A\":0.3,\"B\":0.64,\"D\":0.04}},\"4.1b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.3,\"probabilities\":{\"C\":0.48,\"A\":0.23,\"B\":0.25,\"D\":0.04}},\"4.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.11,\"probabilities\":{\"A\":0.55,\"B\":0.45}},\"4.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.13,\"probabilities\":{\"A\":0.43,\"B\":0.57}},\"4.4\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.84,\"probabilities\":{\"C\":0.12,\"A\":0.88,\"B\":0.0,\"D\":0.0}}},\"usage\":{\"input_tokens\":971,\"output_tokens\":197}}" } ], "response": { "model": "jev-1.13.0", "answers": { "4.1a": { "type": "choice", "choice": "B", "confidence": 0.53, "probabilities": { "C": 0.02, "A": 0.3, "B": 0.64, "D": 0.04 } }, "4.1b": { "type": "choice", "choice": "C", "confidence": 0.3, "probabilities": { "C": 0.48, "A": 0.23, "B": 0.25, "D": 0.04 } }, "4.2": { "type": "choice", "choice": "A", "confidence": 0.11, "probabilities": { "A": 0.55, "B": 0.45 } }, "4.3": { "type": "choice", "choice": "B", "confidence": 0.13, "probabilities": { "A": 0.43, "B": 0.57 } }, "4.4": { "type": "choice", "choice": "A", "confidence": 0.84, "probabilities": { "C": 0.12, "A": 0.88, "B": 0.0, "D": 0.0 } } }, "usage": { "input_tokens": 971, "output_tokens": 197 } } }, { "exam": "midterm_fa25", "section": 5, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "5.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: K4 has an Eulerian tour.", "criteria": { "A": "False", "B": "True" } }, "5.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: K6 is planar.", "criteria": { "A": "False", "B": "True" } }, "5.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A graph has vertex degrees 4,3,3,2,1,1. How many edges?", "criteria": { "A": "8", "B": "14", "C": "6", "D": "7" } }, "5.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: In the 5-dimensional hypercube, what is the distance between 10010 and 01000?", "criteria": { "A": "4", "B": "3", "C": "5", "D": "2" } } } }, "attempts": [ { "status": 200, "full_response_ms": 296.81454099772964, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"5.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.67,\"probabilities\":{\"B\":0.16,\"A\":0.84}},\"5.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"A\":1.0}},\"5.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.44,\"probabilities\":{\"B\":0.0,\"C\":0.59,\"D\":0.34,\"A\":0.07}},\"5.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.62,\"probabilities\":{\"B\":0.71,\"C\":0.0,\"D\":0.09,\"A\":0.2}}},\"usage\":{\"input_tokens\":690,\"output_tokens\":151}}" } ], "response": { "model": "jev-1.13.0", "answers": { "5.1": { "type": "choice", "choice": "A", "confidence": 0.67, "probabilities": { "B": 0.16, "A": 0.84 } }, "5.2": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "B": 0.0, "A": 1.0 } }, "5.3": { "type": "choice", "choice": "C", "confidence": 0.44, "probabilities": { "B": 0.0, "C": 0.59, "D": 0.34, "A": 0.07 } }, "5.4": { "type": "choice", "choice": "B", "confidence": 0.62, "probabilities": { "B": 0.71, "C": 0.0, "D": 0.09, "A": 0.2 } } }, "usage": { "input_tokens": 690, "output_tokens": 151 } } }, { "exam": "midterm_fa25", "section": 6, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "6.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: The undirected train graph has exactly these edges: Oakland–Berkeley, Berkeley–Richmond, Oakland–Orinda, Orinda–Walnut Creek, Oakland–Market Street, Market Street–Mission Street, Mission Street–Daly City, Oakland–Coliseum, Coliseum–Fremont.\nProblem: Is this graph a tree?", "criteria": { "A": "No", "B": "Yes" } }, "6.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: The undirected train graph has exactly these edges: Oakland–Berkeley, Berkeley–Richmond, Oakland–Orinda, Orinda–Walnut Creek, Oakland–Market Street, Market Street–Mission Street, Mission Street–Daly City, Oakland–Coliseum, Coliseum–Fremont.\nProblem: Is this graph bipartite?", "criteria": { "A": "No", "B": "Yes" } }, "6.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: The undirected train graph has exactly these edges: Oakland–Berkeley, Berkeley–Richmond, Oakland–Orinda, Orinda–Walnut Creek, Oakland–Market Street, Market Street–Mission Street, Mission Street–Daly City, Oakland–Coliseum, Coliseum–Fremont.\nProblem: What is the minimum number of colors for a proper edge coloring?", "criteria": { "A": "3", "B": "5", "C": "4", "D": "2" } }, "6.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: The undirected train graph has exactly these edges: Oakland–Berkeley, Berkeley–Richmond, Oakland–Orinda, Orinda–Walnut Creek, Oakland–Market Street, Market Street–Mission Street, Mission Street–Daly City, Oakland–Coliseum, Coliseum–Fremont.\nProblem: Does the graph have an Eulerian walk? If not, choose an edge to add that creates one.", "criteria": { "A": "No; add Berkeley–Coliseum.", "B": "Yes; no added edge needed.", "C": "No; add Oakland–Daly City.", "D": "No; add Walnut Creek–Fremont." } }, "6.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: The undirected train graph has exactly these edges: Oakland–Berkeley, Berkeley–Richmond, Oakland–Orinda, Orinda–Walnut Creek, Oakland–Market Street, Market Street–Mission Street, Mission Street–Daly City, Oakland–Coliseum, Coliseum–Fremont.\nProblem: Does the graph have an Eulerian tour? If not, choose two edges that create one.", "criteria": { "A": "No; add Walnut Creek–Fremont and Daly City–Richmond.", "B": "No; add Oakland–Fremont and Oakland–Richmond.", "C": "Yes; no added edges needed.", "D": "No; add Berkeley–Coliseum and Orinda–Mission Street." } }, "6.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: The undirected train graph has exactly these edges: Oakland–Berkeley, Berkeley–Richmond, Oakland–Orinda, Orinda–Walnut Creek, Oakland–Market Street, Market Street–Mission Street, Mission Street–Daly City, Oakland–Coliseum, Coliseum–Fremont.\nProblem: A new OAK station and edge OAK–Coliseum are added. Further added edges cannot touch OAK. If an Eulerian walk is then possible, what must hold?", "criteria": { "A": "OAK must be an interior vertex, since it is newly added.", "B": "OAK can be anywhere on a closed Eulerian tour.", "C": "OAK must be visited twice, since it has degree one.", "D": "OAK is an endpoint, because its only incident edge cannot be used to both enter and leave without repetition." } } } }, "attempts": [ { "status": 200, "full_response_ms": 561.8327080010204, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"6.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.51,\"probabilities\":{\"A\":0.24,\"B\":0.76}},\"6.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.23,\"probabilities\":{\"A\":0.39,\"B\":0.61}},\"6.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.2,\"probabilities\":{\"D\":0.01,\"A\":0.25,\"B\":0.34,\"C\":0.4}},\"6.4\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.07,\"probabilities\":{\"D\":0.21,\"A\":0.24,\"B\":0.25,\"C\":0.3}},\"6.5\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.46,\"probabilities\":{\"D\":0.12,\"A\":0.59,\"B\":0.23,\"C\":0.06}},\"6.6\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.93,\"probabilities\":{\"D\":0.95,\"A\":0.01,\"B\":0.02,\"C\":0.02}}},\"usage\":{\"input_tokens\":1456,\"output_tokens\":239}}" } ], "response": { "model": "jev-1.13.0", "answers": { "6.1": { "type": "choice", "choice": "B", "confidence": 0.51, "probabilities": { "A": 0.24, "B": 0.76 } }, "6.2": { "type": "choice", "choice": "B", "confidence": 0.23, "probabilities": { "A": 0.39, "B": 0.61 } }, "6.3": { "type": "choice", "choice": "C", "confidence": 0.2, "probabilities": { "D": 0.01, "A": 0.25, "B": 0.34, "C": 0.4 } }, "6.4": { "type": "choice", "choice": "C", "confidence": 0.07, "probabilities": { "D": 0.21, "A": 0.24, "B": 0.25, "C": 0.3 } }, "6.5": { "type": "choice", "choice": "A", "confidence": 0.46, "probabilities": { "D": 0.12, "A": 0.59, "B": 0.23, "C": 0.06 } }, "6.6": { "type": "choice", "choice": "D", "confidence": 0.93, "probabilities": { "D": 0.95, "A": 0.01, "B": 0.02, "C": 0.02 } } }, "usage": { "input_tokens": 1456, "output_tokens": 239 } } }, { "exam": "midterm_fa25", "section": 7, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "7.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Which induction proves the sum of vertex degrees is twice the edge count?", "criteria": { "A": "Induct on vertices. A graph on k vertices has degree sum 2k. Adding any vertex adds exactly two to that sum.", "B": "Removing a vertex always removes one edge and decreases the degree sum by two.", "C": "Every edge touches two vertices, so each vertex has degree two and the sum is twice the vertex count.", "D": "Induct on edges. With zero edges the sum is zero. Remove one edge from a graph with k+1 edges; the remaining degree sum is 2k. Restoring the edge adds one degree at each endpoint, yielding 2k+2." } } } }, "attempts": [ { "status": 200, "full_response_ms": 433.6140000013984, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"7.1\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"A\":0.0,\"C\":0.0,\"B\":0.0}}},\"usage\":{\"input_tokens\":540,\"output_tokens\":47}}" } ], "response": { "model": "jev-1.13.0", "answers": { "7.1": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "A": 0.0, "C": 0.0, "B": 0.0 } } }, "usage": { "input_tokens": 540, "output_tokens": 47 } } }, { "exam": "midterm_fa25", "section": 8, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "8.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For every positive integer n, 5 divides 6^n−1.", "criteria": { "A": "False", "B": "True" } }, "8.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If xy=0 modulo 6, then x=0 or y=0 modulo 6.", "criteria": { "A": "False", "B": "True" } }, "8.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For every positive odd m, 9 divides 4^m+5^m.", "criteria": { "A": "True", "B": "False" } }, "8.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If p>3 and both p and p+2 are prime, what is p modulo 3?", "criteria": { "A": "0", "B": "2", "C": "1", "D": "Not determined" } }, "8.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: What is the last decimal digit of 9^68?", "criteria": { "A": "1", "B": "8", "C": "6", "D": "9" } }, "8.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: RSA has p=5, q=7 and public exponent e=5. Which is a valid decryption exponent d?", "criteria": { "A": "24", "B": "7", "C": "12", "D": "5" } }, "8.7": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: In valid textbook RSA, a=x^e mod N and b=y^e mod N. Does decrypting ab recover xy modulo N? Choose the justification.", "criteria": { "A": "No; RSA only preserves multiplication when x=y.", "B": "Yes; (ab)^d=a^d b^d=xy mod N.", "C": "Yes; because ed=1 as an equality of integers.", "D": "No; decryption recovers x+y because ciphertext multiplication becomes addition." } } } }, "attempts": [ { "status": 200, "full_response_ms": 392.3504590020457, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"8.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.14,\"probabilities\":{\"A\":0.57,\"B\":0.43}},\"8.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"A\":0.99,\"B\":0.01}},\"8.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.64,\"probabilities\":{\"A\":0.82,\"B\":0.18}},\"8.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.39,\"probabilities\":{\"C\":0.41,\"A\":0.04,\"D\":0.01,\"B\":0.54}},\"8.5\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.86,\"probabilities\":{\"C\":0.05,\"A\":0.89,\"D\":0.06,\"B\":0.0}},\"8.6\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.66,\"probabilities\":{\"C\":0.07,\"A\":0.09,\"D\":0.75,\"B\":0.09}},\"8.7\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"C\":0.01,\"A\":0.0,\"D\":0.0,\"B\":0.99}}},\"usage\":{\"input_tokens\":1061,\"output_tokens\":269}}" } ], "response": { "model": "jev-1.13.0", "answers": { "8.1": { "type": "choice", "choice": "A", "confidence": 0.14, "probabilities": { "A": 0.57, "B": 0.43 } }, "8.2": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "A": 0.99, "B": 0.01 } }, "8.3": { "type": "choice", "choice": "A", "confidence": 0.64, "probabilities": { "A": 0.82, "B": 0.18 } }, "8.4": { "type": "choice", "choice": "B", "confidence": 0.39, "probabilities": { "C": 0.41, "A": 0.04, "D": 0.01, "B": 0.54 } }, "8.5": { "type": "choice", "choice": "A", "confidence": 0.86, "probabilities": { "C": 0.05, "A": 0.89, "D": 0.06, "B": 0.0 } }, "8.6": { "type": "choice", "choice": "D", "confidence": 0.66, "probabilities": { "C": 0.07, "A": 0.09, "D": 0.75, "B": 0.09 } }, "8.7": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "C": 0.01, "A": 0.0, "D": 0.0, "B": 0.99 } } }, "usage": { "input_tokens": 1061, "output_tokens": 269 } } }, { "exam": "midterm_fa25", "section": 9, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "9.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Positive integers p,q are coprime; c,d are positive. How many residues x modulo cdpq solve cx=ca mod cp and dx=db mod dq?", "criteria": { "A": "cd: cancel the integer factors to get x=a mod p and x=b mod q; CRT gives one class mod pq, which has cd lifts mod cdpq.", "B": "1: CRT always gives one class regardless of the requested modulus.", "C": "gcd(c,d): the two scaling factors determine all compatibility constraints.", "D": "pq: each residue mod cd gives one solution." } } } }, "attempts": [ { "status": 200, "full_response_ms": 383.8550000000396, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"9.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":1.0,\"C\":0.0}}},\"usage\":{\"input_tokens\":515,\"output_tokens\":47}}" } ], "response": { "model": "jev-1.13.0", "answers": { "9.1": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "D": 0.0, "B": 0.0, "A": 1.0, "C": 0.0 } } }, "usage": { "input_tokens": 515, "output_tokens": 47 } } }, { "exam": "midterm_fa25", "section": 10, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "10.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Polynomials f,g over a field have degrees k>j. What is deg(f+g)?", "criteria": { "A": "Could be any value below k", "B": "k+j", "C": "j", "D": "k" } }, "10.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Nonzero polynomials f,g over a field have degrees k,j. What is deg(fg)?", "criteria": { "A": "k−j", "B": "k*j", "C": "k+j", "D": "max(k,j)" } }, "10.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Over GF(5), which degree-at-most-4 polynomial is equal as a function to 4x^84+x^23+2x^5+3?", "criteria": { "A": "4x^4+x^3+2x+3", "B": "4x^4+x^2+2x+3", "C": "4x+x^3+2x+3", "D": "4+x^3+2x+3" } }, "10.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A degree-2 polynomial over GF(7) passes through (0,1),(3,0),(4,0). Which is it?", "criteria": { "A": "4(x−3)(x−4)", "B": "3(x−3)(x−4)", "C": "3(x−2)(x−4)", "D": "(x−3)(x−4)" } }, "10.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: How many formal polynomials of degree at most 4 over GF(5) pass through two given points with distinct x-coordinates?", "criteria": { "A": "5", "B": "625", "C": "125", "D": "25" } } } }, "attempts": [ { "status": 200, "full_response_ms": 311.4189580010134, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"10.1a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.71,\"probabilities\":{\"D\":0.79,\"B\":0.0,\"A\":0.21,\"C\":0.0}},\"10.1b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":0.0,\"C\":1.0}},\"10.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.79,\"probabilities\":{\"D\":0.03,\"A\":0.84,\"B\":0.05,\"C\":0.08}},\"10.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.45,\"probabilities\":{\"D\":0.17,\"B\":0.58,\"A\":0.25,\"C\":0.0}},\"10.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.58,\"probabilities\":{\"D\":0.05,\"B\":0.6799999999999999,\"A\":0.02,\"C\":0.25}}},\"usage\":{\"input_tokens\":983,\"output_tokens\":230}}" } ], "response": { "model": "jev-1.13.0", "answers": { "10.1a": { "type": "choice", "choice": "D", "confidence": 0.71, "probabilities": { "D": 0.79, "B": 0.0, "A": 0.21, "C": 0.0 } }, "10.1b": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.0, "C": 1.0 } }, "10.2": { "type": "choice", "choice": "A", "confidence": 0.79, "probabilities": { "D": 0.03, "A": 0.84, "B": 0.05, "C": 0.08 } }, "10.3": { "type": "choice", "choice": "B", "confidence": 0.45, "probabilities": { "D": 0.17, "B": 0.58, "A": 0.25, "C": 0.0 } }, "10.4": { "type": "choice", "choice": "B", "confidence": 0.58, "probabilities": { "D": 0.05, "B": 0.6799999999999999, "A": 0.02, "C": 0.25 } } }, "usage": { "input_tokens": 983, "output_tokens": 230 } } }, { "exam": "midterm_fa25", "section": 11, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "11.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: There are 2 professors, 4 head TAs, 7 TAs. Allowed coalitions contain both professors, or one professor and at least three heads, or one professor, two heads and at least four TAs. No other coalition should reconstruct. Each share uses a distinct nonzero field coordinate.\nProblem: Choose a valid Shamir configuration (degree; shares per professor, head TA, TA; sufficient prime bound). Secret s is a nonnegative integer.", "criteria": { "A": "(23; 12,3,1; prime p>=max(s+1,44))", "B": "(23; 11,4,1; prime p>=max(s+1,46))", "C": "(22; 12,4,1; prime p>=max(s+1,48))", "D": "(23; 12,4,1; prime p>=max(s+1,48))" } }, "11.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A polynomial of degree 9 is transmitted by evaluations. At most two values are corrupted. How many evaluations suffice for unique recovery?", "criteria": { "A": "12", "B": "13", "C": "18", "D": "14" } } } }, "attempts": [ { "status": 200, "full_response_ms": 349.7750829992583, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"11.1\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.82,\"probabilities\":{\"B\":0.03,\"A\":0.07,\"C\":0.04,\"D\":0.86}},\"11.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.7,\"probabilities\":{\"B\":0.77,\"A\":0.08,\"C\":0.01,\"D\":0.14}}},\"usage\":{\"input_tokens\":711,\"output_tokens\":93}}" } ], "response": { "model": "jev-1.13.0", "answers": { "11.1": { "type": "choice", "choice": "D", "confidence": 0.82, "probabilities": { "B": 0.03, "A": 0.07, "C": 0.04, "D": 0.86 } }, "11.2": { "type": "choice", "choice": "B", "confidence": 0.7, "probabilities": { "B": 0.77, "A": 0.08, "C": 0.01, "D": 0.14 } } }, "usage": { "input_tokens": 711, "output_tokens": 93 } } }, { "exam": "midterm_fa25", "section": 12, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "12.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: How many length-30 bit strings contain exactly 12 ones?", "criteria": { "A": "C(30,12)", "B": "30^12", "C": "2^12", "D": "30!/12!" } }, "12.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: How many formal polynomials of degree exactly d over GF(p), where 0<=d0 integers, prove some nonempty subset has sum divisible by N. Hint: prefix sums.", "criteria": { "A": "Each integer has a distinct residue, so one must be zero.", "B": "The sum of N integers is always divisible by N.", "C": "If a prefix sum is 0 mod N, use that prefix. Otherwise N prefix sums occupy N−1 nonzero residues, so two coincide; their difference is a nonempty block with sum 0 mod N.", "D": "Take the empty subset, whose sum is zero." } } } }, "attempts": [ { "status": 200, "full_response_ms": 424.6948750005686, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"4.1\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":1.0,\"A\":0.0,\"C\":0.0}},\"4.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":0.0,\"C\":1.0}}},\"usage\":{\"input_tokens\":677,\"output_tokens\":91}}" } ], "response": { "model": "jev-1.13.0", "answers": { "4.1": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 1.0, "A": 0.0, "C": 0.0 } }, "4.2": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 } } }, "usage": { "input_tokens": 677, "output_tokens": 91 } } }, { "exam": "midterm_sp23", "section": 5, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "5.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: F1=F2=1, Fj=F(j−1)+F(j−2) for j>=3. To prove every natural m is a sum of distinct nonconsecutive-index Fibonacci numbers, assume the claim for all smaller naturals and choose largest k with Fk<=m, m>=2.\nProblem: Why is m−Fk=Fk+F(k−1)=F(k+1), contradicting maximality of k.", "C": "Because every Fibonacci number is greater than m.", "D": "Because m=Fk by the definition of largest k." } }, "5.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: F1=F2=1, Fj=F(j−1)+F(j−2) for j>=3. To prove every natural m is a sum of distinct nonconsecutive-index Fibonacci numbers, assume the claim for all smaller naturals and choose largest k with Fk<=m, m>=2. It has been established that m−Fk=2 define Rn=sqrt(n) and Rk=sqrt(k*R(k+1)) for k=2 vertices?", "criteria": { "A": "1", "B": "n−1", "C": "2", "D": "0" } }, "8.5a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Maximum edges in a bipartite graph with fixed parts A,B?", "criteria": { "A": "C(|A|+|B|,2)", "B": "|A|*|B|", "C": "min(|A|,|B|)", "D": "|A|+|B|−1" } }, "8.5b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Every graph of maximum degree at most two is bipartite.", "criteria": { "A": "True", "B": "False" } }, "8.5c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Smallest constant D such that every nonempty bipartite planar graph has a vertex of degree at most D?", "criteria": { "A": "4", "B": "3", "C": "5", "D": "2" } }, "8.5d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For the three-dimensional hypercube, choose one side A of a bipartition.", "criteria": { "A": "{001,011,101,111}", "B": "{000,011,110,101}", "C": "{000,001,010,011}", "D": "{000,111}" } }, "8.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Largest possible vertex degree in a simple planar graph on n vertices?", "criteria": { "A": "6", "B": "5", "C": "3n−6", "D": "n−1" } }, "8.7": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: In a simple n-vertex graph, n>=3, what sharp threshold on du+dv guarantees distinct vertices u,v share a neighbor?", "criteria": { "A": "n−1", "B": "n", "C": "2n−2", "D": "n+1" } }, "8.8": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Must a connected planar graph with more than 100 vertices have at least three vertices of degree below six?", "criteria": { "A": "Yes: every vertex of a planar graph has degree below six.", "B": "Yes: if at most two have degree below six, the degree sum is at least 6(n−2)+2=6n−10, exceeding the planar bound 6n−12.", "C": "No: a planar complete graph on 101 vertices is a counterexample.", "D": "No: any long path has fewer than three such vertices." } }, "8.9": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Remove a degree-d vertex v from G. The rest has a given k-coloring. What bound follows when restoring v?", "criteria": { "A": "k+d", "B": "max(k,1+min(d,k))", "C": "min(k,d)", "D": "k−1" } } } }, "attempts": [ { "status": 200, "full_response_ms": 312.59441699876334, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"8.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"A\":0.99,\"B\":0.01}},\"8.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.96,\"probabilities\":{\"D\":0.03,\"A\":0.0,\"C\":0.0,\"B\":0.97}},\"8.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.47,\"probabilities\":{\"D\":0.37,\"A\":0.02,\"C\":0.01,\"B\":0.6}},\"8.4\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"D\":0.0,\"A\":1.0,\"C\":0.0,\"B\":0.0}},\"8.5a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"C\":0.0,\"B\":1.0}},\"8.5b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.9,\"probabilities\":{\"A\":0.05,\"B\":0.95}},\"8.5c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.46,\"probabilities\":{\"D\":0.02,\"A\":0.21,\"C\":0.17,\"B\":0.6}},\"8.5d\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.47,\"probabilities\":{\"D\":0.01,\"A\":0.38,\"C\":0.01,\"B\":0.6}},\"8.6\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.28,\"probabilities\":{\"D\":0.45,\"A\":0.04,\"C\":0.05,\"B\":0.46}},\"8.7\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.49,\"probabilities\":{\"D\":0.05,\"A\":0.61,\"C\":0.07,\"B\":0.27}},\"8.8\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"C\":0.0,\"B\":1.0}},\"8.9\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.83,\"probabilities\":{\"D\":0.01,\"A\":0.11,\"C\":0.01,\"B\":0.87}}},\"usage\":{\"input_tokens\":1742,\"output_tokens\":507}}" } ], "response": { "model": "jev-1.13.0", "answers": { "8.1": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "A": 0.99, "B": 0.01 } }, "8.2": { "type": "choice", "choice": "B", "confidence": 0.96, "probabilities": { "D": 0.03, "A": 0.0, "C": 0.0, "B": 0.97 } }, "8.3": { "type": "choice", "choice": "B", "confidence": 0.47, "probabilities": { "D": 0.37, "A": 0.02, "C": 0.01, "B": 0.6 } }, "8.4": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "D": 0.0, "A": 1.0, "C": 0.0, "B": 0.0 } }, "8.5a": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.0, "B": 1.0 } }, "8.5b": { "type": "choice", "choice": "B", "confidence": 0.9, "probabilities": { "A": 0.05, "B": 0.95 } }, "8.5c": { "type": "choice", "choice": "B", "confidence": 0.46, "probabilities": { "D": 0.02, "A": 0.21, "C": 0.17, "B": 0.6 } }, "8.5d": { "type": "choice", "choice": "B", "confidence": 0.47, "probabilities": { "D": 0.01, "A": 0.38, "C": 0.01, "B": 0.6 } }, "8.6": { "type": "choice", "choice": "B", "confidence": 0.28, "probabilities": { "D": 0.45, "A": 0.04, "C": 0.05, "B": 0.46 } }, "8.7": { "type": "choice", "choice": "A", "confidence": 0.49, "probabilities": { "D": 0.05, "A": 0.61, "C": 0.07, "B": 0.27 } }, "8.8": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.0, "B": 1.0 } }, "8.9": { "type": "choice", "choice": "B", "confidence": 0.83, "probabilities": { "D": 0.01, "A": 0.11, "C": 0.01, "B": 0.87 } } }, "usage": { "input_tokens": 1742, "output_tokens": 507 } } }, { "exam": "midterm_sp23", "section": 9, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "9.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Every 3-regular graph has an even number of vertices.", "criteria": { "A": "False", "B": "True" } }, "9.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Justify the parity constraint for a 3-regular graph on n vertices.", "criteria": { "A": "Every vertex can be paired with a unique neighbor, without needing a matching theorem.", "B": "Every 3-regular graph is bipartite with equal parts.", "C": "The handshake lemma gives 3n=2e, so n is even.", "D": "All graphs with odd maximum degree have even order." } }, "9.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Every 4-regular graph has an odd number of vertices.", "criteria": { "A": "False", "B": "True" } }, "9.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Give a counterexample to the claim that every 4-regular graph has an odd number of vertices.", "criteria": { "A": "K5: five vertices, each degree four.", "B": "K6 with the three edges of a perfect matching removed: six vertices, each degree four.", "C": "K6: six vertices, each degree five.", "D": "A six-cycle: six vertices, each degree two." } }, "9.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prove any simple n-vertex graph, n>=2, has two equal degrees.", "criteria": { "A": "The degree sum is even, so every individual degree is even.", "B": "Every simple graph has two leaves.", "C": "Degrees lie in 0..n−1, but 0 and n−1 cannot both occur. At most n−1 possible values remain for n vertices, so pigeonhole applies.", "D": "There are n possible degrees and n vertices, which alone forces a repetition." } } } }, "attempts": [ { "status": 200, "full_response_ms": 287.6444579997042, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"9.1a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.93,\"probabilities\":{\"B\":0.97,\"A\":0.03}},\"9.1b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"B\":0.0,\"C\":1.0}},\"9.2a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.87,\"probabilities\":{\"B\":0.06,\"A\":0.94}},\"9.2b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"B\":1.0,\"C\":0.0}},\"9.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"B\":0.0,\"C\":1.0}}},\"usage\":{\"input_tokens\":964,\"output_tokens\":199}}" } ], "response": { "model": "jev-1.13.0", "answers": { "9.1a": { "type": "choice", "choice": "B", "confidence": 0.93, "probabilities": { "B": 0.97, "A": 0.03 } }, "9.1b": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 0.0, "C": 1.0 } }, "9.2a": { "type": "choice", "choice": "A", "confidence": 0.87, "probabilities": { "B": 0.06, "A": 0.94 } }, "9.2b": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 1.0, "C": 0.0 } }, "9.3": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 0.0, "C": 1.0 } } }, "usage": { "input_tokens": 964, "output_tokens": 199 } } }, { "exam": "midterm_sp23", "section": 10, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "10.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A function is k-uniform when each output has either zero or exactly k distinct preimages.\nProblem: A={0,1,2,3}, B={0,1,2}; f(0)=f(1)=0 and f(2)=f(3)=1. Find k.", "criteria": { "A": "Unknown", "B": "1", "C": "2", "D": "3" } }, "10.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A function is k-uniform when each output has either zero or exactly k distinct preimages.\nProblem: g is k1-uniform and h is k2-uniform. What k is guaranteed for g composed with h?", "criteria": { "A": "k1+k2", "B": "k1*k2", "C": "Unknown; the composition need not be uniform.", "D": "max(k1,k2)" } }, "10.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A function is k-uniform when each output has either zero or exactly k distinct preimages.\nProblem: f(x)=ax mod prime m, a nonzero, domain and codomain all residues. Find k.", "criteria": { "A": "1", "B": "a", "C": "m", "D": "Unknown" } }, "10.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A function is k-uniform when each output has either zero or exactly k distinct preimages.\nProblem: f(x)=ax mod m, gcd(a,m)=d, domain and codomain all residues. Find k.", "criteria": { "A": "Unknown", "B": "1", "C": "d", "D": "m/d" } }, "10.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A function is k-uniform when each output has either zero or exactly k distinct preimages.\nProblem: m=pq for distinct primes; f(x)=ax mod m on the unit residues, gcd(a,m)=1. Find k.", "criteria": { "A": "Unknown", "B": "(p−1)(q−1)", "C": "1", "D": "p−1" } }, "10.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A function is k-uniform when each output has either zero or exactly k distinct preimages.\nProblem: m=pq for distinct primes; f(x)=ax mod m on unit residues, gcd(a,m)=p. Codomain all residues. Find k.", "criteria": { "A": "q−1", "B": "p", "C": "p−1", "D": "1" } } } }, "attempts": [ { "status": 200, "full_response_ms": 459.4066669997119, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"10.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.89,\"probabilities\":{\"D\":0.0,\"A\":0.07,\"C\":0.92,\"B\":0.01}},\"10.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.61,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"C\":0.29,\"B\":0.71}},\"10.3a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"D\":0.0,\"A\":1.0,\"C\":0.0,\"B\":0.0}},\"10.3b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.84,\"probabilities\":{\"D\":0.05,\"C\":0.88,\"A\":0.06,\"B\":0.01}},\"10.4\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.92,\"probabilities\":{\"D\":0.0,\"C\":0.94,\"A\":0.04,\"B\":0.02}},\"10.5\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.09,\"probabilities\":{\"D\":0.3,\"A\":0.32,\"C\":0.3,\"B\":0.08}}},\"usage\":{\"input_tokens\":1164,\"output_tokens\":275}}" } ], "response": { "model": "jev-1.13.0", "answers": { "10.1": { "type": "choice", "choice": "C", "confidence": 0.89, "probabilities": { "D": 0.0, "A": 0.07, "C": 0.92, "B": 0.01 } }, "10.2": { "type": "choice", "choice": "B", "confidence": 0.61, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.29, "B": 0.71 } }, "10.3a": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "D": 0.0, "A": 1.0, "C": 0.0, "B": 0.0 } }, "10.3b": { "type": "choice", "choice": "C", "confidence": 0.84, "probabilities": { "D": 0.05, "C": 0.88, "A": 0.06, "B": 0.01 } }, "10.4": { "type": "choice", "choice": "C", "confidence": 0.92, "probabilities": { "D": 0.0, "C": 0.94, "A": 0.04, "B": 0.02 } }, "10.5": { "type": "choice", "choice": "A", "confidence": 0.09, "probabilities": { "D": 0.3, "A": 0.32, "C": 0.3, "B": 0.08 } } }, "usage": { "input_tokens": 1164, "output_tokens": 275 } } }, { "exam": "midterm_sp23", "section": 11, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "11.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: x=3 mod 5 and x=2 mod 11. Find x mod 55 in 0..54.", "criteria": { "A": "13", "B": "23", "C": "2", "D": "3" } }, "11.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: x=a mod 10 and x=b mod 15, where b−a is divisible by 5. Number of solutions modulo 150?", "criteria": { "A": "0", "B": "5", "C": "1", "D": "10" } }, "11.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: x=a mod 10 and x=b mod 15, where b−a is not divisible by 5. Number of solutions modulo 150?", "criteria": { "A": "0", "B": "5", "C": "1", "D": "15" } }, "11.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For every integer a and prime p, p divides a^p−a.", "criteria": { "A": "False", "B": "True" } }, "11.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For distinct primes p,q and every integer a, choose a guaranteed divisor of a*(a^((p−1)(q−1))−1).", "criteria": { "A": "pq", "B": "p²q", "C": "p+q", "D": "p²q²" } }, "11.5a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: RSA ciphertexts are y1=x1^e and y2=x2^e modulo N. Ciphertext of x1*x2?", "criteria": { "A": "(y1*y2)^d mod N", "B": "y1*y2 mod N", "C": "y1+y2 mod N", "D": "(y1+y2)^e mod N" } }, "11.5b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: In valid RSA, y1=x1^e and y2=x2^e modulo N, with decryption exponent d. Express x1*x2 modulo N.", "criteria": { "A": "(y1*y2)^e mod N", "B": "(y1*y2)^d mod N", "C": "y1*y2 mod N", "D": "(y1+y2)^d mod N" } } } }, "attempts": [ { "status": 200, "full_response_ms": 341.4315000009083, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"11.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.65,\"probabilities\":{\"C\":0.01,\"D\":0.02,\"A\":0.23,\"B\":0.74}},\"11.2a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.16,\"probabilities\":{\"C\":0.37,\"D\":0.33,\"A\":0.02,\"B\":0.28}},\"11.2b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.89,\"probabilities\":{\"C\":0.05,\"D\":0.01,\"A\":0.91,\"B\":0.03}},\"11.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.91,\"probabilities\":{\"A\":0.04,\"B\":0.96}},\"11.4\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"C\":0.0,\"D\":0.0,\"A\":0.99,\"B\":0.01}},\"11.5a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"C\":0.0,\"D\":0.0,\"A\":0.01,\"B\":0.99}},\"11.5b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"A\":0.0,\"D\":0.0,\"B\":1.0}}},\"usage\":{\"input_tokens\":1175,\"output_tokens\":308}}" } ], "response": { "model": "jev-1.13.0", "answers": { "11.1": { "type": "choice", "choice": "B", "confidence": 0.65, "probabilities": { "C": 0.01, "D": 0.02, "A": 0.23, "B": 0.74 } }, "11.2a": { "type": "choice", "choice": "C", "confidence": 0.16, "probabilities": { "C": 0.37, "D": 0.33, "A": 0.02, "B": 0.28 } }, "11.2b": { "type": "choice", "choice": "A", "confidence": 0.89, "probabilities": { "C": 0.05, "D": 0.01, "A": 0.91, "B": 0.03 } }, "11.3": { "type": "choice", "choice": "B", "confidence": 0.91, "probabilities": { "A": 0.04, "B": 0.96 } }, "11.4": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "C": 0.0, "D": 0.0, "A": 0.99, "B": 0.01 } }, "11.5a": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "C": 0.0, "D": 0.0, "A": 0.01, "B": 0.99 } }, "11.5b": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "C": 0.0, "A": 0.0, "D": 0.0, "B": 1.0 } } }, "usage": { "input_tokens": 1175, "output_tokens": 308 } } }, { "exam": "midterm_sp23", "section": 12, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "12.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: p=2^n−1 is a Mersenne prime, n>=2.\nProblem: Least nonnegative residue of 2^n modulo p?", "criteria": { "A": "2", "B": "0", "C": "1", "D": "p−1" } }, "12.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: p=2^n−1 is a Mersenne prime, n>=2.\nProblem: For positive integer a, least nonnegative residue of (2^a)^n modulo p?", "criteria": { "A": "0", "B": "1", "C": "a", "D": "2^a" } }, "12.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: p=2^n−1 is a Mersenne prime, n>=2.\nProblem: Show there are at least n distinct residues x with x^n=1 mod p.", "criteria": { "A": "Use x=2^j for j=0,...,n−1. They are distinct positive integers below p and each nth power is (2^n)^j=1 mod p.", "B": "Use x=2^(jn), j=1,...,n; these are all distinct modulo p.", "C": "A degree-n polynomial must have at least n roots over every field.", "D": "Use all residues 0,...,n−1; every small residue has nth power one." } } } }, "attempts": [ { "status": 200, "full_response_ms": 302.1441249984491, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"12.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.69,\"probabilities\":{\"B\":0.0,\"D\":0.01,\"C\":0.22,\"A\":0.77}},\"12.2\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.48,\"probabilities\":{\"B\":0.34,\"D\":0.61,\"C\":0.02,\"A\":0.03}},\"12.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"C\":0.0,\"A\":1.0}}},\"usage\":{\"input_tokens\":781,\"output_tokens\":138}}" } ], "response": { "model": "jev-1.13.0", "answers": { "12.1": { "type": "choice", "choice": "A", "confidence": 0.69, "probabilities": { "B": 0.0, "D": 0.01, "C": 0.22, "A": 0.77 } }, "12.2": { "type": "choice", "choice": "D", "confidence": 0.48, "probabilities": { "B": 0.34, "D": 0.61, "C": 0.02, "A": 0.03 } }, "12.3": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "C": 0.0, "A": 1.0 } } }, "usage": { "input_tokens": 781, "output_tokens": 138 } } }, { "exam": "midterm_sp23", "section": 13, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "13.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Choose a polynomial over GF(7) through (1,0),(2,5).", "criteria": { "A": "5x+1", "B": "5x+2", "C": "x−1", "D": "2x+5" } }, "13.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Over GF(p), p>d, how many formal polynomials of degree at most d pass through d−1 points with distinct x-coordinates?", "criteria": { "A": "p²", "B": "p^(d−1)", "C": "p", "D": "1" } }, "13.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: The normalized Lagrange basis polynomials Delta1,...,Delta(d+1) correspond to d+1 points with distinct x-coordinates over a sufficiently large field, d>=1.\nProblem: How many distinct roots does Delta1 have?", "criteria": { "A": "d", "B": "d+1", "C": "d−1", "D": "Unknown" } }, "13.3bi": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: The normalized Lagrange basis polynomials Delta1,...,Delta(d+1) correspond to d+1 points with distinct x-coordinates over a sufficiently large field, d>=1.\nProblem: Formal degree of Delta1*Delta2?", "criteria": { "A": "d+1", "B": "2d", "C": "d", "D": "Unknown" } }, "13.3bii": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: The normalized Lagrange basis polynomials Delta1,...,Delta(d+1) correspond to d+1 points with distinct x-coordinates over a sufficiently large field, d>=1.\nProblem: Number of distinct roots of Delta1*Delta2?", "criteria": { "A": "d+1", "B": "2d", "C": "d", "D": "Unknown" } }, "13.3c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Every real polynomial of degree d has exactly d distinct real roots.", "criteria": { "A": "True", "B": "False" } } } }, "attempts": [ { "status": 200, "full_response_ms": 414.7680419991957, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"13.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.79,\"probabilities\":{\"A\":0.1,\"C\":0.01,\"B\":0.84,\"D\":0.05}},\"13.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.81,\"probabilities\":{\"A\":0.86,\"C\":0.09,\"B\":0.05,\"D\":0.0}},\"13.3a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.97,\"probabilities\":{\"A\":0.98,\"C\":0.0,\"B\":0.0,\"D\":0.02}},\"13.3bi\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.78,\"probabilities\":{\"A\":0.02,\"C\":0.13,\"B\":0.83,\"D\":0.02}},\"13.3bii\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.42,\"probabilities\":{\"A\":0.09,\"C\":0.27,\"B\":0.56,\"D\":0.07}},\"13.3c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"A\":0.0,\"B\":1.0}}},\"usage\":{\"input_tokens\":1061,\"output_tokens\":264}}" } ], "response": { "model": "jev-1.13.0", "answers": { "13.1": { "type": "choice", "choice": "B", "confidence": 0.79, "probabilities": { "A": 0.1, "C": 0.01, "B": 0.84, "D": 0.05 } }, "13.2": { "type": "choice", "choice": "A", "confidence": 0.81, "probabilities": { "A": 0.86, "C": 0.09, "B": 0.05, "D": 0.0 } }, "13.3a": { "type": "choice", "choice": "A", "confidence": 0.97, "probabilities": { "A": 0.98, "C": 0.0, "B": 0.0, "D": 0.02 } }, "13.3bi": { "type": "choice", "choice": "B", "confidence": 0.78, "probabilities": { "A": 0.02, "C": 0.13, "B": 0.83, "D": 0.02 } }, "13.3bii": { "type": "choice", "choice": "B", "confidence": 0.42, "probabilities": { "A": 0.09, "C": 0.27, "B": 0.56, "D": 0.07 } }, "13.3c": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "A": 0.0, "B": 1.0 } } }, "usage": { "input_tokens": 1061, "output_tokens": 264 } } }, { "exam": "midterm_sp23", "section": 14, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "14.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Packets sufficient to recover n message symbols with at most e erased packets and k corruptions?", "criteria": { "A": "n+2e+k", "B": "n+e+2k", "C": "n+e+k", "D": "n+2(e+k)" } }, "14.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Minimal error locator is x²−1 over GF(13). Give the error coordinates in 0..12.", "criteria": { "A": "{1,2}", "B": "{0,1}", "C": "{1,12}", "D": "{1,6}" } }, "14.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Shamir shares secret 5 with ten participants, any three reconstruct. Smallest prime modulus, with nonzero distinct share coordinates?", "criteria": { "A": "7", "B": "5", "C": "11", "D": "13" } }, "14.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Design a scheme requiring a strict majority of both a five-person group and a seven-person group.", "criteria": { "A": "Use a random degree-one master polynomial with secret at zero. Share one of its nonzero-coordinate values using degree two among the five people, and another using degree three among the seven; use one sufficiently large field.", "B": "Use one degree-two polynomial and give all twelve people one share each.", "C": "Use a degree-one master polynomial and give every participant both required master points.", "D": "Use two independent polynomials of degrees two and three, each with the full secret at zero; either group then suffices." } } } }, "attempts": [ { "status": 200, "full_response_ms": 311.50424999941606, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"14.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.44,\"probabilities\":{\"D\":0.2,\"A\":0.21,\"C\":0.01,\"B\":0.58}},\"14.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"C\":1.0,\"B\":0.0}},\"14.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.24,\"probabilities\":{\"D\":0.42,\"A\":0.14,\"C\":0.43,\"B\":0.01}},\"14.4\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.57,\"probabilities\":{\"C\":0.05,\"A\":0.68,\"D\":0.24,\"B\":0.03}}},\"usage\":{\"input_tokens\":895,\"output_tokens\":183}}" } ], "response": { "model": "jev-1.13.0", "answers": { "14.1": { "type": "choice", "choice": "B", "confidence": 0.44, "probabilities": { "D": 0.2, "A": 0.21, "C": 0.01, "B": 0.58 } }, "14.2": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "D": 0.0, "A": 0.0, "C": 1.0, "B": 0.0 } }, "14.3": { "type": "choice", "choice": "C", "confidence": 0.24, "probabilities": { "D": 0.42, "A": 0.14, "C": 0.43, "B": 0.01 } }, "14.4": { "type": "choice", "choice": "A", "confidence": 0.57, "probabilities": { "C": 0.05, "A": 0.68, "D": 0.24, "B": 0.03 } } }, "usage": { "input_tokens": 895, "output_tokens": 183 } } }, { "exam": "midterm_sp23", "section": 15, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "15.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: The set of finite subsets of a countably infinite set is uncountable.", "criteria": { "A": "False", "B": "True" } }, "15.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: The set of all subsets of a countably infinite set is uncountable.", "criteria": { "A": "True", "B": "False" } }, "15.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Every real number has a finite program that halts on input n and returns its nth digit.", "criteria": { "A": "False", "B": "True" } }, "15.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Why are some real numbers not computable digit-by-digit?", "criteria": { "A": "Every real has infinitely many digits, so none is computable.", "B": "There are countably many finite programs but uncountably many reals, so programs cannot cover all reals.", "C": "A program can output only finitely many digits over all inputs.", "D": "Uncomputable reals must all be rational." } } } }, "attempts": [ { "status": 200, "full_response_ms": 337.7193750020524, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"15.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"B\":0.01,\"A\":0.99}},\"15.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"A\":1.0}},\"15.3a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.95,\"probabilities\":{\"B\":0.03,\"A\":0.97}},\"15.3b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"B\":1.0,\"D\":0.0,\"C\":0.0,\"A\":0.0}}},\"usage\":{\"input_tokens\":725,\"output_tokens\":143}}" } ], "response": { "model": "jev-1.13.0", "answers": { "15.1": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "B": 0.01, "A": 0.99 } }, "15.2": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "A": 1.0 } }, "15.3a": { "type": "choice", "choice": "A", "confidence": 0.95, "probabilities": { "B": 0.03, "A": 0.97 } }, "15.3b": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "B": 1.0, "D": 0.0, "C": 0.0, "A": 0.0 } } }, "usage": { "input_tokens": 725, "output_tokens": 143 } } }, { "exam": "midterm_sp23", "section": 16, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "16.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct anagrams of ABRACADABRA?", "criteria": { "A": "11!/5!", "B": "11!/(5!2!2!)", "C": "11!/(4!2!2!)", "D": "10!/(5!2!2!)" } }, "16.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Strings with exactly four As and three Bs?", "criteria": { "A": "4!*3!", "B": "C(7,3)", "C": "2^7", "D": "7!" } }, "16.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distribute n1 identical apples and n2 identical bananas among k named people, allowing zero. Count?", "criteria": { "A": "C(n1+n2+k−1,k−1)", "B": "C(n1+k−1,k−1)*C(n2+k−1,k−1)", "C": "k^(n1+n2)", "D": "C(n1,k)*C(n2,k)" } }, "16.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Walk starts at 0 and ends at n>=0 after n+2k unit steps, each ±1, unrestricted positions. Number of step sequences?", "criteria": { "A": "2^(n+2k)", "B": "C(n+2k,n)", "C": "C(n+k,k)", "D": "C(n+2k,k)" } }, "16.5a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Sn counts ordered sums of ones and twos equaling n. Give a summation.", "criteria": { "A": "sum(i=0..n) C(n,i)", "B": "sum(i=0..floor(n/2)) C(n−i,i)", "C": "sum(i=0..floor(n/2)) 2^i", "D": "sum(i=0..floor(n/2)) C(n,i)" } }, "16.5b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: With S0=S1=1, recurrence for ordered sums of ones and twos equaling n, n>=2?", "criteria": { "A": "Sn=2S(n−1)", "B": "Sn=S(n−1)+S(n−2)", "C": "Sn=S(n−1)+1", "D": "Sn=S(n−1)*S(n−2)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 352.52466699967044, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"16.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.97,\"probabilities\":{\"D\":0.0,\"A\":0.01,\"C\":0.02,\"B\":0.97}},\"16.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"B\":0.99,\"A\":0.01,\"D\":0.0,\"C\":0.0}},\"16.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"B\":1.0,\"A\":0.0,\"D\":0.0,\"C\":0.0}},\"16.4\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.29,\"probabilities\":{\"B\":0.29,\"A\":0.21,\"D\":0.45999999999999996,\"C\":0.04}},\"16.5a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"A\":0.0,\"B\":1.0,\"D\":0.0,\"C\":0.0}},\"16.5b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"B\":1.0,\"A\":0.0,\"D\":0.0,\"C\":0.0}}},\"usage\":{\"input_tokens\":1154,\"output_tokens\":275}}" } ], "response": { "model": "jev-1.13.0", "answers": { "16.1": { "type": "choice", "choice": "B", "confidence": 0.97, "probabilities": { "D": 0.0, "A": 0.01, "C": 0.02, "B": 0.97 } }, "16.2": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "B": 0.99, "A": 0.01, "D": 0.0, "C": 0.0 } }, "16.3": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "B": 1.0, "A": 0.0, "D": 0.0, "C": 0.0 } }, "16.4": { "type": "choice", "choice": "D", "confidence": 0.29, "probabilities": { "B": 0.29, "A": 0.21, "D": 0.45999999999999996, "C": 0.04 } }, "16.5a": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "A": 0.0, "B": 1.0, "D": 0.0, "C": 0.0 } }, "16.5b": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "B": 1.0, "A": 0.0, "D": 0.0, "C": 0.0 } } }, "usage": { "input_tokens": 1154, "output_tokens": 275 } } }, { "exam": "midterm_sp24", "section": 1, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "1.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A judge says: if Bob stole the diamonds, he did not act alone. A truthful attorney denies that statement. What must be true?", "criteria": { "A": "Bob stole the diamonds, and did so alone.", "B": "Bob may have stolen the diamonds, but we do not know for sure.", "C": "Bob stole the diamonds with someone else.", "D": "Bob did not steal the diamonds.", "E": "None of the above." } } } }, "attempts": [ { "status": 200, "full_response_ms": 397.95404100004816, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"1.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.95,\"probabilities\":{\"C\":0.0,\"E\":0.01,\"A\":0.96,\"B\":0.0,\"D\":0.03}}},\"usage\":{\"input_tokens\":494,\"output_tokens\":54}}" } ], "response": { "model": "jev-1.13.0", "answers": { "1.1": { "type": "choice", "choice": "A", "confidence": 0.95, "probabilities": { "C": 0.0, "E": 0.01, "A": 0.96, "B": 0.0, "D": 0.03 } } }, "usage": { "input_tokens": 494, "output_tokens": 54 } } }, { "exam": "midterm_sp24", "section": 2, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "2.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If the halting problem is computable, then pigs can fly.", "criteria": { "A": "False", "B": "True" } }, "2.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: forall x [P(x) implies Q(x)] is equivalent to NOT exists x [NOT P(x) AND Q(x)].", "criteria": { "A": "False", "B": "True" } }, "2.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: [Q(0) AND forall n>=0 ((P(n) implies Q(n+1)) AND (Q(n) implies P(n)))] implies forall n>=0 P(n).", "criteria": { "A": "False", "B": "True" } }, "2.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: [Q(0) AND forall n>=0 ((P(n) implies Q(n+1)) AND (Q(n) implies P(n+1)))] implies forall n>=0 P(n).", "criteria": { "A": "True", "B": "False" } }, "2.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: There is an algorithm deciding whether an input finite decimal string is the corresponding prefix of pi’s decimal expansion.", "criteria": { "A": "False", "B": "True" } }, "2.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: There is an algorithm deciding, for arbitrary program P and strings x,y, whether P on x outputs y.", "criteria": { "A": "True", "B": "False" } } } }, "attempts": [ { "status": 200, "full_response_ms": 318.3907499987981, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"2.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.66,\"probabilities\":{\"A\":0.17,\"B\":0.83}},\"2.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"A\":0.99,\"B\":0.01}},\"2.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.43,\"probabilities\":{\"A\":0.29,\"B\":0.71}},\"2.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.65,\"probabilities\":{\"A\":0.18,\"B\":0.82}},\"2.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.63,\"probabilities\":{\"A\":0.19,\"B\":0.81}},\"2.6\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.98,\"probabilities\":{\"A\":0.01,\"B\":0.99}}},\"usage\":{\"input_tokens\":861,\"output_tokens\":183}}" } ], "response": { "model": "jev-1.13.0", "answers": { "2.1": { "type": "choice", "choice": "B", "confidence": 0.66, "probabilities": { "A": 0.17, "B": 0.83 } }, "2.2": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "A": 0.99, "B": 0.01 } }, "2.3": { "type": "choice", "choice": "B", "confidence": 0.43, "probabilities": { "A": 0.29, "B": 0.71 } }, "2.4": { "type": "choice", "choice": "B", "confidence": 0.65, "probabilities": { "A": 0.18, "B": 0.82 } }, "2.5": { "type": "choice", "choice": "B", "confidence": 0.63, "probabilities": { "A": 0.19, "B": 0.81 } }, "2.6": { "type": "choice", "choice": "B", "confidence": 0.98, "probabilities": { "A": 0.01, "B": 0.99 } } }, "usage": { "input_tokens": 861, "output_tokens": 183 } } }, { "exam": "midterm_sp24", "section": 3, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "3.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A pair present in both the job-proposing and candidate-proposing stable matchings is present in every stable matching.", "criteria": { "A": "True", "B": "False" } }, "3.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For every n, there can be a job-optimal stable matching containing a pair who rank each other last.", "criteria": { "A": "True", "B": "False" } }, "3.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Two participants who rank each other first must be paired in every stable matching.", "criteria": { "A": "True", "B": "False" } }, "3.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Two disjoint pairs, each pair ranking one another last, cannot both occur in a stable matching.", "criteria": { "A": "False", "B": "True" } }, "3.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If job J is ranked last by every candidate other than C, and C is ranked last by every job other than J, then J and C must be matched in every stable matching.", "criteria": { "A": "False", "B": "True" } } } }, "attempts": [ { "status": 200, "full_response_ms": 349.6895000025688, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"3.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.82,\"probabilities\":{\"B\":0.09,\"A\":0.91}},\"3.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.62,\"probabilities\":{\"B\":0.19,\"A\":0.81}},\"3.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.95,\"probabilities\":{\"B\":0.03,\"A\":0.97}},\"3.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.15,\"probabilities\":{\"B\":0.57,\"A\":0.43}},\"3.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.59,\"probabilities\":{\"B\":0.79,\"A\":0.21}}},\"usage\":{\"input_tokens\":759,\"output_tokens\":153}}" } ], "response": { "model": "jev-1.13.0", "answers": { "3.1": { "type": "choice", "choice": "A", "confidence": 0.82, "probabilities": { "B": 0.09, "A": 0.91 } }, "3.2": { "type": "choice", "choice": "A", "confidence": 0.62, "probabilities": { "B": 0.19, "A": 0.81 } }, "3.3": { "type": "choice", "choice": "A", "confidence": 0.95, "probabilities": { "B": 0.03, "A": 0.97 } }, "3.4": { "type": "choice", "choice": "B", "confidence": 0.15, "probabilities": { "B": 0.57, "A": 0.43 } }, "3.5": { "type": "choice", "choice": "B", "confidence": 0.59, "probabilities": { "B": 0.79, "A": 0.21 } } }, "usage": { "input_tokens": 759, "output_tokens": 153 } } }, { "exam": "midterm_sp24", "section": 4, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "4.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P: powered on. M: in motion. A: alarm activated. R: rerouting. S: senses an obstacle.\nProblem: Express: the robot is not in motion unless powered on.", "criteria": { "A": "P AND M", "B": "NOT M implies NOT P", "C": "P implies M", "D": "M implies P" } }, "4.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P: powered on. M: in motion. A: alarm activated. R: rerouting. S: senses an obstacle.\nProblem: Contrapositive of: if powered on, it is in motion or its alarm is activated.", "criteria": { "A": "NOT P implies (NOT M AND NOT A)", "B": "(NOT M AND NOT A) implies NOT P", "C": "(NOT M OR NOT A) implies NOT P", "D": "(M OR A) implies P" } }, "4.c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P: powered on. M: in motion. A: alarm activated. R: rerouting. S: senses an obstacle. Rules: M=>P; (S AND A)=>(NOT R OR NOT P); (P AND M AND R)=>S; (P AND S)=>(A OR NOT M).\nProblem: Given M true, what follows about R,S?", "criteria": { "A": "R true; S could be either", "B": "R false; S true", "C": "R false; S could be either", "D": "R true; S true", "E": "R false; S false", "F": "R true; S false" } }, "4.d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: L(t) means low battery at t, C(t) means charging. Express: whenever battery is low, it charges at some strictly later time.", "criteria": { "A": "forall t>=0, C(t) implies exists u>t L(u)", "B": "forall t>=0, L(t) implies exists u>t C(u)", "C": "forall t>=0, L(t) implies C(t)", "D": "exists u>=0, forall t>=0, L(t) implies C(u)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 323.58054199721664, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"4.a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"B\":0.0,\"C\":0.0,\"A\":0.0}},\"4.b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"C\":0.0,\"B\":1.0}},\"4.c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.38,\"probabilities\":{\"D\":0.1,\"E\":0.18,\"B\":0.03,\"F\":0.08,\"A\":0.14,\"C\":0.47}},\"4.d\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"C\":0.0,\"B\":1.0}}},\"usage\":{\"input_tokens\":1010,\"output_tokens\":189}}" } ], "response": { "model": "jev-1.13.0", "answers": { "4.a": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "C": 0.0, "A": 0.0 } }, "4.b": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.0, "B": 1.0 } }, "4.c": { "type": "choice", "choice": "C", "confidence": 0.38, "probabilities": { "D": 0.1, "E": 0.18, "B": 0.03, "F": 0.08, "A": 0.14, "C": 0.47 } }, "4.d": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.0, "B": 1.0 } } }, "usage": { "input_tokens": 1010, "output_tokens": 189 } } }, { "exam": "midterm_sp24", "section": 5, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "5.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Find the fatal error and explanation in this attempted induction that 2^n=1 for all n>=0: base 2^0=1; write n+1=i+j with positive i,j<=n; apply strong induction to i,j; multiply 2^i 2^j=1.", "criteria": { "A": "The base case is wrong because 2^0=0.", "B": "The multiplication identity 2^(i+j)=2^i*2^j is false.", "C": "Strong induction cannot assume two earlier cases.", "D": "The decomposition fails at n=0: 1 cannot be the sum of two positive integers." } }, "5.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Find the error: base triangle angle sum is 180; assume all n-gons have angle sum 180(n−2); add a triangle to an n-gon to make an (n+1)-gon; the angle sum increases by 180; conclude all (n+1)-gons satisfy the formula.", "criteria": { "A": "The construction has not shown that every (n+1)-gon is obtainable this way; start with an arbitrary larger polygon and justify removing a triangle.", "B": "The base case fails because triangles have angle sum 360.", "C": "Adding the three angles of a triangle gives 360, not 180.", "D": "Induction can never prove a geometric statement." } } } }, "attempts": [ { "status": 200, "full_response_ms": 294.62887499903445, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"5.a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":0.0,\"A\":0.0,\"D\":1.0}},\"5.b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"C\":0.0,\"A\":1.0,\"B\":0.0}}},\"usage\":{\"input_tokens\":760,\"output_tokens\":89}}" } ], "response": { "model": "jev-1.13.0", "answers": { "5.a": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 0.0, "D": 1.0 } }, "5.b": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "C": 0.0, "A": 1.0, "B": 0.0 } } }, "usage": { "input_tokens": 760, "output_tokens": 89 } } }, { "exam": "midterm_sp24", "section": 6, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "6.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: F0=0,F1=1,Fn=F(n−1)+F(n−2). Which induction proves sum(Fi²,i=0..n)=Fn*F(n+1)?", "criteria": { "A": "Every Fibonacci number equals its index, so the claim reduces to the sum of squares formula.", "B": "Square the recurrence and discard its cross term to make the sum telescope.", "C": "Base n=0 gives zero on both sides. Add F(k+1)² to Fk*F(k+1), factor to F(k+1)[Fk+F(k+1)]=F(k+1)F(k+2).", "D": "Base n=0 holds. Add F(k+1) instead of its square, obtaining the next product." } }, "6.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prove sum(i³,i=1..n)=[sum(i,i=1..n)]². You may use sum(i)=n(n+1)/2.", "criteria": { "A": "Base n=1 holds. By induction, the next sum is [k(k+1)/2]²+(k+1)³=(k+1)²(k+2)²/4, the required next square.", "B": "Cube the formula for sum(i), because the cube of a sum is the sum of cubes.", "C": "The next difference between the squared sums is (k+1)², equal to the next cube.", "D": "Checking n=1 and n=2 suffices, since both sides are polynomials of arbitrary degree." } } } }, "attempts": [ { "status": 200, "full_response_ms": 581.02966700244, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"6.a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":0.0,\"C\":1.0}},\"6.b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":1.0,\"C\":0.0}}},\"usage\":{\"input_tokens\":783,\"output_tokens\":89}}" } ], "response": { "model": "jev-1.13.0", "answers": { "6.a": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 } }, "6.b": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 1.0, "C": 0.0 } } }, "usage": { "input_tokens": 783, "output_tokens": 89 } } }, { "exam": "midterm_sp24", "section": 7, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "7.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Vertices a,b,c,d,e,f. Edges ab,af,bf,bc,fc,fe,ce,cd,de. There are no other edges.\nProblem: Which one new edge creates an Eulerian tour?", "criteria": { "A": "b–d", "B": "a–e", "C": "a–d", "D": "b–e" } }, "7.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Vertices a,b,c,d,e,f,g. Edges ab,ad,bc,bf,ce,de,ef,fg,eg. There are no other edges.\nProblem: Which one vertex must be removed so the remaining graph has a Hamiltonian cycle?", "criteria": { "A": "g", "B": "b", "C": "c", "D": "a" } }, "7.c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prove a tree with a degree-k vertex has at least k leaves.", "criteria": { "A": "The handshake lemma says every tree has exactly k leaves regardless of its other degrees.", "B": "All k neighbors must be leaves because trees have no cycles.", "C": "For k>=2, remove that vertex; its k components each contain a leaf of the original tree, by following a longest path away from the removed vertex. The k=0,1 cases hold directly.", "D": "Remove the vertex; all remaining vertices become isolated and therefore leaves." } }, "7.d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A connected graph has 80 vertices, 135 edges and girth 5. Is it planar? Justify.", "criteria": { "A": "No. Euler and 5f<=2e imply e<=5(v−2)/3=130, violated by 135.", "B": "Yes. The inequality e<=3v−6 holds and is sufficient for planarity.", "C": "No. Any graph with a five-cycle is nonplanar.", "D": "Yes. Odd girth guarantees a planar embedding." } } } }, "attempts": [ { "status": 200, "full_response_ms": 420.4808750000666, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"7.a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.03,\"probabilities\":{\"C\":0.28,\"D\":0.2,\"A\":0.25,\"B\":0.27}},\"7.b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.22,\"probabilities\":{\"C\":0.16,\"D\":0.41,\"A\":0.1,\"B\":0.33}},\"7.c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"C\":1.0,\"D\":0.0,\"A\":0.0,\"B\":0.0}},\"7.d\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"D\":0.0,\"A\":1.0,\"B\":0.0}}},\"usage\":{\"input_tokens\":968,\"output_tokens\":175}}" } ], "response": { "model": "jev-1.13.0", "answers": { "7.a": { "type": "choice", "choice": "C", "confidence": 0.03, "probabilities": { "C": 0.28, "D": 0.2, "A": 0.25, "B": 0.27 } }, "7.b": { "type": "choice", "choice": "D", "confidence": 0.22, "probabilities": { "C": 0.16, "D": 0.41, "A": 0.1, "B": 0.33 } }, "7.c": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "C": 1.0, "D": 0.0, "A": 0.0, "B": 0.0 } }, "7.d": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "C": 0.0, "D": 0.0, "A": 1.0, "B": 0.0 } } }, "usage": { "input_tokens": 968, "output_tokens": 175 } } }, { "exam": "midterm_sp24", "section": 8, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "8.ai": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Compute 1!+3!+5!+...+99! modulo 24.", "criteria": { "A": "1", "B": "6", "C": "7", "D": "0" } }, "8.aii": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Solve 5x=7 mod 11, x in 0..10.", "criteria": { "A": "2", "B": "5", "C": "9", "D": "8" } }, "8.aiii": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Units digit of 7^96?", "criteria": { "A": "1", "B": "3", "C": "9", "D": "7" } }, "8.aiv": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: RSA uses primes p=7,q=11 and public exponent e=11.\nProblem: Give the public key (N,e).", "criteria": { "A": "(77,7)", "B": "(77,11)", "C": "(60,11)", "D": "(18,11)" } }, "8.av": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: RSA uses primes p=7,q=11 and public exponent e=11.\nProblem: Encrypt message 3.", "criteria": { "A": "27", "B": "11", "C": "33", "D": "47" } }, "8.avi": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: RSA uses primes p=7,q=11 and public exponent e=11.\nProblem: Decrypt ciphertext 47.", "criteria": { "A": "3", "B": "7", "C": "47", "D": "11" } }, "8.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Three products appear every 5,3,11 days and were last available 3,1,2 days ago respectively. Earliest number of days from today when all are available? Hint: CRT.", "criteria": { "A": "97", "B": "42", "C": "165", "D": "152" } } } }, "attempts": [ { "status": 200, "full_response_ms": 364.05420799928834, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"8.ai\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.49,\"probabilities\":{\"B\":0.18,\"D\":0.05,\"C\":0.61,\"A\":0.16}},\"8.aii\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.26,\"probabilities\":{\"B\":0.1,\"D\":0.37,\"C\":0.45,\"A\":0.08}},\"8.aiii\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.86,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"C\":0.1,\"A\":0.9}},\"8.aiv\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"B\":1.0,\"D\":0.0,\"C\":0.0,\"A\":0.0}},\"8.av\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.31,\"probabilities\":{\"B\":0.06,\"D\":0.48,\"C\":0.13,\"A\":0.33}},\"8.avi\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.23,\"probabilities\":{\"B\":0.21,\"D\":0.04,\"C\":0.43,\"A\":0.32}},\"8.b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.32,\"probabilities\":{\"B\":0.16,\"D\":0.23,\"C\":0.12,\"A\":0.49}}},\"usage\":{\"input_tokens\":1129,\"output_tokens\":308}}" } ], "response": { "model": "jev-1.13.0", "answers": { "8.ai": { "type": "choice", "choice": "C", "confidence": 0.49, "probabilities": { "B": 0.18, "D": 0.05, "C": 0.61, "A": 0.16 } }, "8.aii": { "type": "choice", "choice": "C", "confidence": 0.26, "probabilities": { "B": 0.1, "D": 0.37, "C": 0.45, "A": 0.08 } }, "8.aiii": { "type": "choice", "choice": "A", "confidence": 0.86, "probabilities": { "B": 0.0, "D": 0.0, "C": 0.1, "A": 0.9 } }, "8.aiv": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "B": 1.0, "D": 0.0, "C": 0.0, "A": 0.0 } }, "8.av": { "type": "choice", "choice": "D", "confidence": 0.31, "probabilities": { "B": 0.06, "D": 0.48, "C": 0.13, "A": 0.33 } }, "8.avi": { "type": "choice", "choice": "C", "confidence": 0.23, "probabilities": { "B": 0.21, "D": 0.04, "C": 0.43, "A": 0.32 } }, "8.b": { "type": "choice", "choice": "A", "confidence": 0.32, "probabilities": { "B": 0.16, "D": 0.23, "C": 0.12, "A": 0.49 } } }, "usage": { "input_tokens": 1129, "output_tokens": 308 } } }, { "exam": "midterm_sp24", "section": 9, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "9.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Over GF(p), prime p>20, a polynomial P has degree at most 2. Treasure coordinates are (P(10),P(20)). Hints give values at distinct field coordinates other than 10 and 20.\nProblem: Hints needed to determine the treasure location?", "criteria": { "A": "3", "B": "4", "C": "1", "D": "2" } }, "9.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Over GF(p), prime p>20, a polynomial P has degree at most 2. Treasure coordinates are (P(10),P(20)). Hints give values at distinct field coordinates other than 10 and 20.\nProblem: After one hint, number of still-possible treasure locations?", "criteria": { "A": "p−1", "B": "p²", "C": "p", "D": "1" } }, "9.c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Over GF(p), prime p>20, a polynomial P has degree at most 2. Treasure coordinates are (P(10),P(20)). Hints give values at distinct field coordinates other than 10 and 20.\nProblem: After two hints, number of still-possible treasure locations?", "criteria": { "A": "p−1", "B": "1", "C": "p²", "D": "p" } }, "9.d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Over GF(p), prime p>20, a polynomial P has degree at most 2. Treasure coordinates are (P(10),P(20)). Hints give values at distinct field coordinates other than 10 and 20.\nProblem: Hints P(0)=5,P(1)=5,P(3)=11. Find P. Hint: solve for coefficients rather than use Lagrange.", "criteria": { "A": "x²−x+5", "B": "x²+x+5", "C": "x+5", "D": "2x²−2x+5" } }, "9.e": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Over GF(p), prime p>20, a polynomial P has degree at most 2. Treasure coordinates are (P(10),P(20)). Hints give values at distinct field coordinates other than 10 and 20.\nProblem: For p=89 and hints P(0)=5,P(1)=5,P(3)=11, find the treasure coordinates.", "criteria": { "A": "(16,69)", "B": "(6,29)", "C": "(5,11)", "D": "(95,385)" } }, "9.f": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Over GF(p), prime p>20, a polynomial P has degree at most 2. Treasure coordinates are (P(10),P(20)). Hints give values at distinct field coordinates other than 10 and 20.\nProblem: At most one hint is corrupted. Minimum hints to detect whether a corruption occurred, using only the hints and no visits to treasure locations?", "criteria": { "A": "5", "B": "2", "C": "3", "D": "4" } }, "9.g": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Over GF(p), prime p>20, a polynomial P has degree at most 2. Treasure coordinates are (P(10),P(20)). Hints give values at distinct field coordinates other than 10 and 20.\nProblem: One hint is corrupted at an unknown location. Minimum hints guaranteeing recovery of the correct polynomial?", "criteria": { "A": "6", "B": "4", "C": "3", "D": "5" } } } }, "attempts": [ { "status": 200, "full_response_ms": 369.33124999995925, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"9.a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.31,\"probabilities\":{\"B\":0.01,\"C\":0.16,\"A\":0.35,\"D\":0.48}},\"9.b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.57,\"probabilities\":{\"B\":0.12,\"C\":0.68,\"A\":0.18,\"D\":0.02}},\"9.c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.79,\"probabilities\":{\"B\":0.85,\"C\":0.02,\"A\":0.03,\"D\":0.1}},\"9.d\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.65,\"probabilities\":{\"B\":0.05,\"C\":0.07,\"A\":0.73,\"D\":0.15}},\"9.e\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.32,\"probabilities\":{\"B\":0.23,\"C\":0.21,\"A\":0.48,\"D\":0.08}},\"9.f\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.44,\"probabilities\":{\"B\":0.12,\"C\":0.58,\"A\":0.12,\"D\":0.18}},\"9.g\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.26,\"probabilities\":{\"B\":0.33,\"C\":0.11,\"A\":0.12,\"D\":0.44}}},\"usage\":{\"input_tokens\":1479,\"output_tokens\":304}}" } ], "response": { "model": "jev-1.13.0", "answers": { "9.a": { "type": "choice", "choice": "D", "confidence": 0.31, "probabilities": { "B": 0.01, "C": 0.16, "A": 0.35, "D": 0.48 } }, "9.b": { "type": "choice", "choice": "C", "confidence": 0.57, "probabilities": { "B": 0.12, "C": 0.68, "A": 0.18, "D": 0.02 } }, "9.c": { "type": "choice", "choice": "B", "confidence": 0.79, "probabilities": { "B": 0.85, "C": 0.02, "A": 0.03, "D": 0.1 } }, "9.d": { "type": "choice", "choice": "A", "confidence": 0.65, "probabilities": { "B": 0.05, "C": 0.07, "A": 0.73, "D": 0.15 } }, "9.e": { "type": "choice", "choice": "A", "confidence": 0.32, "probabilities": { "B": 0.23, "C": 0.21, "A": 0.48, "D": 0.08 } }, "9.f": { "type": "choice", "choice": "C", "confidence": 0.44, "probabilities": { "B": 0.12, "C": 0.58, "A": 0.12, "D": 0.18 } }, "9.g": { "type": "choice", "choice": "D", "confidence": 0.26, "probabilities": { "B": 0.33, "C": 0.11, "A": 0.12, "D": 0.44 } } }, "usage": { "input_tokens": 1479, "output_tokens": 304 } } }, { "exam": "midterm_sp24", "section": 10, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "10.a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Is the set of integer grid points (x,y) with gcd(x,y)=1 countable? Justify.", "criteria": { "A": "No; the plane is uncountable.", "B": "No; coprimality prevents enumeration.", "C": "Yes; it is a subset of the countable set Z×Z.", "D": "Yes; it is finite." } }, "10.b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A vehicle starts at the origin facing up and chooses left, straight or right at each grid intersection. Are its finite routes countable?", "criteria": { "A": "No; three choices at every step make even finite strings uncountable.", "B": "No; routes may have arbitrary finite length.", "C": "Yes; finite routes are finite strings over a three-symbol alphabet.", "D": "Yes; there are only three possible routes." } }, "10.c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A vehicle starts at the origin facing up and chooses left, straight or right at each grid intersection. Are all infinite routes countable?", "criteria": { "A": "No; there are only finitely many infinite routes.", "B": "Yes; the union of all finite routes contains every infinite route.", "C": "No; restricting to two turn choices embeds all infinite binary sequences.", "D": "Yes; each route visits only countably many intersections." } }, "10.d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: At each (x,y) in Z² there are |x|+|y| parking spaces. Is the set of all spaces countable?", "criteria": { "A": "Yes; there is a finite total number of spaces.", "B": "No; the number at an intersection is unbounded across the grid.", "C": "Yes; a countable union of finite sets is countable.", "D": "No; a union of infinitely many finite sets is always uncountable." } } } }, "attempts": [ { "status": 200, "full_response_ms": 299.58137499852455, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"10.a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":0.0,\"C\":1.0}},\"10.b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":0.0,\"C\":1.0}},\"10.c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.98,\"probabilities\":{\"B\":0.0,\"D\":0.01,\"A\":0.0,\"C\":0.99}},\"10.d\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.99,\"probabilities\":{\"B\":0.01,\"D\":0.0,\"A\":0.0,\"C\":0.99}}},\"usage\":{\"input_tokens\":962,\"output_tokens\":179}}" } ], "response": { "model": "jev-1.13.0", "answers": { "10.a": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 } }, "10.b": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 } }, "10.c": { "type": "choice", "choice": "C", "confidence": 0.98, "probabilities": { "B": 0.0, "D": 0.01, "A": 0.0, "C": 0.99 } }, "10.d": { "type": "choice", "choice": "C", "confidence": 0.99, "probabilities": { "B": 0.01, "D": 0.0, "A": 0.0, "C": 0.99 } } }, "usage": { "input_tokens": 962, "output_tokens": 179 } } }, { "exam": "midterm_sp25", "section": 2, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "2.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Another exam question says a valid RSA signature with public key (55,3) is attached to one of the possible TA counts, 12 or 13, with signature 23.\nProblem: How many TAs are there this semester?", "criteria": { "A": "13", "B": "14", "C": "12", "D": "11" } } } }, "attempts": [ { "status": 200, "full_response_ms": 347.33437500108266, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"2.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.38,\"probabilities\":{\"C\":0.43,\"D\":0.01,\"A\":0.55,\"B\":0.01}}},\"usage\":{\"input_tokens\":467,\"output_tokens\":47}}" } ], "response": { "model": "jev-1.13.0", "answers": { "2.1": { "type": "choice", "choice": "A", "confidence": 0.38, "probabilities": { "C": 0.43, "D": 0.01, "A": 0.55, "B": 0.01 } } }, "usage": { "input_tokens": 467, "output_tokens": 47 } } }, { "exam": "midterm_sp25", "section": 3, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "3.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: (NOT P AND R) OR (NOT P AND Q) is equivalent to NOT P AND (R OR Q).", "criteria": { "A": "False", "B": "True" } }, "3.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: (P AND Q) AND (NOT P OR NOT Q) is equivalent to P AND NOT P.", "criteria": { "A": "False", "B": "True" } }, "3.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: (P implies Q) AND (P implies NOT Q) is equivalent to NOT P.", "criteria": { "A": "True", "B": "False" } }, "3.1d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: (P implies R) AND (P implies NOT R) is equivalent to NOT R.", "criteria": { "A": "False", "B": "True" } }, "3.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Negate forall n in N, P(n), without using forall.", "criteria": { "A": "exists n in N, P(n)", "B": "NOT exists n in N, P(n)", "C": "exists n in N, NOT P(n)", "D": "exists n in N, P(n) AND NOT P(n)" } }, "3.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: N includes zero. For every predicate P, [NOT(forall n P(n)) AND P(0)] implies exists n [P(n) AND NOT P(n+1)].", "criteria": { "A": "True", "B": "False" } }, "3.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: forall n [Q(n) AND P(n)] is equivalent to (forall n Q(n)) AND (forall n P(n)).", "criteria": { "A": "True", "B": "False" } }, "3.2d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: forall n [Q(n) implies P(n)] is equivalent to (forall n NOT Q(n)) OR (forall n P(n)).", "criteria": { "A": "False", "B": "True" } } } }, "attempts": [ { "status": 200, "full_response_ms": 393.1597920018248, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"3.1a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"B\":1.0,\"A\":0.0}},\"3.1b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.12,\"probabilities\":{\"B\":0.44,\"A\":0.56}},\"3.1c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"B\":0.01,\"A\":0.99}},\"3.1d\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.97,\"probabilities\":{\"B\":0.02,\"A\":0.98}},\"3.2a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":0.0,\"C\":1.0}},\"3.2b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.42,\"probabilities\":{\"B\":0.29,\"A\":0.71}},\"3.2c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"A\":1.0}},\"3.2d\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.97,\"probabilities\":{\"B\":0.01,\"A\":0.99}}},\"usage\":{\"input_tokens\":1068,\"output_tokens\":265}}" } ], "response": { "model": "jev-1.13.0", "answers": { "3.1a": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "B": 1.0, "A": 0.0 } }, "3.1b": { "type": "choice", "choice": "A", "confidence": 0.12, "probabilities": { "B": 0.44, "A": 0.56 } }, "3.1c": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "B": 0.01, "A": 0.99 } }, "3.1d": { "type": "choice", "choice": "A", "confidence": 0.97, "probabilities": { "B": 0.02, "A": 0.98 } }, "3.2a": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 } }, "3.2b": { "type": "choice", "choice": "A", "confidence": 0.42, "probabilities": { "B": 0.29, "A": 0.71 } }, "3.2c": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "B": 0.0, "A": 1.0 } }, "3.2d": { "type": "choice", "choice": "A", "confidence": 0.97, "probabilities": { "B": 0.01, "A": 0.99 } } }, "usage": { "input_tokens": 1068, "output_tokens": 265 } } }, { "exam": "midterm_sp25", "section": 4, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "4.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Three people each like exactly one other person. Must someone be liked by nobody? Give a proof or counterexample.", "criteria": { "A": "Yes: three preferences among three people force two preferences to share a target.", "B": "No: A likes B, B likes C, and C likes A; everyone is liked once.", "C": "Yes: whoever is least popular receives no preference.", "D": "No: let everyone like themselves." } }, "4.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Positive integers a,b,c satisfy a²+b²=c² and gcd(a,b,c)=1. Why must c be odd?", "criteria": { "A": "Every square is odd, so the sum of two squares is odd.", "B": "If c is even, then a+b=c and both legs are automatically even.", "C": "Squares are 0 or 1 mod 4. If c were even, both a,b would have to be even (two odd squares sum to 2 mod 4), contradicting the common gcd 1.", "D": "Since the gcd is 1, a,b,c must all be odd, proving the claim." } } } }, "attempts": [ { "status": 200, "full_response_ms": 342.6203749986598, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"4.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"D\":0.0,\"A\":0.0,\"B\":1.0}},\"4.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"C\":1.0,\"A\":0.0,\"D\":0.0,\"B\":0.0}}},\"usage\":{\"input_tokens\":692,\"output_tokens\":91}}" } ], "response": { "model": "jev-1.13.0", "answers": { "4.1": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "C": 0.0, "D": 0.0, "A": 0.0, "B": 1.0 } }, "4.2": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "C": 1.0, "A": 0.0, "D": 0.0, "B": 0.0 } } }, "usage": { "input_tokens": 692, "output_tokens": 91 } } }, { "exam": "midterm_sp25", "section": 5, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "5.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prove sum(i³,i=1..n)=[n(n+1)/2]² by induction. Which step is valid? You may use sum(i)=n(n+1)/2.", "criteria": { "A": "Cube the identity sum(i)=n(n+1)/2; sums commute with cubing.", "B": "Base n=1 holds. The next cube is (n+1)², so adding it gives the next square of a triangular number.", "C": "Base n=1 holds. Adding (n+1)³ to [n(n+1)/2]² gives (n+1)²[n²+4(n+1)]/4=[(n+1)(n+2)/2]².", "D": "Because the formula holds for n=1 and n=2, it holds for all n without another step." } } } }, "attempts": [ { "status": 200, "full_response_ms": 479.07479100103956, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"5.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"C\":1.0,\"B\":0.0}}},\"usage\":{\"input_tokens\":575,\"output_tokens\":47}}" } ], "response": { "model": "jev-1.13.0", "answers": { "5.1": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 1.0, "B": 0.0 } } }, "usage": { "input_tokens": 575, "output_tokens": 47 } } }, { "exam": "midterm_sp25", "section": 6, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "6.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Among all preference instances with n jobs and n candidates, what is the maximum possible number of blocking pairs in a complete pairing?", "criteria": { "A": "n(n−1)", "B": "n−1", "C": "n²", "D": "n(n−1)/2" } }, "6.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If a stable matching gives one particular job its best attainable stable partner, must that candidate have its worst attainable stable partner?", "criteria": { "A": "False", "B": "True" } }, "6.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: In job-proposing deferred acceptance, if a candidate rejects n−1 distinct jobs, that candidate ends with its favorite job.", "criteria": { "A": "True", "B": "False" } }, "6.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If job J is rejected by C, and J′ ends with C, then J’s eventual partner has a worse rank in J’s list than C has in J′’s list.", "criteria": { "A": "True", "B": "False" } }, "6.5a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If a job and candidate rank each other first, they are paired in every stable matching.", "criteria": { "A": "True", "B": "False" } }, "6.5b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Construct, for each n>=2, an instance with exactly one stable matching.", "criteria": { "A": "Let rankings be arbitrary; the deferred acceptance output proves uniqueness.", "B": "Make every job rank C1 first; this alone uniquely determines every candidate’s partner.", "C": "Make Ji and Ci rank each other first for all i. Every such pair must be matched, otherwise it blocks; their combined matching is stable.", "D": "Let each Ji rank Ci last and each Ci rank Ji last; these pairs must be stable." } }, "6.5c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Construct an instance with exactly two stable matchings for every n>=2.", "criteria": { "A": "Take two arbitrary complete pairings and declare both stable; preferences do not affect stability.", "B": "Give all jobs the same ranking and all candidates the same ranking; this yields exactly two stable matchings.", "C": "On the first two pairs use J1:C2>C1, J2:C1>C2, C1:J1>J2, C2:J2>J1; rank outsiders below these. For i>2, Ji and Ci rank each other first. The forced outer pairs leave exactly the two stable pairings of the first block.", "D": "Make every Ji and Ci rank each other first; then both the identity and any swap are stable." } } } }, "attempts": [ { "status": 200, "full_response_ms": 421.0574999997334, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"6.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.71,\"probabilities\":{\"B\":0.0,\"C\":0.06,\"D\":0.16,\"A\":0.78}},\"6.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.16,\"probabilities\":{\"B\":0.58,\"A\":0.42}},\"6.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.16,\"probabilities\":{\"B\":0.58,\"A\":0.42}},\"6.4\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.2,\"probabilities\":{\"B\":0.4,\"A\":0.6}},\"6.5a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.96,\"probabilities\":{\"B\":0.02,\"A\":0.98}},\"6.5b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.98,\"probabilities\":{\"B\":0.0,\"C\":0.99,\"A\":0.01,\"D\":0.0}},\"6.5c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"C\":1.0,\"A\":0.0,\"D\":0.0}}},\"usage\":{\"input_tokens\":1211,\"output_tokens\":258}}" } ], "response": { "model": "jev-1.13.0", "answers": { "6.1": { "type": "choice", "choice": "A", "confidence": 0.71, "probabilities": { "B": 0.0, "C": 0.06, "D": 0.16, "A": 0.78 } }, "6.2": { "type": "choice", "choice": "B", "confidence": 0.16, "probabilities": { "B": 0.58, "A": 0.42 } }, "6.3": { "type": "choice", "choice": "B", "confidence": 0.16, "probabilities": { "B": 0.58, "A": 0.42 } }, "6.4": { "type": "choice", "choice": "A", "confidence": 0.2, "probabilities": { "B": 0.4, "A": 0.6 } }, "6.5a": { "type": "choice", "choice": "A", "confidence": 0.96, "probabilities": { "B": 0.02, "A": 0.98 } }, "6.5b": { "type": "choice", "choice": "C", "confidence": 0.98, "probabilities": { "B": 0.0, "C": 0.99, "A": 0.01, "D": 0.0 } }, "6.5c": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "C": 1.0, "A": 0.0, "D": 0.0 } } }, "usage": { "input_tokens": 1211, "output_tokens": 258 } } }, { "exam": "midterm_sp25", "section": 7, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "7.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A connected simple planar graph has v>=3 vertices. Maximum number of faces?", "criteria": { "A": "3v−6", "B": "2v−4", "C": "2v−2", "D": "v−2" } }, "7.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A connected simple planar graph contains a cycle and has girth at least g>=3. Complete the bound e <= [g/(g−2)]v − __.", "criteria": { "A": "g/(g−2)", "B": "g−2", "C": "2", "D": "2g/(g−2)" } }, "7.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Why is every planar graph with no cycle shorter than six edges 3-colorable?", "criteria": { "A": "The average degree is below three, so every vertex has degree at most two and the graph is a disjoint union of paths.", "B": "Every nonempty subgraph has a vertex of degree at most two: forests have leaves, and the planar girth bound gives average degree below three for cyclic components. Remove such a vertex, inductively 3-color the rest, then restore it using a color absent from its at most two neighbors.", "C": "Euler’s formula gives exactly three faces, which can be used as vertex colors.", "D": "Every planar graph is bipartite, so two colors suffice." } }, "7.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: How many edges are in the complement of a tree on n vertices?", "criteria": { "A": "C(n,2)−n+1", "B": "C(n−1,2)−1", "C": "C(n,2)−n", "D": "n−1" } }, "7.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: How many edges are in the complement of a d-dimensional hypercube?", "criteria": { "A": "C(2^d,2)−d*2^(d−1)", "B": "2^d−d", "C": "C(2^d,2)−d*2^d", "D": "C(d,2)−2^d" } }, "7.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Doubling a graph means taking two disjoint copies and adding an edge between each original vertex and its corresponding copy.\nProblem: If G has n vertices and m edges, how many edges does its double have?", "criteria": { "A": "m+2n", "B": "2m+n", "C": "2m+2n", "D": "2m" } }, "7.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Doubling a graph means taking two disjoint copies and adding an edge between each original vertex and its corresponding copy.\nProblem: If G is bipartite, its double is bipartite.", "criteria": { "A": "True", "B": "False" } }, "7.3c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Doubling a graph means taking two disjoint copies and adding an edge between each original vertex and its corresponding copy.\nProblem: How many doublings turn a k-dimensional hypercube into a d-dimensional hypercube, k=2, what is the smallest universal upper bound on colors needed for its double?", "criteria": { "A": "2c", "B": "c", "C": "c+1", "D": "c−1" } }, "7.3f": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Doubling a graph means taking two disjoint copies and adding an edge between each original vertex and its corresponding copy.\nProblem: If G contains a Hamiltonian cycle, its double also contains a Hamiltonian cycle.", "criteria": { "A": "True", "B": "False" } }, "7.3g": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Doubling a graph means taking two disjoint copies and adding an edge between each original vertex and its corresponding copy.\nProblem: If a graph with at least two vertices contains an Eulerian tour, its double contains an Eulerian tour.", "criteria": { "A": "False", "B": "True" } } } }, "attempts": [ { "status": 200, "full_response_ms": 311.75304200223763, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"7.1a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.89,\"probabilities\":{\"B\":0.91,\"C\":0.06,\"A\":0.02,\"D\":0.01}},\"7.1b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.4,\"probabilities\":{\"B\":0.03,\"C\":0.06,\"A\":0.36,\"D\":0.55}},\"7.1c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":1.0,\"A\":0.0,\"D\":0.0}},\"7.2a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":0.0,\"A\":1.0,\"D\":0.0}},\"7.2b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"C\":0.0,\"B\":0.0,\"A\":1.0,\"D\":0.0}},\"7.3a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"B\":1.0,\"C\":0.0,\"A\":0.0,\"D\":0.0}},\"7.3b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.25,\"probabilities\":{\"B\":0.62,\"A\":0.38}},\"7.3c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.93,\"probabilities\":{\"B\":0.0,\"C\":0.95,\"A\":0.05,\"D\":0.0}},\"7.3d\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.5,\"probabilities\":{\"B\":0.03,\"C\":0.62,\"A\":0.31,\"D\":0.04}},\"7.3e\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.27,\"probabilities\":{\"B\":0.46,\"C\":0.31,\"A\":0.23,\"D\":0.0}},\"7.3f\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.71,\"probabilities\":{\"B\":0.15,\"A\":0.85}},\"7.3g\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.51,\"probabilities\":{\"B\":0.76,\"A\":0.24}}},\"usage\":{\"input_tokens\":1938,\"output_tokens\":501}}" } ], "response": { "model": "jev-1.13.0", "answers": { "7.1a": { "type": "choice", "choice": "B", "confidence": 0.89, "probabilities": { "B": 0.91, "C": 0.06, "A": 0.02, "D": 0.01 } }, "7.1b": { "type": "choice", "choice": "D", "confidence": 0.4, "probabilities": { "B": 0.03, "C": 0.06, "A": 0.36, "D": 0.55 } }, "7.1c": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 1.0, "A": 0.0, "D": 0.0 } }, "7.2a": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 1.0, "D": 0.0 } }, "7.2b": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "C": 0.0, "B": 0.0, "A": 1.0, "D": 0.0 } }, "7.3a": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "B": 1.0, "C": 0.0, "A": 0.0, "D": 0.0 } }, "7.3b": { "type": "choice", "choice": "B", "confidence": 0.25, "probabilities": { "B": 0.62, "A": 0.38 } }, "7.3c": { "type": "choice", "choice": "C", "confidence": 0.93, "probabilities": { "B": 0.0, "C": 0.95, "A": 0.05, "D": 0.0 } }, "7.3d": { "type": "choice", "choice": "C", "confidence": 0.5, "probabilities": { "B": 0.03, "C": 0.62, "A": 0.31, "D": 0.04 } }, "7.3e": { "type": "choice", "choice": "B", "confidence": 0.27, "probabilities": { "B": 0.46, "C": 0.31, "A": 0.23, "D": 0.0 } }, "7.3f": { "type": "choice", "choice": "A", "confidence": 0.71, "probabilities": { "B": 0.15, "A": 0.85 } }, "7.3g": { "type": "choice", "choice": "B", "confidence": 0.51, "probabilities": { "B": 0.76, "A": 0.24 } } }, "usage": { "input_tokens": 1938, "output_tokens": 501 } } }, { "exam": "midterm_sp25", "section": 8, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "8.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prove the complement of a disconnected graph is connected.", "criteria": { "A": "For vertices u,v in different original components, uv is a complement edge. For u,v in the same component, choose w in another component; uw and wv are complement edges, giving a path.", "B": "Every original component becomes a complete graph, so all components join automatically.", "C": "A disconnected graph always has an isolated vertex, which becomes adjacent to all vertices in the complement.", "D": "The complement has more edges than the original, and any denser graph is connected." } } } }, "attempts": [ { "status": 200, "full_response_ms": 384.1220419999445, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"8.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"C\":0.0,\"A\":1.0,\"D\":0.0}}},\"usage\":{\"input_tokens\":512,\"output_tokens\":47}}" } ], "response": { "model": "jev-1.13.0", "answers": { "8.1": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "A": 1.0, "D": 0.0 } } }, "usage": { "input_tokens": 512, "output_tokens": 47 } } }, { "exam": "midterm_sp25", "section": 9, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "9.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Inverse of 5 modulo 24, in 0..23?", "criteria": { "A": "19", "B": "1", "C": "7", "D": "5" } }, "9.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Compute 2^75 modulo 35. Hint: consider RSA with p=7,q=5,e=5.", "criteria": { "A": "2", "B": "1", "C": "8", "D": "32" } }, "9.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Public key is (N,e)=(55,3). Received signed messages are (12,23) and (13,23), where the second value is the signature m^d mod N. Which verifies, and why? Hint: CRT may help.", "criteria": { "A": "Both, because they carry the same signature.", "B": "Neither, because a signature must equal the message.", "C": "12, because 23³=12 mod 55.", "D": "13, because 23³=13 mod 55." } }, "9.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For positive integers y=0, prove a^k=a^(k mod (p−1)) mod p.", "criteria": { "A": "Write k=q(p−1)+r. Fermat gives a^(p−1)=1, so a^k=(a^(p−1))^q*a^r=a^r.", "B": "Reduce the base modulo p−1 and retain the exponent; these are equivalent operations.", "C": "Fermat gives a^p=1, so exponents repeat modulo p.", "D": "All powers of nonzero residues are equal, so the exponent is irrelevant." } }, "9.6c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If gcd(k,p−1)=1, multiplication by k permutes the residue classes modulo p−1.", "criteria": { "A": "True", "B": "False" } }, "9.6d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prime p, positive k, and a^k=1 mod p. Why must gcd(k,p−1)>1 or a=1 mod p?", "criteria": { "A": "If gcd(k,p−1)=1, choose t with kt=1 mod p−1. Since a is nonzero, Fermat gives a=a^(kt)=(a^k)^t=1 mod p.", "B": "If a is not one, it is a primitive root by definition.", "C": "Fermat implies every k divides p−1, so their gcd exceeds one.", "D": "The equation a^k=1 implies a=k=1 as integers." } }, "9.6e": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If a^k=1 mod prime p, k>0, and a is not 1 mod p, then gcd(k,p−1)>1.", "criteria": { "A": "False", "B": "True" } } } }, "attempts": [ { "status": 200, "full_response_ms": 379.47658299890463, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"9.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.8,\"probabilities\":{\"C\":0.01,\"B\":0.01,\"D\":0.13,\"A\":0.85}},\"9.2\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.18,\"probabilities\":{\"A\":0.17,\"B\":0.1,\"C\":0.35,\"D\":0.38}},\"9.3\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.29,\"probabilities\":{\"C\":0.42,\"B\":0.04,\"D\":0.47,\"A\":0.07}},\"9.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.79,\"probabilities\":{\"B\":0.89,\"A\":0.11}},\"9.5a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.91,\"probabilities\":{\"D\":0.92,\"A\":0.02,\"C\":0.05,\"B\":0.01}},\"9.5b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.65,\"probabilities\":{\"C\":0.13,\"B\":0.74,\"D\":0.13,\"A\":0.0}},\"9.5c\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.6,\"probabilities\":{\"C\":0.0,\"B\":0.0,\"D\":0.7,\"A\":0.3}},\"9.6a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.97,\"probabilities\":{\"A\":0.02,\"B\":0.98,\"C\":0.0,\"D\":0.0}},\"9.6b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"A\":1.0,\"B\":0.0,\"C\":0.0,\"D\":0.0}},\"9.6c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.98,\"probabilities\":{\"B\":0.01,\"A\":0.99}},\"9.6d\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":0.0,\"D\":0.0,\"A\":1.0}},\"9.6e\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.17,\"probabilities\":{\"A\":0.42,\"B\":0.58}}},\"usage\":{\"input_tokens\":1874,\"output_tokens\":497}}" } ], "response": { "model": "jev-1.13.0", "answers": { "9.1": { "type": "choice", "choice": "A", "confidence": 0.8, "probabilities": { "C": 0.01, "B": 0.01, "D": 0.13, "A": 0.85 } }, "9.2": { "type": "choice", "choice": "D", "confidence": 0.18, "probabilities": { "A": 0.17, "B": 0.1, "C": 0.35, "D": 0.38 } }, "9.3": { "type": "choice", "choice": "D", "confidence": 0.29, "probabilities": { "C": 0.42, "B": 0.04, "D": 0.47, "A": 0.07 } }, "9.4": { "type": "choice", "choice": "B", "confidence": 0.79, "probabilities": { "B": 0.89, "A": 0.11 } }, "9.5a": { "type": "choice", "choice": "D", "confidence": 0.91, "probabilities": { "D": 0.92, "A": 0.02, "C": 0.05, "B": 0.01 } }, "9.5b": { "type": "choice", "choice": "B", "confidence": 0.65, "probabilities": { "C": 0.13, "B": 0.74, "D": 0.13, "A": 0.0 } }, "9.5c": { "type": "choice", "choice": "D", "confidence": 0.6, "probabilities": { "C": 0.0, "B": 0.0, "D": 0.7, "A": 0.3 } }, "9.6a": { "type": "choice", "choice": "B", "confidence": 0.97, "probabilities": { "A": 0.02, "B": 0.98, "C": 0.0, "D": 0.0 } }, "9.6b": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "A": 1.0, "B": 0.0, "C": 0.0, "D": 0.0 } }, "9.6c": { "type": "choice", "choice": "A", "confidence": 0.98, "probabilities": { "B": 0.01, "A": 0.99 } }, "9.6d": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "D": 0.0, "A": 1.0 } }, "9.6e": { "type": "choice", "choice": "B", "confidence": 0.17, "probabilities": { "A": 0.42, "B": 0.58 } } }, "usage": { "input_tokens": 1874, "output_tokens": 497 } } }, { "exam": "midterm_sp25", "section": 10, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "10.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Over GF(5), ax²+bx+c passes through (0,1),(1,2),(2,0). Give (a,b,c).", "criteria": { "A": "(1,0,1)", "B": "(1,1,0)", "C": "(2,4,1)", "D": "(0,1,1)" } }, "10.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A degree-one polynomial over GF(5) was evaluated; received (0,1),(1,3),(2,1),(3,2), with at most one corrupted value.\nProblem: Recover the polynomial.", "criteria": { "A": "3x+1", "B": "2x+1", "C": "x+1", "D": "2x+3" } }, "10.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: A degree-one polynomial over GF(5) was evaluated; received (0,1),(1,3),(2,1),(3,2), with at most one corrupted value.\nProblem: Find the monic minimal error locator.", "criteria": { "A": "x", "B": "x−3", "C": "x−2", "D": "x−1" } }, "10.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Over GF(p), P has degree exactly d with 02d and d>=1, P and Q have degree exactly d, with r and s distinct roots respectively.\nProblem: Maximum distinct roots of PQ?", "criteria": { "A": "max(r,s)", "B": "r*s", "C": "min(r,s)", "D": "r+s" } }, "10.4b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Over GF(p) with p>2d and d>=1, P and Q have degree exactly d, with r and s distinct roots respectively.\nProblem: Minimum distinct roots of PQ?", "criteria": { "A": "r+s", "B": "max(r,s)", "C": "min(r,s)", "D": "0" } }, "10.4c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Over GF(p) with p>2d and d>=1, P and Q have degree exactly d, with r and s distinct roots respectively.\nProblem: Degree of PQ?", "criteria": { "A": "d²", "B": "d", "C": "At most d−1", "D": "2d" } }, "10.4d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Over GF(p) with p>2d and d>=1, P and Q have degree exactly d, with r and s distinct roots respectively.\nProblem: Maximum degree of P+Q?", "criteria": { "A": "0", "B": "2d", "C": "d−1", "D": "d" } }, "10.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Why does a degree-one polynomial over a field have exactly one root?", "criteria": { "A": "All field elements have inverses, including zero, so every polynomial has a unique root.", "B": "The root is always the constant coefficient b.", "C": "Write P(x)=mx+b with m nonzero. The equation has the unique solution x=−m^(-1)b.", "D": "A degree-one polynomial has at most one root, which alone proves existence." } }, "10.6a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Over GF(p), count degree-exactly-two polynomials that split into linear factors, including repeated roots.", "criteria": { "A": "p²", "B": "(p−1)C(p,2)", "C": "(p−1)(C(p,2)+p)", "D": "(p−1)p²" } }, "10.6b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Over GF(p), count all formal polynomials of degree exactly two.", "criteria": { "A": "(p−1)(C(p,2)+p)", "B": "(p−1)³", "C": "p³", "D": "(p−1)p²" } } } }, "attempts": [ { "status": 200, "full_response_ms": 343.57004200137453, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"10.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.67,\"probabilities\":{\"B\":0.01,\"D\":0.01,\"A\":0.23,\"C\":0.75}},\"10.2a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.35,\"probabilities\":{\"B\":0.51,\"D\":0.09,\"A\":0.36,\"C\":0.04}},\"10.2b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.3,\"probabilities\":{\"D\":0.26,\"B\":0.18,\"A\":0.09,\"C\":0.47}},\"10.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.97,\"probabilities\":{\"B\":0.0,\"D\":0.01,\"A\":0.98,\"C\":0.01}},\"10.4a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.98,\"probabilities\":{\"B\":0.01,\"D\":0.98,\"A\":0.01,\"C\":0.0}},\"10.4b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.33,\"probabilities\":{\"B\":0.44,\"D\":0.03,\"A\":0.49,\"C\":0.04}},\"10.4c\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"D\":1.0,\"A\":0.0,\"C\":0.0}},\"10.4d\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.77,\"probabilities\":{\"B\":0.07,\"D\":0.84,\"A\":0.0,\"C\":0.09}},\"10.5\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":0.0,\"C\":1.0}},\"10.6a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.8,\"probabilities\":{\"B\":0.02,\"D\":0.06,\"A\":0.07,\"C\":0.85}},\"10.6b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.85,\"probabilities\":{\"B\":0.06,\"D\":0.89,\"A\":0.05,\"C\":0.0}}},\"usage\":{\"input_tokens\":1780,\"output_tokens\":506}}" } ], "response": { "model": "jev-1.13.0", "answers": { "10.1": { "type": "choice", "choice": "C", "confidence": 0.67, "probabilities": { "B": 0.01, "D": 0.01, "A": 0.23, "C": 0.75 } }, "10.2a": { "type": "choice", "choice": "B", "confidence": 0.35, "probabilities": { "B": 0.51, "D": 0.09, "A": 0.36, "C": 0.04 } }, "10.2b": { "type": "choice", "choice": "C", "confidence": 0.3, "probabilities": { "D": 0.26, "B": 0.18, "A": 0.09, "C": 0.47 } }, "10.3": { "type": "choice", "choice": "A", "confidence": 0.97, "probabilities": { "B": 0.0, "D": 0.01, "A": 0.98, "C": 0.01 } }, "10.4a": { "type": "choice", "choice": "D", "confidence": 0.98, "probabilities": { "B": 0.01, "D": 0.98, "A": 0.01, "C": 0.0 } }, "10.4b": { "type": "choice", "choice": "A", "confidence": 0.33, "probabilities": { "B": 0.44, "D": 0.03, "A": 0.49, "C": 0.04 } }, "10.4c": { "type": "choice", "choice": "D", "confidence": 0.99, "probabilities": { "B": 0.0, "D": 1.0, "A": 0.0, "C": 0.0 } }, "10.4d": { "type": "choice", "choice": "D", "confidence": 0.77, "probabilities": { "B": 0.07, "D": 0.84, "A": 0.0, "C": 0.09 } }, "10.5": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 } }, "10.6a": { "type": "choice", "choice": "C", "confidence": 0.8, "probabilities": { "B": 0.02, "D": 0.06, "A": 0.07, "C": 0.85 } }, "10.6b": { "type": "choice", "choice": "D", "confidence": 0.85, "probabilities": { "B": 0.06, "D": 0.89, "A": 0.05, "C": 0.0 } } }, "usage": { "input_tokens": 1780, "output_tokens": 506 } } }, { "exam": "midterm_sp25", "section": 11, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "11.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: 70 groups each hold one point of a degree-49 shared polynomial. Within each group, 10 delegates hold evaluations of a degree-5 polynomial encoding that group’s point. Adversaries know all polynomials and choose their groups. Berlekamp–Welch is applied at both levels.\nProblem: What is the largest number of adversarial delegates guaranteed tolerable? Give the threshold argument.", "criteria": { "A": "32: each group corrects two bad shares, so corrupting a group needs three delegates; the outer code corrects ten bad groups, and defeating that guarantee needs eleven groups, costing 33 delegates.", "B": "30: three delegates per group times ten groups is the first failure.", "C": "22: the outer code tolerates eleven groups and each needs two delegates.", "D": "20: each of ten bad groups needs two delegates, so the 21st defeats the scheme." } } } }, "attempts": [ { "status": 200, "full_response_ms": 283.5174169995298, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"11.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.79,\"probabilities\":{\"C\":0.01,\"D\":0.11,\"A\":0.85,\"B\":0.03}}},\"usage\":{\"input_tokens\":593,\"output_tokens\":48}}" } ], "response": { "model": "jev-1.13.0", "answers": { "11.1": { "type": "choice", "choice": "A", "confidence": 0.79, "probabilities": { "C": 0.01, "D": 0.11, "A": 0.85, "B": 0.03 } } }, "usage": { "input_tokens": 593, "output_tokens": 48 } } }, { "exam": "midterm_sp25", "section": 12, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "12.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct arrangements of BAA?", "criteria": { "A": "3", "B": "1", "C": "6", "D": "2" } }, "12.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Distinct arrangements of 126?", "criteria": { "A": "3", "B": "6", "C": "1", "D": "9" } }, "12.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: n distinct bears, k distinct children, n>k. Assign exactly one distinct bear to each child. How many assignments?", "criteria": { "A": "k^n", "B": "C(n,k)", "C": "n^k", "D": "n!/(n−k)!" } }, "12.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of positive integer n-tuples summing to k, k>=n>=1?", "criteria": { "A": "C(k+n−1,n−1)", "B": "C(k−1,n−1)", "C": "n^k", "D": "C(k,n)" } }, "12.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Number of n-bit strings with exactly k ones, 0<=k<=n?", "criteria": { "A": "n!/k!", "B": "C(n,k)", "C": "2^k", "D": "n^k" } }, "12.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Circular arrangements of n distinct beads, identifying rotations but not reflections?", "criteria": { "A": "(n−1)!", "B": "(n−1)!/2", "C": "2^n", "D": "n!" } } } }, "attempts": [ { "status": 200, "full_response_ms": 400.84479200231726, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"12.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.82,\"probabilities\":{\"D\":0.12,\"C\":0.01,\"A\":0.86,\"B\":0.01}},\"12.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.34,\"probabilities\":{\"D\":0.05,\"C\":0.3,\"A\":0.13,\"B\":0.52}},\"12.3\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.99,\"probabilities\":{\"D\":1.0,\"C\":0.0,\"A\":0.0,\"B\":0.0}},\"12.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"D\":0.0,\"C\":0.0,\"A\":0.0,\"B\":1.0}},\"12.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":0.0,\"C\":0.0,\"B\":1.0}},\"12.6\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"C\":0.0,\"A\":1.0,\"B\":0.0}}},\"usage\":{\"input_tokens\":987,\"output_tokens\":273}}" } ], "response": { "model": "jev-1.13.0", "answers": { "12.1": { "type": "choice", "choice": "A", "confidence": 0.82, "probabilities": { "D": 0.12, "C": 0.01, "A": 0.86, "B": 0.01 } }, "12.2": { "type": "choice", "choice": "B", "confidence": 0.34, "probabilities": { "D": 0.05, "C": 0.3, "A": 0.13, "B": 0.52 } }, "12.3": { "type": "choice", "choice": "D", "confidence": 0.99, "probabilities": { "D": 1.0, "C": 0.0, "A": 0.0, "B": 0.0 } }, "12.4": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "D": 0.0, "C": 0.0, "A": 0.0, "B": 1.0 } }, "12.5": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.0, "B": 1.0 } }, "12.6": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "C": 0.0, "A": 1.0, "B": 0.0 } } }, "usage": { "input_tokens": 987, "output_tokens": 273 } } }, { "exam": "midterm_su25", "section": 2, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "2.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: What is the truth status of (P AND NOT P) implies Q?", "criteria": { "A": "Could be either", "B": "True", "C": "False" } }, "2.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: What is the truth status of P implies (Q OR NOT Q)?", "criteria": { "A": "Could be either", "B": "False", "C": "True" } }, "2.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: What is the truth status of (P OR NOT P) implies (Q AND NOT Q)?", "criteria": { "A": "Could be either", "B": "False", "C": "True" } }, "2.1d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: What is the truth status of (P OR Q) implies (P AND Q)?", "criteria": { "A": "False", "B": "Could be either", "C": "True" } }, "2.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Over nonempty domains, are forall x [P(x) implies exists y Q(x,y)] and forall x exists y [P(x) implies Q(x,y)] equivalent?", "criteria": { "A": "Equivalent", "B": "Not equivalent" } }, "2.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Are exists x [P(x) implies forall y R(x,y)] and forall x [P(x) OR exists y R(x,y)] equivalent?", "criteria": { "A": "Not equivalent", "B": "Equivalent" } }, "2.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: R(x)=NOT P(x), S(x,y)=NOT Q(x,y). Rewrite NOT[(forall x P(x)) AND (exists x forall y Q(x,y))] without NOT.", "criteria": { "A": "(exists x R(x)) AND (forall x exists y S(x,y))", "B": "(exists x R(x)) OR (forall x exists y S(x,y))", "C": "(forall x R(x)) OR (exists x forall y S(x,y))", "D": "(forall x R(x)) AND (exists x forall y S(x,y))" } } } }, "attempts": [ { "status": 200, "full_response_ms": 357.25633300171467, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"2.1a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.98,\"probabilities\":{\"B\":0.99,\"C\":0.0,\"A\":0.01}},\"2.1b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.98,\"probabilities\":{\"B\":0.0,\"C\":0.99,\"A\":0.01}},\"2.1c\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.82,\"probabilities\":{\"B\":0.88,\"C\":0.01,\"A\":0.11}},\"2.1d\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.35,\"probabilities\":{\"B\":0.43,\"C\":0.0,\"A\":0.5700000000000001}},\"2.2a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.81,\"probabilities\":{\"B\":0.1,\"A\":0.9}},\"2.2b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.83,\"probabilities\":{\"B\":0.09,\"A\":0.91}},\"2.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.93,\"probabilities\":{\"B\":0.95,\"C\":0.05,\"D\":0.0,\"A\":0.0}}},\"usage\":{\"input_tokens\":1075,\"output_tokens\":261}}" } ], "response": { "model": "jev-1.13.0", "answers": { "2.1a": { "type": "choice", "choice": "B", "confidence": 0.98, "probabilities": { "B": 0.99, "C": 0.0, "A": 0.01 } }, "2.1b": { "type": "choice", "choice": "C", "confidence": 0.98, "probabilities": { "B": 0.0, "C": 0.99, "A": 0.01 } }, "2.1c": { "type": "choice", "choice": "B", "confidence": 0.82, "probabilities": { "B": 0.88, "C": 0.01, "A": 0.11 } }, "2.1d": { "type": "choice", "choice": "A", "confidence": 0.35, "probabilities": { "B": 0.43, "C": 0.0, "A": 0.5700000000000001 } }, "2.2a": { "type": "choice", "choice": "A", "confidence": 0.81, "probabilities": { "B": 0.1, "A": 0.9 } }, "2.2b": { "type": "choice", "choice": "A", "confidence": 0.83, "probabilities": { "B": 0.09, "A": 0.91 } }, "2.3": { "type": "choice", "choice": "B", "confidence": 0.93, "probabilities": { "B": 0.95, "C": 0.05, "D": 0.0, "A": 0.0 } } }, "usage": { "input_tokens": 1075, "output_tokens": 261 } } }, { "exam": "midterm_su25", "section": 3, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "3.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For integers a,b,c, prove a does not divide 2b+3c implies a does not divide b or a does not divide c.", "criteria": { "A": "Contrapositive: b=ak and c=al imply 2b+3c=a(2k+3l), so a divides the sum.", "B": "Since a does not divide the sum, it cannot divide either 2b or 3c individually.", "C": "If a fails to divide b and c, it also fails to divide every linear combination.", "D": "If a divides 2b+3c, then it divides b and c separately; use the converse." } }, "3.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: State AM–GM for two nonnegative numbers x,y.", "criteria": { "A": "sqrt(x+y) >= x*y", "B": "x+y >= xy", "C": "(x+y)/2 >= sqrt(xy)", "D": "(x+y)/2 <= sqrt(xy)" } }, "3.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Which proves AM–GM for nonnegative x,y?", "criteria": { "A": "(x−y)^2>=0 gives (x+y)^2>=4xy; both sides of x+y>=2sqrt(xy) are nonnegative, so taking square roots is valid.", "B": "Since x>=y and y>=x, adding yields equality in AM–GM.", "C": "Squaring any inequality preserves its direction, even for negative sides; apply this to sqrt(xy).", "D": "x^2+y^2>=0 gives x+y>=2sqrt(xy) directly by distributing the square root over a sum." } } } }, "attempts": [ { "status": 200, "full_response_ms": 447.66695799989975, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"3.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"D\":0.0,\"B\":0.0,\"A\":1.0}},\"3.2a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"C\":1.0,\"D\":0.0,\"B\":0.0,\"A\":0.0}},\"3.2b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"D\":0.0,\"B\":0.0,\"A\":1.0}}},\"usage\":{\"input_tokens\":855,\"output_tokens\":137}}" } ], "response": { "model": "jev-1.13.0", "answers": { "3.1": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "C": 0.0, "D": 0.0, "B": 0.0, "A": 1.0 } }, "3.2a": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "C": 1.0, "D": 0.0, "B": 0.0, "A": 0.0 } }, "3.2b": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "C": 0.0, "D": 0.0, "B": 0.0, "A": 1.0 } } }, "usage": { "input_tokens": 855, "output_tokens": 137 } } }, { "exam": "midterm_su25", "section": 4, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "4.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: S0=3, S1=2, S2=1; for n>=3, Sn=S(n−1)+S(n−2)+2S(n−3).\nProblem: Smallest integer c with Sn<=c*2^n for all n>=0?", "criteria": { "A": "3", "B": "2", "C": "4", "D": "1" } }, "4.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: S0=3, S1=2, S2=1; for n>=3, Sn=S(n−1)+S(n−2)+2S(n−3).\nProblem: Which is a valid induction proving Sn<=3*2^n?", "criteria": { "A": "Check n=0,1,2. Strong induction gives S(k+1)<=3*2^k+3*2^(k−1)+6*2^(k−2)=3*2^(k+1).", "B": "Check the first three terms and use S(k+1)=2Sk.", "C": "Every term is at most twice its predecessor, including S3, so ordinary induction is immediate.", "D": "Check n=0 only. Assume Sk<=3*2^k; this also establishes bounds on earlier terms without any additional hypothesis." } }, "4.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: S0=3, S1=2, S2=1; for n>=3, Sn=S(n−1)+S(n−2)+2S(n−3).\nProblem: Smallest integer c with Sn<=c*2^n for every n>=4?", "criteria": { "A": "1", "B": "3", "C": "2", "D": "0" } }, "4.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Which induction proves 7 divides 11^n−4^n for n>=0?", "criteria": { "A": "Base n=1 is 7. Since both bases are odd, the difference must be divisible by 7.", "B": "The first seven values suffice because every exponential sequence has period seven.", "C": "Base n=0 is zero. Write 11^(k+1)−4^(k+1)=7*11^k+4(11^k−4^k), so the inductive divisibility persists.", "D": "Base n=0 is zero. Multiply the difference by 7 to obtain the next difference." } } } }, "attempts": [ { "status": 200, "full_response_ms": 336.0027919989079, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"4.1a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.1,\"probabilities\":{\"C\":0.07,\"B\":0.33,\"A\":0.31,\"D\":0.29}},\"4.1b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"A\":1.0,\"B\":0.0,\"D\":0.0}},\"4.1c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.49,\"probabilities\":{\"C\":0.19,\"B\":0.16,\"A\":0.62,\"D\":0.03}},\"4.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"C\":1.0,\"B\":0.0,\"A\":0.0,\"D\":0.0}}},\"usage\":{\"input_tokens\":1059,\"output_tokens\":182}}" } ], "response": { "model": "jev-1.13.0", "answers": { "4.1a": { "type": "choice", "choice": "B", "confidence": 0.1, "probabilities": { "C": 0.07, "B": 0.33, "A": 0.31, "D": 0.29 } }, "4.1b": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "C": 0.0, "A": 1.0, "B": 0.0, "D": 0.0 } }, "4.1c": { "type": "choice", "choice": "A", "confidence": 0.49, "probabilities": { "C": 0.19, "B": 0.16, "A": 0.62, "D": 0.03 } }, "4.2": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "C": 1.0, "B": 0.0, "A": 0.0, "D": 0.0 } } }, "usage": { "input_tokens": 1059, "output_tokens": 182 } } }, { "exam": "midterm_su25", "section": 5, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "5.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Jobs: A:3>1>2, B:2>3>1, C:3>2>1. Candidates: 1:B>C>A, 2:C>A>B, 3:A>C>B.\nProblem: Find the job-optimal stable matching.", "criteria": { "A": "A–3, B–1, C–2", "B": "A–2, B–1, C–3", "C": "A–3, B–2, C–1", "D": "A–1, B–2, C–3" } }, "5.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Jobs: A:3>1>2, B:2>3>1, C:3>2>1. Candidates: 1:B>C>A, 2:C>A>B, 3:A>C>B.\nProblem: Find the candidate-optimal stable matching.", "criteria": { "A": "A–3, B–2, C–1", "B": "A–2, B–1, C–3", "C": "A–1, B–2, C–3", "D": "A–3, B–1, C–2" } }, "5.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Jobs: A:3>1>2, B:2>3>1, C:3>2>1. Candidates: 1:B>C>A, 2:C>A>B, 3:A>C>B.\nProblem: Is there any other stable matching beyond the outputs of the two proposer directions?", "criteria": { "A": "No; the two extremes coincide.", "B": "Yes: A–2, B–1, C–3.", "C": "Yes: A–1, B–2, C–3.", "D": "Yes: A–3, B–2, C–1." } }, "5.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If every job has a different first-choice candidate, there must be exactly one stable matching.", "criteria": { "A": "False", "B": "True" } }, "5.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: In the job-optimal stable matching, at least one job must get its first-choice candidate.", "criteria": { "A": "True", "B": "False" } }, "5.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A job can receive its least-preferred candidate in the job-optimal stable matching.", "criteria": { "A": "False", "B": "True" } }, "5.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If a job is matched to its favorite candidate, that candidate cannot participate in a blocking pair.", "criteria": { "A": "True", "B": "False" } }, "5.6": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For every preference instance, modifying only candidate preferences can change the job-proposing algorithm’s output.", "criteria": { "A": "False", "B": "True" } }, "5.7": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For every n>=2, construct preferences where job-proposing matches every job to its favorite and every candidate to its least favorite.", "criteria": { "A": "All jobs and candidates have identical ranking lists; everyone therefore receives their first choice on one side and last on the other.", "B": "Each Ji and Ci rank one another first; jobs get their favorite while candidates get their least favorite.", "C": "Each job ranks a different candidate last and proposes to that candidate first, as required by deferred acceptance.", "D": "Each Ji ranks Ci first, and each Ci ranks Ji last; complete all remaining rankings strictly in any order. All jobs propose to distinct first choices, so those proposals are accepted." } } } }, "attempts": [ { "status": 200, "full_response_ms": 329.3755000013334, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"5.1a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.24,\"probabilities\":{\"B\":0.06,\"D\":0.11,\"A\":0.42,\"C\":0.41}},\"5.1b\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.52,\"probabilities\":{\"D\":0.08,\"B\":0.06,\"A\":0.63,\"C\":0.23}},\"5.1c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.26,\"probabilities\":{\"B\":0.14,\"D\":0.14,\"A\":0.45,\"C\":0.27}},\"5.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.09,\"probabilities\":{\"B\":0.45,\"A\":0.55}},\"5.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.22,\"probabilities\":{\"B\":0.61,\"A\":0.39}},\"5.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.29,\"probabilities\":{\"B\":0.65,\"A\":0.35}},\"5.5\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.02,\"probabilities\":{\"B\":0.49,\"A\":0.51}},\"5.6\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.48,\"probabilities\":{\"B\":0.26,\"A\":0.74}},\"5.7\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.95,\"probabilities\":{\"B\":0.01,\"D\":0.97,\"A\":0.02,\"C\":0.0}}},\"usage\":{\"input_tokens\":1490,\"output_tokens\":332}}" } ], "response": { "model": "jev-1.13.0", "answers": { "5.1a": { "type": "choice", "choice": "A", "confidence": 0.24, "probabilities": { "B": 0.06, "D": 0.11, "A": 0.42, "C": 0.41 } }, "5.1b": { "type": "choice", "choice": "A", "confidence": 0.52, "probabilities": { "D": 0.08, "B": 0.06, "A": 0.63, "C": 0.23 } }, "5.1c": { "type": "choice", "choice": "A", "confidence": 0.26, "probabilities": { "B": 0.14, "D": 0.14, "A": 0.45, "C": 0.27 } }, "5.2": { "type": "choice", "choice": "A", "confidence": 0.09, "probabilities": { "B": 0.45, "A": 0.55 } }, "5.3": { "type": "choice", "choice": "B", "confidence": 0.22, "probabilities": { "B": 0.61, "A": 0.39 } }, "5.4": { "type": "choice", "choice": "B", "confidence": 0.29, "probabilities": { "B": 0.65, "A": 0.35 } }, "5.5": { "type": "choice", "choice": "A", "confidence": 0.02, "probabilities": { "B": 0.49, "A": 0.51 } }, "5.6": { "type": "choice", "choice": "A", "confidence": 0.48, "probabilities": { "B": 0.26, "A": 0.74 } }, "5.7": { "type": "choice", "choice": "D", "confidence": 0.95, "probabilities": { "B": 0.01, "D": 0.97, "A": 0.02, "C": 0.0 } } }, "usage": { "input_tokens": 1490, "output_tokens": 332 } } }, { "exam": "midterm_su25", "section": 6, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "6.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A six-vertex graph with four edges is connected.", "criteria": { "A": "False", "B": "Could be either", "C": "True" } }, "6.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A six-vertex graph with six edges is connected.", "criteria": { "A": "Could be either", "B": "True", "C": "False" } }, "6.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A six-vertex graph with six edges is acyclic.", "criteria": { "A": "Could be either", "B": "True", "C": "False" } }, "6.1d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A six-vertex graph with thirteen edges is connected.", "criteria": { "A": "True", "B": "False", "C": "Could be either" } }, "6.1e": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A six-vertex graph with thirteen edges is planar.", "criteria": { "A": "False", "B": "Could be either", "C": "True" } }, "6.1f": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Maximum edges of a simple graph on six vertices?", "criteria": { "A": "15", "B": "12", "C": "18", "D": "30" } }, "6.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Minimum edges of an undirected graph with V vertices and C connected components?", "criteria": { "A": "C−1", "B": "V+C", "C": "V−1", "D": "V−C" } }, "6.3a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: K7 has an Eulerian tour.", "criteria": { "A": "True", "B": "False" } }, "6.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Why does K7 have an Eulerian tour?", "criteria": { "A": "Every complete graph has all degrees odd.", "B": "It is planar and connected.", "C": "It has an odd number of vertices, which alone guarantees a tour.", "D": "It is connected and each vertex has even degree 6." } }, "6.4a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: K8 has an Eulerian tour.", "criteria": { "A": "True", "B": "False" } }, "6.4b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Why does K8 lack an Eulerian tour?", "criteria": { "A": "It is disconnected.", "B": "It has too many edges to be bipartite, which forbids Eulerian tours.", "C": "It has an even number of vertices, and no even-order graph has a tour.", "D": "Each vertex has odd degree 7, whereas a connected Eulerian graph requires all degrees even." } }, "6.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A connected planar embedding has 15 vertices and 4 faces including the outer face. How many edges?", "criteria": { "A": "13", "B": "17", "C": "19", "D": "11" } } } }, "attempts": [ { "status": 200, "full_response_ms": 367.22316699888324, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"6.1a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.24,\"probabilities\":{\"C\":0.03,\"B\":0.49,\"A\":0.48}},\"6.1b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.26,\"probabilities\":{\"C\":0.51,\"B\":0.09,\"A\":0.4}},\"6.1c\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.68,\"probabilities\":{\"C\":0.79,\"B\":0.01,\"A\":0.2}},\"6.1d\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.55,\"probabilities\":{\"C\":0.13,\"B\":0.18,\"A\":0.69}},\"6.1e\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.85,\"probabilities\":{\"C\":0.08,\"B\":0.02,\"A\":0.9}},\"6.1f\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":0.0,\"A\":1.0,\"D\":0.0}},\"6.2\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":0.0,\"A\":0.0,\"D\":1.0}},\"6.3a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.62,\"probabilities\":{\"B\":0.19,\"A\":0.81}},\"6.3b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":0.0,\"A\":0.0,\"D\":1.0}},\"6.4a\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.59,\"probabilities\":{\"B\":0.79,\"A\":0.21}},\"6.4b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":0.0,\"A\":0.0,\"D\":1.0}},\"6.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.42,\"probabilities\":{\"C\":0.37,\"B\":0.57,\"A\":0.02,\"D\":0.04}}},\"usage\":{\"input_tokens\":1523,\"output_tokens\":478}}" } ], "response": { "model": "jev-1.13.0", "answers": { "6.1a": { "type": "choice", "choice": "B", "confidence": 0.24, "probabilities": { "C": 0.03, "B": 0.49, "A": 0.48 } }, "6.1b": { "type": "choice", "choice": "C", "confidence": 0.26, "probabilities": { "C": 0.51, "B": 0.09, "A": 0.4 } }, "6.1c": { "type": "choice", "choice": "C", "confidence": 0.68, "probabilities": { "C": 0.79, "B": 0.01, "A": 0.2 } }, "6.1d": { "type": "choice", "choice": "A", "confidence": 0.55, "probabilities": { "C": 0.13, "B": 0.18, "A": 0.69 } }, "6.1e": { "type": "choice", "choice": "A", "confidence": 0.85, "probabilities": { "C": 0.08, "B": 0.02, "A": 0.9 } }, "6.1f": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 1.0, "D": 0.0 } }, "6.2": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 0.0, "D": 1.0 } }, "6.3a": { "type": "choice", "choice": "A", "confidence": 0.62, "probabilities": { "B": 0.19, "A": 0.81 } }, "6.3b": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 0.0, "D": 1.0 } }, "6.4a": { "type": "choice", "choice": "B", "confidence": 0.59, "probabilities": { "B": 0.79, "A": 0.21 } }, "6.4b": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 0.0, "D": 1.0 } }, "6.5": { "type": "choice", "choice": "B", "confidence": 0.42, "probabilities": { "C": 0.37, "B": 0.57, "A": 0.02, "D": 0.04 } } }, "usage": { "input_tokens": 1523, "output_tokens": 478 } } }, { "exam": "midterm_su25", "section": 7, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "7.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Prove that a graph all of whose vertices have odd degree has an even number of vertices.", "criteria": { "A": "Every vertex must have an odd number of neighbors, so all vertices can be paired along disjoint edges.", "B": "Each odd degree is at least one, so the vertex count equals twice the edge count.", "C": "The degree sum is twice the number of edges and therefore even. A sum of an odd number of odd integers is odd, so the number of vertices must be even.", "D": "All odd-degree graphs are complete, and complete graphs have even order." } }, "7.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: In a finite tree with at least two vertices, start walking along unused edges until stuck. Why does this reach a degree-one vertex?", "criteria": { "A": "A tree has no edges, so the starting vertex already has degree one.", "B": "Every vertex of a tree has degree at most two; hence any arbitrary walk ends at a leaf.", "C": "A repeated vertex would create a cycle. The walk is finite and its terminal vertex was not previously visited; only its arrival edge has been used there, so being stuck means it has no other incident edge.", "D": "Each step decreases the total degree by two, so eventually every vertex has degree one." } } } }, "attempts": [ { "status": 200, "full_response_ms": 281.9140419996984, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"7.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"A\":0.0,\"C\":1.0,\"D\":0.0}},\"7.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"A\":0.0,\"C\":1.0,\"D\":0.0}}},\"usage\":{\"input_tokens\":719,\"output_tokens\":91}}" } ], "response": { "model": "jev-1.13.0", "answers": { "7.1": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "A": 0.0, "C": 1.0, "D": 0.0 } }, "7.2": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "B": 0.0, "A": 0.0, "C": 1.0, "D": 0.0 } } }, "usage": { "input_tokens": 719, "output_tokens": 91 } } }, { "exam": "midterm_su25", "section": 8, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "8.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Compute 5^42 modulo 41.", "criteria": { "A": "1", "B": "0", "C": "5", "D": "25" } }, "8.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Last digit of 519^487?", "criteria": { "A": "5", "B": "9", "C": "7", "D": "1" } }, "8.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Compute 43^21 modulo 55.", "criteria": { "A": "43", "B": "1", "C": "23", "D": "12" } }, "8.4": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Choose nonzero a modulo m for which ax=0 modulo m has a nonzero solution.", "criteria": { "A": "a=4,m=9", "B": "a=2,m=5", "C": "a=3,m=7", "D": "a=2,m=4" } }, "8.5": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For positive integers k1?", "criteria": { "A": "3", "B": "2", "C": "7", "D": "5" } }, "8.8b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: RSA has p=11,q=7,e=29. What is the least positive decryption exponent?", "criteria": { "A": "31", "B": "29", "C": "59", "D": "7" } } } }, "attempts": [ { "status": 200, "full_response_ms": 757.2612500007381, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"8.1\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.27,\"probabilities\":{\"D\":0.46,\"B\":0.01,\"A\":0.29,\"C\":0.24}},\"8.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.63,\"probabilities\":{\"B\":0.72,\"D\":0.05,\"A\":0.07,\"C\":0.16}},\"8.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.15,\"probabilities\":{\"B\":0.08,\"D\":0.25,\"A\":0.36,\"C\":0.31}},\"8.4\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"B\":0.0,\"A\":0.0,\"C\":0.0}},\"8.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.98,\"probabilities\":{\"B\":0.99,\"A\":0.01}},\"8.6\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"A\":1.0}},\"8.7\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.71,\"probabilities\":{\"B\":0.15,\"A\":0.85}},\"8.8a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.64,\"probabilities\":{\"B\":0.2,\"D\":0.06,\"A\":0.73,\"C\":0.01}},\"8.8b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.48,\"probabilities\":{\"B\":0.2,\"D\":0.09,\"A\":0.1,\"C\":0.61}}},\"usage\":{\"input_tokens\":1225,\"output_tokens\":359}}" } ], "response": { "model": "jev-1.13.0", "answers": { "8.1": { "type": "choice", "choice": "D", "confidence": 0.27, "probabilities": { "D": 0.46, "B": 0.01, "A": 0.29, "C": 0.24 } }, "8.2": { "type": "choice", "choice": "B", "confidence": 0.63, "probabilities": { "B": 0.72, "D": 0.05, "A": 0.07, "C": 0.16 } }, "8.3": { "type": "choice", "choice": "A", "confidence": 0.15, "probabilities": { "B": 0.08, "D": 0.25, "A": 0.36, "C": 0.31 } }, "8.4": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "A": 0.0, "C": 0.0 } }, "8.5": { "type": "choice", "choice": "B", "confidence": 0.98, "probabilities": { "B": 0.99, "A": 0.01 } }, "8.6": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "B": 0.0, "A": 1.0 } }, "8.7": { "type": "choice", "choice": "A", "confidence": 0.71, "probabilities": { "B": 0.15, "A": 0.85 } }, "8.8a": { "type": "choice", "choice": "A", "confidence": 0.64, "probabilities": { "B": 0.2, "D": 0.06, "A": 0.73, "C": 0.01 } }, "8.8b": { "type": "choice", "choice": "C", "confidence": 0.48, "probabilities": { "B": 0.2, "D": 0.09, "A": 0.1, "C": 0.61 } } }, "usage": { "input_tokens": 1225, "output_tokens": 359 } } }, { "exam": "midterm_su25", "section": 9, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "9.1": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Without induction, why is 11^n−4^n divisible by 7 for all nonnegative n?", "criteria": { "A": "11−4=7, so 11^n−4^n=7^n as integers.", "B": "11=4 mod 7, so exponentiating preserves congruence: 11^n=4^n mod 7.", "C": "All differences of powers with odd and even bases are multiples of 7.", "D": "Both 11 and 4 are zero modulo 7." } }, "9.2": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: If a has multiplicative order p−1 modulo prime p, why do a^0,...,a^(p−2) cover all nonzero residues?", "criteria": { "A": "They are nonzero. A repeated pair i1.", "criteria": { "A": "2", "B": "0", "C": "m−1", "D": "1" } }, "9.3b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: The residue m−1 is a nontrivial square root of 1 modulo m, where trivial roots mean ±1.", "criteria": { "A": "True", "B": "False" } }, "9.3c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Why can a prime modulus p have no square root of 1 other than ±1?", "criteria": { "A": "x^2−1=(x−1)(x+1); a prime dividing this product divides one factor, so x=1 or −1 mod p.", "B": "A polynomial of degree two always has exactly one root.", "C": "One can cancel x−1 from the equation even when x=1.", "D": "Every composite modulus has no roots of 1, so primes have only the trivial ones." } }, "9.3d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: Which is a nontrivial square root of 1 modulo 15?", "criteria": { "A": "4", "B": "14", "C": "0", "D": "1" } } } }, "attempts": [ { "status": 200, "full_response_ms": 288.97745900030714, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"9.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"A\":0.0,\"D\":0.0,\"B\":1.0}},\"9.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":1.0,\"B\":0.0,\"C\":0.0}},\"9.3a\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.85,\"probabilities\":{\"D\":0.89,\"A\":0.01,\"B\":0.0,\"C\":0.1}},\"9.3b\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.55,\"probabilities\":{\"A\":0.22,\"B\":0.78}},\"9.3c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"A\":1.0,\"B\":0.0,\"C\":0.0}},\"9.3d\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.73,\"probabilities\":{\"D\":0.0,\"A\":0.8,\"B\":0.2,\"C\":0.0}}},\"usage\":{\"input_tokens\":1210,\"output_tokens\":257}}" } ], "response": { "model": "jev-1.13.0", "answers": { "9.1": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "C": 0.0, "A": 0.0, "D": 0.0, "B": 1.0 } }, "9.2": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 1.0, "B": 0.0, "C": 0.0 } }, "9.3a": { "type": "choice", "choice": "D", "confidence": 0.85, "probabilities": { "D": 0.89, "A": 0.01, "B": 0.0, "C": 0.1 } }, "9.3b": { "type": "choice", "choice": "B", "confidence": 0.55, "probabilities": { "A": 0.22, "B": 0.78 } }, "9.3c": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "A": 1.0, "B": 0.0, "C": 0.0 } }, "9.3d": { "type": "choice", "choice": "A", "confidence": 0.73, "probabilities": { "D": 0.0, "A": 0.8, "B": 0.2, "C": 0.0 } } }, "usage": { "input_tokens": 1210, "output_tokens": 257 } } }, { "exam": "midterm_su25", "section": 10, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "10.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P,Q are distinct formal polynomials over GF(p), with degrees d1,d2=2. Smallest possible degree of P−Q?", "criteria": { "A": "d1", "B": "Undefined because it must be zero", "C": "0", "D": "d1−1" } }, "10.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: P,Q are distinct formal polynomials over GF(p), with degrees d1,d2=2. Smallest possible number of x values where P(x)=Q(x)?", "criteria": { "A": "1", "B": "d1", "C": "0", "D": "p" } }, "10.2a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Shamir sharing over GF(7), where any two of three participants can reconstruct the secret P(0).\nProblem: What polynomial degree is used?", "criteria": { "A": "0", "B": "2", "C": "1", "D": "3" } }, "10.2b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Shamir sharing over GF(7), where any two of three participants can reconstruct the secret P(0).\nProblem: Shares are (1,6) and (2,2). What is the secret?", "criteria": { "A": "6", "B": "5", "C": "4", "D": "3" } }, "10.2c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: Shamir sharing over GF(7), where any two of three participants can reconstruct the secret P(0).\nProblem: Shares are (1,6) and (2,2). What should a third share at x=3 contain?", "criteria": { "A": "4", "B": "5", "C": "3", "D": "1" } }, "10.3": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: A k-threshold scheme is reconstructed from k shares. An attacker controls coordinate x1 and knows the other k−1 shares. How can it force the reconstructed secret to v?", "criteria": { "A": "Interpolate the original k shares first, then multiply only its own share by v.", "B": "Interpolate the unique degree-at-most-(k−1) polynomial through the other shares and (0,v), then announce its value at x1.", "C": "Announce the pair (0,v) as its share even though its assigned coordinate is x1.", "D": "Set its share value to v; interpolation always copies any participant’s value to the constant term." } } } }, "attempts": [ { "status": 200, "full_response_ms": 364.7012920009729, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"10.1a\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":1.0,\"C\":0.0}},\"10.1b\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.73,\"probabilities\":{\"B\":0.0,\"D\":0.15,\"A\":0.05,\"C\":0.8}},\"10.1c\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.96,\"probabilities\":{\"B\":0.01,\"D\":0.97,\"A\":0.01,\"C\":0.01}},\"10.1d\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.87,\"probabilities\":{\"B\":0.05,\"D\":0.0,\"A\":0.05,\"C\":0.9}},\"10.2a\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.95,\"probabilities\":{\"B\":0.03,\"D\":0.0,\"A\":0.0,\"C\":0.97}},\"10.2b\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.11,\"probabilities\":{\"B\":0.29,\"D\":0.33,\"A\":0.07,\"C\":0.31}},\"10.2c\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.18,\"probabilities\":{\"B\":0.23,\"D\":0.24,\"A\":0.38,\"C\":0.15}},\"10.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.98,\"probabilities\":{\"B\":0.99,\"D\":0.0,\"A\":0.0,\"C\":0.01}}},\"usage\":{\"input_tokens\":1429,\"output_tokens\":370}}" } ], "response": { "model": "jev-1.13.0", "answers": { "10.1a": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 1.0, "C": 0.0 } }, "10.1b": { "type": "choice", "choice": "C", "confidence": 0.73, "probabilities": { "B": 0.0, "D": 0.15, "A": 0.05, "C": 0.8 } }, "10.1c": { "type": "choice", "choice": "D", "confidence": 0.96, "probabilities": { "B": 0.01, "D": 0.97, "A": 0.01, "C": 0.01 } }, "10.1d": { "type": "choice", "choice": "C", "confidence": 0.87, "probabilities": { "B": 0.05, "D": 0.0, "A": 0.05, "C": 0.9 } }, "10.2a": { "type": "choice", "choice": "C", "confidence": 0.95, "probabilities": { "B": 0.03, "D": 0.0, "A": 0.0, "C": 0.97 } }, "10.2b": { "type": "choice", "choice": "D", "confidence": 0.11, "probabilities": { "B": 0.29, "D": 0.33, "A": 0.07, "C": 0.31 } }, "10.2c": { "type": "choice", "choice": "A", "confidence": 0.18, "probabilities": { "B": 0.23, "D": 0.24, "A": 0.38, "C": 0.15 } }, "10.3": { "type": "choice", "choice": "B", "confidence": 0.98, "probabilities": { "B": 0.99, "D": 0.0, "A": 0.0, "C": 0.01 } } }, "usage": { "input_tokens": 1429, "output_tokens": 370 } } }, { "exam": "midterm_su25", "section": 11, "request": { "model": "jev-1.13.0", "state": { "instructions": "Solve the mathematical problem using only the supplied context. Select one best complete answer. All graphs are simple and undirected unless specified. Standard first-order logic uses nonempty domains. Candidate answers are alternatives, not hints or evidence. No calculator, external tools, answer key, or feedback is available." }, "questions": { "11.1a": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: To transmit n field symbols by Reed–Solomon and correct up to k corrupt evaluations, how many packets suffice?", "criteria": { "A": "n+2k−1", "B": "n+2k", "C": "2n+k", "D": "n+k" } }, "11.1b": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: For n message symbols used as coefficients, what is the degree bound of the message polynomial?", "criteria": { "A": "At most n−1", "B": "At most n+2k", "C": "Exactly n", "D": "Exactly k" } }, "11.1c": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: With at most k errors, what is the degree bound of the minimal error locator E(x), the product over actual error locations?", "criteria": { "A": "At most n−1", "B": "At most k", "C": "Exactly 2k", "D": "Exactly n+k" } }, "11.1d": { "type": "choice", "instructions": "Follow the exam instructions in state. Select the best complete answer to this problem. Context: \nProblem: There are exactly m0 vertices?", "response_mode": "single_choice", "options": { "A": "2n−2", "B": "n+1", "C": "n", "D": "n−1" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "5.2a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 1.0, "A": 0.0, "B": 0.0, "C": 0.0 }, "batch_response_ms": 337.21541599879856, "repaired": false }, { "id": "5.2b", "section": 5, "context": "", "prompt": "Delete k edges from a tree, keeping all vertices. Component count?", "response_mode": "single_choice", "options": { "A": "k", "B": "k−1", "C": "2k", "D": "k+1" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "5.2b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 1.0, "A": 0.0, "B": 0.0, "C": 0.0 }, "batch_response_ms": 337.21541599879856, "repaired": false }, { "id": "5.2c", "section": 5, "context": "", "prompt": "A tree has exactly two degree-five vertices. Sharp minimum number of leaves?", "response_mode": "single_choice", "options": { "A": "10", "B": "6", "C": "5", "D": "8" }, "correct": true, "kind": "multi_step", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "5.2c" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.69, "probabilities": { "A": 0.07, "D": 0.77, "B": 0.14, "C": 0.02 }, "batch_response_ms": 337.21541599879856, "repaired": false }, { "id": "5.3a", "section": 5, "context": "", "prompt": "After removing a simple cycle’s edges from a connected Eulerian graph, every component is Eulerian, allowing the trivial zero-edge tour at an isolated vertex.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "5.3a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.82, "probabilities": { "A": 0.91, "B": 0.09 }, "batch_response_ms": 337.21541599879856, "repaired": false }, { "id": "5.3b", "section": 5, "context": "", "prompt": "After removing a simple cycle’s edges from a connected Eulerian simple graph, c components remain. Sharp universal lower bound on removed cycle length?", "response_mode": "single_choice", "options": { "A": "c", "B": "c+1", "C": "max(3,c)", "D": "2c" }, "correct": true, "kind": "multi_step", "topic": "graphs", "original_format": "written", "repairs": [ "Include minimum length three of a simple cycle; source c alone is not tight for c=1 or2." ], "source_parts": [ "5.3b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.33, "probabilities": { "D": 0.07, "A": 0.18, "B": 0.25, "C": 0.5 }, "batch_response_ms": 337.21541599879856, "repaired": true }, { "id": "5.4", "section": 5, "context": "", "prompt": "The four-dimensional hypercube has an Eulerian tour.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "5.4" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.66, "probabilities": { "A": 0.17, "B": 0.83 }, "batch_response_ms": 337.21541599879856, "repaired": false }, { "id": "5.5", "section": 5, "context": "", "prompt": "G is k-colorable and H is bipartite on the same vertices. Sharp universal color bound for their edge union, in terms of k alone?", "response_mode": "single_choice", "options": { "A": "k", "B": "2k", "C": "k+1", "D": "k²" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "5.5" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.56, "probabilities": { "D": 0.16, "A": 0.05, "B": 0.68, "C": 0.11 }, "batch_response_ms": 337.21541599879856, "repaired": false }, { "id": "5.6a", "section": 5, "context": "", "prompt": "Connected planar embedding: n vertices,2n edges. Faces?", "response_mode": "single_choice", "options": { "A": "n", "B": "2n", "C": "n−2", "D": "n+2" }, "correct": false, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "5.6a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.25, "probabilities": { "D": 0.18, "A": 0.37, "B": 0.01, "C": 0.44 }, "batch_response_ms": 337.21541599879856, "repaired": false }, { "id": "5.6b", "section": 5, "context": "", "prompt": "Connected planar embedding: n vertices, average face boundary length five. Edges?", "response_mode": "single_choice", "options": { "A": "2n−4", "B": "3n−6", "C": "5(n−2)/3", "D": "5n/2" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "5.6b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.68, "probabilities": { "D": 0.06, "A": 0.16, "B": 0.02, "C": 0.76 }, "batch_response_ms": 337.21541599879856, "repaired": false }, { "id": "6.1", "section": 6, "context": "", "prompt": "Choose integers (a,b) solving 47a+49b=1.", "response_mode": "single_choice", "options": { "A": "(−24,23)", "B": "(24,−23)", "C": "(23,−22)", "D": "(49,−47)" }, "correct": true, "kind": "multi_step", "topic": "number_theory", "original_format": "written", "repairs": [], "source_parts": [ "6.1" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.38, "probabilities": { "B": 0.53, "D": 0.02, "A": 0.34, "C": 0.11 }, "batch_response_ms": 419.9863750000077, "repaired": false }, { "id": "6.2", "section": 6, "context": "", "prompt": "2^30 modulo 35?", "response_mode": "single_choice", "options": { "A": "8", "B": "29", "C": "1", "D": "16" }, "correct": true, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "6.2" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.21, "probabilities": { "B": 0.41, "D": 0.34, "A": 0.17, "C": 0.08 }, "batch_response_ms": 419.9863750000077, "repaired": false }, { "id": "6.3a", "section": 6, "context": "", "prompt": "Distinct primes p,q. Simplify q^p mod pq.", "response_mode": "single_choice", "options": { "A": "p", "B": "1", "C": "q", "D": "0" }, "correct": false, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "6.3a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.34, "probabilities": { "C": 0.36, "B": 0.04, "A": 0.09, "D": 0.51 }, "batch_response_ms": 419.9863750000077, "repaired": false }, { "id": "6.3b", "section": 6, "context": "", "prompt": "There are pq−__ invertible residues modulo pq for distinct primes p,q. Fill the blank.", "response_mode": "single_choice", "options": { "A": "p+q−1", "B": "p+q", "C": "p+q+1", "D": "pq−1" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "6.3b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.93, "probabilities": { "B": 0.03, "D": 0.0, "A": 0.95, "C": 0.02 }, "batch_response_ms": 419.9863750000077, "repaired": false }, { "id": "7.1", "section": 7, "context": "", "prompt": "Prime p: why is no a in 2..p−2 its own inverse?", "response_mode": "single_choice", "options": { "A": "The inverse operation has no fixed points in any field.", "B": "a=a^(-1) would imply (a−1)(a+1)=0 modp. A field has no zero divisors, forcing a=1 or−1, outside the range.", "C": "All nonzero residues have inverse one.", "D": "Every inverse is larger than its original residue." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "7.1" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 1.0, "C": 0.0 }, "batch_response_ms": 427.8359579984681, "repaired": false }, { "id": "7.2", "section": 7, "context": "", "prompt": "Product of all residues 1,...,p−1 modulo prime p?", "response_mode": "single_choice", "options": { "A": "0", "B": "p−2", "C": "p−1", "D": "1" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "7.2" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.98, "probabilities": { "C": 0.99, "A": 0.0, "B": 0.0, "D": 0.01 }, "batch_response_ms": 427.8359579984681, "repaired": false }, { "id": "7.3", "section": 7, "context": "", "prompt": "Distinct odd primes p,q: construct a nontrivial self-inverse unit modulo pq.", "response_mode": "single_choice", "options": { "A": "Set a=p, since any prime is its own inverse.", "B": "Set a=0, whose inverse equals itself.", "C": "CRT choose a=1 modp and a=−1 modq. Then a²=1 modulo both primes, but a is neither 1 nor−1 modulo pq.", "D": "Use a=pq−1, which lies in the required nontrivial range 2..pq−2." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "7.3" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 0.0, "C": 1.0 }, "batch_response_ms": 427.8359579984681, "repaired": false }, { "id": "8.1", "section": 8, "context": "", "prompt": "Give a degree-two polynomial over GF(5) with roots 0 and1.", "response_mode": "single_choice", "options": { "A": "x(x−1)", "B": "(x−1)²", "C": "x²+1", "D": "x(x+1)" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "8.1" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 1.0, "C": 0.0 }, "batch_response_ms": 386.2307499985036, "repaired": false }, { "id": "8.2", "section": 8, "context": "", "prompt": "Distinct formal P,Q have degrees dP,dQ over GF(p),p>max(dP,dQ). General tight upper bound on coordinates where P=Q?", "response_mode": "single_choice", "options": { "A": "min(dP,dQ)", "B": "dP+dQ", "C": "max(dP,dQ)", "D": "p" }, "correct": false, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [ "Specify field and size for a sharp degree-based bound." ], "source_parts": [ "8.2" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.33, "probabilities": { "B": 0.21, "D": 0.01, "A": 0.49, "C": 0.29 }, "batch_response_ms": 386.2307499985036, "repaired": true }, { "id": "8.3a", "section": 8, "context": "P(x)=mx+b over GF(p), prime p>2. One child receives P(1), another P(2), a third m; b is secret.", "prompt": "Any two children can reconstruct b.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "secret_sharing", "original_format": "choice", "repairs": [], "source_parts": [ "8.3a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.43, "probabilities": { "B": 0.28, "A": 0.72 }, "batch_response_ms": 386.2307499985036, "repaired": false }, { "id": "8.3b", "section": 8, "context": "P(x)=mx+b over GF(p), prime p>2. One child receives P(1), another P(2), a third m; b is secret.", "prompt": "Recover b given m and P(1).", "response_mode": "single_choice", "options": { "A": "P(1)+m modp", "B": "P(1)−m modp", "C": "P(1)/m modp", "D": "m−P(1) modp" }, "correct": true, "kind": "recognition", "topic": "secret_sharing", "original_format": "written", "repairs": [], "source_parts": [ "8.3b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 1.0, "D": 0.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 386.2307499985036, "repaired": false }, { "id": "9.1", "section": 9, "context": "", "prompt": "The set of all subsets of the primes is countable.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "9.1" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.93, "probabilities": { "B": 0.97, "A": 0.03 }, "batch_response_ms": 328.3510840010422, "repaired": false }, { "id": "9.2", "section": 9, "context": "", "prompt": "For sets A1,...,An, let D={i in 1..n: i is not in Ai}. Then D differs from every Ai.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "9.2" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.93, "probabilities": { "B": 0.03, "A": 0.97 }, "batch_response_ms": 328.3510840010422, "repaired": false }, { "id": "9.3", "section": 9, "context": "", "prompt": "Programs that halt on input1 form a countable set.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "9.3" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.9, "probabilities": { "B": 0.95, "A": 0.05 }, "batch_response_ms": 328.3510840010422, "repaired": false }, { "id": "9.4", "section": 9, "context": "", "prompt": "A total algorithm can decide for any P,x and natural n whether P(x) runs for more than n steps.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": false, "kind": "recognition", "topic": "computability", "original_format": "choice", "repairs": [], "source_parts": [ "9.4" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.85, "probabilities": { "B": 0.93, "A": 0.07 }, "batch_response_ms": 328.3510840010422, "repaired": false }, { "id": "10.1", "section": 10, "context": "", "prompt": "Distinct arrangements of CAT?", "response_mode": "single_choice", "options": { "A": "6", "B": "1", "C": "9", "D": "3" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "10.1" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "A": 1.0, "D": 0.0, "B": 0.0 }, "batch_response_ms": 344.4564169985824, "repaired": false }, { "id": "10.2", "section": 10, "context": "", "prompt": "Histograms with k named bins and n observations: nonnegative counts sum to n. Count?", "response_mode": "single_choice", "options": { "A": "k^n", "B": "C(n+k−1,k−1)", "C": "C(n−1,k−1)", "D": "C(n,k)" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "10.2" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 0.0, "A": 0.0, "D": 0.0, "B": 1.0 }, "batch_response_ms": 344.4564169985824, "repaired": false }, { "id": "10.3a", "section": 10, "context": "", "prompt": "A staircase has k steps of heights one or two and total height n,k<=n<=2k. Number of height-two steps?", "response_mode": "single_choice", "options": { "A": "n−k", "B": "2k−n", "C": "k", "D": "n−2k" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "10.3a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "C": 0.0, "A": 0.99, "D": 0.0, "B": 0.01 }, "batch_response_ms": 344.4564169985824, "repaired": false }, { "id": "10.3b", "section": 10, "context": "", "prompt": "Number of ordered staircases of k steps, each height one or two, total n?", "response_mode": "single_choice", "options": { "A": "C(k,n−k)", "B": "C(n−k,k)", "C": "C(n,k)", "D": "2^k" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "10.3b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.53, "probabilities": { "C": 0.01, "A": 0.64, "D": 0.0, "B": 0.35 }, "batch_response_ms": 344.4564169985824, "repaired": false }, { "id": "11.1a", "section": 11, "context": "Uniformly choose bag A:4 red,4 blue or bag B:3 red,5 blue. Draw two marbles without replacement. E:both blue.", "prompt": "P(E given A)?", "response_mode": "single_choice", "options": { "A": "1/2", "B": "5/14", "C": "3/14", "D": "1/4" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "11.1a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.77, "probabilities": { "D": 0.02, "B": 0.15, "A": 0.01, "C": 0.82 }, "batch_response_ms": 374.01329199929023, "repaired": false }, { "id": "11.1b", "section": 11, "context": "Uniformly choose bag A:4 red,4 blue or bag B:3 red,5 blue. Draw two marbles without replacement. E:both blue.", "prompt": "P(A given E)?", "response_mode": "single_choice", "options": { "A": "3/8", "B": "5/8", "C": "1/2", "D": "3/14" }, "correct": true, "kind": "multi_step", "topic": "bayes", "original_format": "written", "repairs": [], "source_parts": [ "11.1b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.3, "probabilities": { "B": 0.37, "C": 0.03, "A": 0.48, "D": 0.12 }, "batch_response_ms": 374.01329199929023, "repaired": false }, { "id": "11.2a", "section": 11, "context": "Roll two independent fair six-sided dice. X=sum modulo3,Y=sum modulo4.", "prompt": "P(X=1)?", "response_mode": "single_choice", "options": { "A": "1/4", "B": "1/3", "C": "5/18", "D": "1/6" }, "correct": true, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "11.2a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.63, "probabilities": { "B": 0.72, "C": 0.24, "A": 0.02, "D": 0.02 }, "batch_response_ms": 374.01329199929023, "repaired": false }, { "id": "11.2b", "section": 11, "context": "Roll two independent fair six-sided dice. X=sum modulo3,Y=sum modulo4.", "prompt": "P(Y=3)?", "response_mode": "single_choice", "options": { "A": "1/4", "B": "1/6", "C": "5/18", "D": "1/3" }, "correct": true, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "11.2b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.23, "probabilities": { "C": 0.43, "D": 0.04, "A": 0.35, "B": 0.18 }, "batch_response_ms": 374.01329199929023, "repaired": false }, { "id": "11.2c", "section": 11, "context": "Roll two independent fair six-sided dice. X=sum modulo3,Y=sum modulo4.", "prompt": "P(X=1 and Y=3)?", "response_mode": "single_choice", "options": { "A": "1/3", "B": "1/6", "C": "5/54", "D": "1/12" }, "correct": false, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "11.2c" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.36, "probabilities": { "B": 0.12, "C": 0.51, "A": 0.01, "D": 0.36 }, "batch_response_ms": 374.01329199929023, "repaired": false }, { "id": "11.2d", "section": 11, "context": "Roll two independent fair six-sided dice. X=sum modulo3,Y=sum modulo4.", "prompt": "P(X=1 or Y=3)?", "response_mode": "single_choice", "options": { "A": "11/18", "B": "4/9", "C": "5/54", "D": "1/2" }, "correct": false, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "11.2d" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.43, "probabilities": { "B": 0.23, "C": 0.01, "A": 0.57, "D": 0.19 }, "batch_response_ms": 374.01329199929023, "repaired": false }, { "id": "11.3a", "section": 11, "context": "Prize uniformly behind one of three doors. You choose door1. Host, unaware of prize, uniformly opens door2 or3; if prize revealed you lose. A:host reveals no prize; B:prize is behind door1.", "prompt": "P(A)?", "response_mode": "single_choice", "options": { "A": "2/3", "B": "1/2", "C": "1", "D": "1/3" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "11.3a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.89, "probabilities": { "B": 0.04, "C": 0.03, "A": 0.91, "D": 0.02 }, "batch_response_ms": 374.01329199929023, "repaired": false }, { "id": "11.3b", "section": 11, "context": "Prize uniformly behind one of three doors. You choose door1. Host, unaware of prize, uniformly opens door2 or3; if prize revealed you lose. A:host reveals no prize; B:prize is behind door1.", "prompt": "P(B given A)?", "response_mode": "single_choice", "options": { "A": "1/3", "B": "2/3", "C": "1", "D": "1/2" }, "correct": true, "kind": "multi_step", "topic": "bayes", "original_format": "written", "repairs": [], "source_parts": [ "11.3b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.26, "probabilities": { "B": 0.24, "C": 0.01, "A": 0.3, "D": 0.45 }, "batch_response_ms": 374.01329199929023, "repaired": false }, { "id": "12.1a", "section": 12, "context": "n distinct balls independently choose uniform bins among n named bins. X counts balls in bin1.", "prompt": "P(X=0)?", "response_mode": "single_choice", "options": { "A": "exp(−n)", "B": "(1−1/n)^n", "C": "(1/n)^n", "D": "1−1/n" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "12.1a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "B": 1.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 330.14270900093834, "repaired": false }, { "id": "12.1b", "section": 12, "context": "n distinct balls independently choose uniform bins among n named bins. X counts balls in bin1.", "prompt": "Distribution of X?", "response_mode": "single_choice", "options": { "A": "Poisson(n)", "B": "Uniform{0,...,n}", "C": "Geom(1/n)", "D": "Bin(n,1/n)" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "12.1b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 1.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 330.14270900093834, "repaired": false }, { "id": "12.1c", "section": 12, "context": "n distinct balls independently choose uniform bins among n named bins. X counts balls in bin1.", "prompt": "Limit of P(X=10) as n grows?", "response_mode": "single_choice", "options": { "A": "1/10", "B": "0", "C": "exp(−1)/10!", "D": "exp(−10)" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "12.1c" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.52, "probabilities": { "D": 0.0, "B": 0.36, "A": 0.0, "C": 0.64 }, "batch_response_ms": 330.14270900093834, "repaired": false }, { "id": "12.2a", "section": 12, "context": "100 independent fair flips. X=number of heads minus number of tails; Y=X².", "prompt": "P(X=0)?", "response_mode": "single_choice", "options": { "A": "1/2", "B": "C(100,0)/2^100", "C": "1/101", "D": "C(100,50)/2^100" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "12.2a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 330.14270900093834, "repaired": false }, { "id": "12.2b", "section": 12, "context": "100 independent fair flips. X=number of heads minus number of tails; Y=X².", "prompt": "E[X]?", "response_mode": "single_choice", "options": { "A": "50", "B": "0", "C": "1/2", "D": "100" }, "correct": true, "kind": "recognition", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "12.2b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 1.0, "D": 0.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 330.14270900093834, "repaired": false }, { "id": "12.2c", "section": 12, "context": "100 independent fair flips. X=number of heads minus number of tails; Y=X².", "prompt": "Are X,Y independent? Justify.", "response_mode": "single_choice", "options": { "A": "Yes; their covariance is zero.", "B": "No; E[Y] is negative.", "C": "Yes; independent flips imply any functions of them are independent.", "D": "No; Y=0 forces X=0, whereas P(X=0)<1." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "12.2c" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 330.14270900093834, "repaired": false }, { "id": "12.2d", "section": 12, "context": "100 independent fair flips. X=number of heads minus number of tails; Y=X².", "prompt": "Cov(X,Y)?", "response_mode": "single_choice", "options": { "A": "0", "B": "100", "C": "−100", "D": "10000" }, "correct": true, "kind": "multi_step", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "12.2d" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.83, "probabilities": { "D": 0.01, "B": 0.11, "A": 0.87, "C": 0.01 }, "batch_response_ms": 330.14270900093834, "repaired": false }, { "id": "13.1a", "section": 13, "context": "Uniform random linear permutation of 2n distinct socks, n>=2 labeled left-right matching pairs. X counts pairs whose two socks are adjacent.", "prompt": "Probability pair1 is adjacent?", "response_mode": "single_choice", "options": { "A": "1/n", "B": "1/(2n−1)", "C": "2/n", "D": "2/(2n−1)" }, "correct": false, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "13.1a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.53, "probabilities": { "D": 0.64, "A": 0.03, "B": 0.32, "C": 0.01 }, "batch_response_ms": 418.6213749999297, "repaired": false }, { "id": "13.1b", "section": 13, "context": "Uniform random linear permutation of 2n distinct socks, n>=2 labeled left-right matching pairs. X counts pairs whose two socks are adjacent.", "prompt": "E[X]?", "response_mode": "single_choice", "options": { "A": "1", "B": "n", "C": "n/2", "D": "2" }, "correct": true, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "13.1b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.5, "probabilities": { "D": 0.05, "A": 0.62, "B": 0.11, "C": 0.22 }, "batch_response_ms": 418.6213749999297, "repaired": false }, { "id": "13.1c", "section": 13, "context": "Uniform random linear permutation of 2n distinct socks, n>=2 labeled left-right matching pairs. X counts pairs whose two socks are adjacent.", "prompt": "Probability both pairs1 and2 are adjacent?", "response_mode": "single_choice", "options": { "A": "2/((2n)(2n−1))", "B": "4/((2n−1)(2n−2))", "C": "1/n²", "D": "4/((2n)(2n−1))" }, "correct": false, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "13.1c" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.34, "probabilities": { "D": 0.25, "A": 0.23, "B": 0.51, "C": 0.01 }, "batch_response_ms": 418.6213749999297, "repaired": false }, { "id": "13.1d", "section": 13, "context": "Uniform random linear permutation of 2n distinct socks, n>=2 labeled left-right matching pairs. X counts pairs whose two socks are adjacent.", "prompt": "Let p=P(pair1 adjacent),q=P(pairs1,2 adjacent). Var(X)?", "response_mode": "single_choice", "options": { "A": "nq−n²p²", "B": "np(1−p)", "C": "np+n(n−1)q−n²p²", "D": "np+n(n−1)q" }, "correct": true, "kind": "multi_step", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "13.1d" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.98, "probabilities": { "D": 0.0, "A": 0.0, "B": 0.01, "C": 0.99 }, "batch_response_ms": 418.6213749999297, "repaired": false }, { "id": "13.2a", "section": 13, "context": "k students have independent uniform birthdays on 365 days.", "prompt": "Expected unordered pairs sharing a birthday?", "response_mode": "single_choice", "options": { "A": "C(k,2)/365", "B": "C(k,2)/365²", "C": "365/C(k,2)", "D": "k/365" }, "correct": true, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "13.2a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 1.0, "B": 0.0, "C": 0.0 }, "batch_response_ms": 418.6213749999297, "repaired": false }, { "id": "13.2b", "section": 13, "context": "k students have independent uniform birthdays on 365 days.", "prompt": "Expected unordered triples sharing a birthday?", "response_mode": "single_choice", "options": { "A": "C(k,3)/365³", "B": "C(k,3)/365²", "C": "k/365²", "D": "C(k,3)/365" }, "correct": true, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "13.2b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.85, "probabilities": { "D": 0.04, "A": 0.07, "B": 0.89, "C": 0.0 }, "batch_response_ms": 418.6213749999297, "repaired": false }, { "id": "13.2c", "section": 13, "context": "k students have independent uniform birthdays on 365 days.", "prompt": "Variance of number of matching-birthday pairs?", "response_mode": "single_choice", "options": { "A": "C(k,2)*(1/365)*(364/365)", "B": "C(k,2)/365²", "C": "C(k,2)/365", "D": "C(k,2)²*(1/365)*(364/365)" }, "correct": true, "kind": "multi_step", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "13.2c" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.66, "probabilities": { "D": 0.11, "A": 0.75, "B": 0.02, "C": 0.12 }, "batch_response_ms": 418.6213749999297, "repaired": false }, { "id": "14.1", "section": 14, "context": "Independent biased coin flips with heads probability p>0. X counts flips through the kth head, k>=1; Y=X/k.", "prompt": "Why is Y unbiased for 1/p?", "response_mode": "single_choice", "options": { "A": "The inverse of any unbiased estimator of p is unbiased for1/p.", "B": "X is geometric with parameter kp, so dividing by k yields1/p.", "C": "X always equals k/p, so Y has no sampling variability.", "D": "Split X into k independent geometric waiting times, each mean1/p; E[X]=k/p and E[Y]=1/p." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "14.1" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "A": 0.0, "D": 1.0, "B": 0.0 }, "batch_response_ms": 309.46979200234637, "repaired": false }, { "id": "14.2", "section": 14, "context": "Independent biased coin flips with heads probability p>0. X counts flips through the kth head, k>=1; Y=X/k.", "prompt": "Var(Y)?", "response_mode": "single_choice", "options": { "A": "k*(1−p)/p²", "B": "(1−p)/(k²*p²)", "C": "(1−p)/p²", "D": "(1−p)/(k*p²)" }, "correct": true, "kind": "calculation", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "14.2" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.95, "probabilities": { "C": 0.0, "B": 0.03, "A": 0.0, "D": 0.97 }, "batch_response_ms": 309.46979200234637, "repaired": false }, { "id": "14.3", "section": 14, "context": "Independent biased coin flips with heads probability p>0. X counts flips through the kth head, k>=1; Y=X/k.", "prompt": "Chebyshev bound for P(|Y−1/p|>=epsilon/p), clipped at1?", "response_mode": "single_choice", "options": { "A": "min(1,(1−p)/(epsilon²*k*p²))", "B": "min(1,(1−p)/(epsilon*k))", "C": "min(1,p/(epsilon²*k))", "D": "min(1,(1−p)/(epsilon²*k))" }, "correct": true, "kind": "multi_step", "topic": "bounds", "original_format": "written", "repairs": [], "source_parts": [ "14.3" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.92, "probabilities": { "D": 0.95, "B": 0.0, "C": 0.0, "A": 0.05 }, "batch_response_ms": 309.46979200234637, "repaired": false }, { "id": "14.4", "section": 14, "context": "Independent biased coin flips with heads probability p>0. X counts flips through the kth head, k>=1; Y=X/k.", "prompt": "Only p>=1/2 is known. Smallest k certified by that Chebyshev bound for relative error strictly below10% with probability at least95%?", "response_mode": "single_choice", "options": { "A": "100", "B": "2000", "C": "500", "D": "1000" }, "correct": false, "kind": "multi_step", "topic": "bounds", "original_format": "written", "repairs": [], "source_parts": [ "14.4" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.1, "probabilities": { "C": 0.23, "B": 0.33, "A": 0.15, "D": 0.29 }, "batch_response_ms": 309.46979200234637, "repaired": false }, { "id": "15.1", "section": 15, "context": "", "prompt": "Drive 360 miles at one speed S chosen uniformly in [40,60] mph. Expected hours? Give a valid integral and value.", "response_mode": "single_choice", "options": { "A": "E[S]/360=50/360", "B": "(1/20)integral(40..60)360/s ds=18ln(3/2)", "C": "360/E[S]=360/50", "D": "integral(40..60)360/s ds=360ln(3/2)" }, "correct": true, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "15.1" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 0.0, "B": 0.99, "A": 0.0, "D": 0.01 }, "batch_response_ms": 312.2406250004133, "repaired": false }, { "id": "15.2a", "section": 15, "context": "Independent X,Y,Z~Uniform[0,1].", "prompt": "CDF of max(X,Y,Z) on [0,1]?", "response_mode": "single_choice", "options": { "A": "t³", "B": "1−(1−t)³", "C": "3t²", "D": "t" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "15.2a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.89, "probabilities": { "D": 0.0, "B": 0.08, "C": 0.0, "A": 0.92 }, "batch_response_ms": 312.2406250004133, "repaired": false }, { "id": "15.2b", "section": 15, "context": "Independent X,Y,Z~Uniform[0,1].", "prompt": "PDF of max(X,Y,Z) on [0,1]?", "response_mode": "single_choice", "options": { "A": "3(1−t)²", "B": "3t²", "C": "1", "D": "t³" }, "correct": true, "kind": "calculation", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "15.2b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.0, "B": 1.0 }, "batch_response_ms": 312.2406250004133, "repaired": false }, { "id": "15.2c", "section": 15, "context": "Independent X,Y,Z~Uniform[0,1].", "prompt": "E[max(X,Y,Z)]?", "response_mode": "single_choice", "options": { "A": "3/4", "B": "1/2", "C": "1/4", "D": "2/3" }, "correct": true, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "15.2c" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "D": 0.0, "A": 1.0, "C": 0.0, "B": 0.0 }, "batch_response_ms": 312.2406250004133, "repaired": false }, { "id": "15.2d", "section": 15, "context": "Independent X,Y,Z~Uniform[0,1].", "prompt": "PDF of min(X,Y,Z) on [0,1]?", "response_mode": "single_choice", "options": { "A": "3t²", "B": "1−t³", "C": "(1−t)³", "D": "3(1−t)²" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "15.2d" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 0.0, "B": 0.0, "A": 0.01, "D": 0.99 }, "batch_response_ms": 312.2406250004133, "repaired": false }, { "id": "15.2e", "section": 15, "context": "Independent X,Y,Z~Uniform[0,1].", "prompt": "Expected median of X,Y,Z?", "response_mode": "single_choice", "options": { "A": "1/4", "B": "1/2", "C": "2/3", "D": "3/4" }, "correct": true, "kind": "recognition", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "15.2e" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "D": 0.0, "B": 1.0, "C": 0.0, "A": 0.0 }, "batch_response_ms": 312.2406250004133, "repaired": false }, { "id": "15.3a", "section": 15, "context": "Independent TS~N(mean4,variance0.2),TJ~N(mean4,variance0.3).", "prompt": "Distribution of TS+TJ, second parameter variance?", "response_mode": "single_choice", "options": { "A": "N(8,0.13)", "B": "N(4,0.5)", "C": "N(8,0.1)", "D": "N(8,0.5)" }, "correct": true, "kind": "calculation", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "15.3a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 312.2406250004133, "repaired": false }, { "id": "15.3b", "section": 15, "context": "Independent TS~N(mean4,variance0.2),TJ~N(mean4,variance0.3).", "prompt": "Distribution of TS−TJ, second parameter variance?", "response_mode": "single_choice", "options": { "A": "N(0,0.5)", "B": "N(0,0.1)", "C": "N(8,0.5)", "D": "N(0,−0.1)" }, "correct": true, "kind": "calculation", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "15.3b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.91, "probabilities": { "D": 0.0, "A": 0.93, "C": 0.0, "B": 0.07 }, "batch_response_ms": 312.2406250004133, "repaired": false }, { "id": "15.3c", "section": 15, "context": "Independent TS~N(mean4,variance0.2),TJ~N(mean4,variance0.3).", "prompt": "P(TS>TJ)?", "response_mode": "single_choice", "options": { "A": "2/5", "B": "0", "C": "1/2", "D": "3/5" }, "correct": true, "kind": "recognition", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "15.3c" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.76, "probabilities": { "C": 0.82, "B": 0.01, "A": 0.11, "D": 0.06 }, "batch_response_ms": 312.2406250004133, "repaired": false }, { "id": "16.1a", "section": 16, "context": "Uniformly choose one of three circular boards with positive radii a,b,c, then throw a dart uniformly over the chosen board’s area.", "prompt": "Expected distance to center?", "response_mode": "single_choice", "options": { "A": "(a+b+c)/6", "B": "(a+b+c)/3", "C": "2(a²+b²+c²)/9", "D": "2(a+b+c)/9" }, "correct": true, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "16.1a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.97, "probabilities": { "B": 0.01, "D": 0.98, "A": 0.01, "C": 0.0 }, "batch_response_ms": 354.43083299833233, "repaired": false }, { "id": "16.1b", "section": 16, "context": "Uniformly choose one of three circular boards with positive radii a,b,c, then throw a dart uniformly over the chosen board’s area.", "prompt": "Justify the expected distance.", "response_mode": "single_choice", "options": { "A": "The mean distance equals the radius by rotational symmetry.", "B": "Boards must be weighted proportional to their area, despite uniform board selection.", "C": "Distance is uniform on[0,r], so each conditional mean is r/2.", "D": "At fixed radius r, radial density2x/r² yields conditional mean2r/3. Average these means with weight1/3 over the three boards." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "16.1b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "D": 1.0, "C": 0.0, "A": 0.0 }, "batch_response_ms": 354.43083299833233, "repaired": false }, { "id": "16.2a", "section": 16, "context": "Y has mean7,variance2. Independent Z~N(mean3,variance5). X=Y+Z.", "prompt": "Given Y=y, give the minimum-MSE estimate of X and its conditional MSE.", "response_mode": "single_choice", "options": { "A": "(y+3,7)", "B": "(y+7,2)", "C": "(y+3,5)", "D": "(10,7)" }, "correct": true, "kind": "recognition", "topic": "regression", "original_format": "written", "repairs": [], "source_parts": [ "16.2a.estimate", "16.2a.mse" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.85, "probabilities": { "B": 0.0, "D": 0.0, "C": 0.89, "A": 0.11 }, "batch_response_ms": 354.43083299833233, "repaired": false }, { "id": "16.2b", "section": 16, "context": "Y has mean7,variance2. Independent Z~N(mean3,variance5). X=Y+Z.", "prompt": "Best affine estimator of X from Y?", "response_mode": "single_choice", "options": { "A": "(2/7)Y+8", "B": "Y+7", "C": "10", "D": "Y+3" }, "correct": true, "kind": "recognition", "topic": "regression", "original_format": "written", "repairs": [], "source_parts": [ "16.2b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.96, "probabilities": { "B": 0.01, "D": 0.97, "C": 0.0, "A": 0.02 }, "batch_response_ms": 354.43083299833233, "repaired": false }, { "id": "16.3", "section": 16, "context": "", "prompt": "X~N(69,3),Y1,Y2~N(0,1) mutually independent, second parameters variances. Twins have Z1=X+Y1,Z2=X+Y2. Best affine estimator of Z2 given Z1?", "response_mode": "single_choice", "options": { "A": "Z1", "B": "69+(4/3)(Z1−69)", "C": "69+(3/4)(Z1−69)", "D": "69+(1/4)(Z1−69)" }, "correct": true, "kind": "multi_step", "topic": "regression", "original_format": "written", "repairs": [], "source_parts": [ "16.3" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.48, "probabilities": { "B": 0.37, "D": 0.01, "C": 0.6, "A": 0.02 }, "batch_response_ms": 354.43083299833233, "repaired": false }, { "id": "17.1a", "section": 17, "context": "Chain states0,1,2. Row0:[1−p,p,0]; row1:[1−p,0,p]; row2:[0,1−p,p].", "prompt": "p=1/3. Choose stationary equations plus normalization.", "response_mode": "single_choice", "options": { "A": "pi0=pi1=pi2=1/3", "B": "pi0=2pi0/3; pi1=pi0/3; pi2=pi1/3+pi2; sum pi=1", "C": "pi0=2(pi0+pi1)/3; pi1=pi0/3+2pi2/3; pi2=(pi1+pi2)/3; sum pi=1", "D": "pi0=2pi0/3+pi1/3; pi1=2pi0/3+pi2/3; pi2=2pi1/3+pi2/3; sum pi=1" }, "correct": true, "kind": "recognition", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "17.1a" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.21, "probabilities": { "C": 0.4, "B": 0.02, "D": 0.37, "A": 0.21 }, "batch_response_ms": 368.06929200247396, "repaired": false }, { "id": "17.1b", "section": 17, "context": "Chain states0,1,2. Row0:[1−p,p,0]; row1:[1−p,0,p]; row2:[0,1−p,p]. Now choose p such that stationary masses are (1,3,9)/13. X has that stationary law and Y=0 for X=0 or1, Y=1 for X=2.", "prompt": "E[X given Y]?", "response_mode": "single_choice", "options": { "A": "3/4 if Y=0;2 if Y=1", "B": "1/2 if Y=0;2 if Y=1", "C": "3/13 if Y=0;18/13 if Y=1", "D": "0 if Y=0;1 if Y=1" }, "correct": true, "kind": "multi_step", "topic": "regression", "original_format": "written", "repairs": [], "source_parts": [ "17.1b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.66, "probabilities": { "C": 0.01, "D": 0.01, "B": 0.23, "A": 0.75 }, "batch_response_ms": 368.06929200247396, "repaired": false }, { "id": "17.1c", "section": 17, "context": "Chain states0,1,2. Row0:[1−p,p,0]; row1:[1−p,0,p]; row2:[0,1−p,p]. Now choose p such that stationary masses are (1,3,9)/13. X has that stationary law and Y=0 for X=0 or1, Y=1 for X=2.", "prompt": "Best affine estimate of X as aY+b?", "response_mode": "single_choice", "options": { "A": "(5/4)Y+3/4", "B": "(3/4)Y+5/4", "C": "(18/13)Y+3/13", "D": "2Y" }, "correct": true, "kind": "calculation", "topic": "regression", "original_format": "written", "repairs": [], "source_parts": [ "17.1c" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.23, "probabilities": { "C": 0.32, "B": 0.16, "D": 0.09, "A": 0.43 }, "batch_response_ms": 368.06929200247396, "repaired": false }, { "id": "17.2a", "section": 17, "context": "Roll a fair six-sided die until the product of the last two rolls is15. StateA=start or previous roll in{1,2,4,6}; B=previous in{3,5} but target not hit; C=target hit, then stop and remain atC.", "prompt": "Give (PAA,PAB,PBA,PBB,PBC,PCC).", "response_mode": "single_choice", "options": { "A": "(1/3,2/3,1/3,1/3,1/3,1)", "B": "(2/3,1/3,2/3,0,1/3,1)", "C": "(2/3,1/3,1/6,2/3,1/6,1)", "D": "(2/3,1/3,2/3,1/6,1/6,1)" }, "correct": true, "kind": "multi_step", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "17.2a.AA", "17.2a.AB", "17.2a.BA", "17.2a.BB", "17.2a.BC", "17.2a.CC" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.54, "probabilities": { "C": 0.14, "B": 0.08, "D": 0.66, "A": 0.12 }, "batch_response_ms": 368.06929200247396, "repaired": false }, { "id": "17.2b", "section": 17, "context": "Roll a fair six-sided die until the product of the last two rolls is15. StateA=start or previous roll in{1,2,4,6}; B=previous in{3,5} but target not hit; C=target hit, then stop and remain atC.", "prompt": "This stopped chain is irreducible.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "markov", "original_format": "choice", "repairs": [], "source_parts": [ "17.2b" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.99, "A": 0.01 }, "batch_response_ms": 368.06929200247396, "repaired": false }, { "id": "17.2c", "section": 17, "context": "Roll a fair six-sided die until the product of the last two rolls is15. StateA=start or previous roll in{1,2,4,6}; B=previous in{3,5} but target not hit; C=target hit, then stop and remain atC.", "prompt": "Every state of this chain has period one.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "markov", "original_format": "choice", "repairs": [], "source_parts": [ "17.2c" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.19, "probabilities": { "B": 0.6, "A": 0.4 }, "batch_response_ms": 368.06929200247396, "repaired": false }, { "id": "17.2d", "section": 17, "context": "Roll a fair six-sided die until the product of the last two rolls is15. StateA=start or previous roll in{1,2,4,6}; B=previous in{3,5} but target not hit; C=target hit, then stop and remain atC.", "prompt": "Let a0=PAA,a1=PAB,b1=PBA,b0=PBB,b2=PBC,c0=PCC. Equations for expected remaining rolls beta to hit C?", "response_mode": "single_choice", "options": { "A": "betaA=a0*betaA+a1*betaB; betaB=b1*betaA+b0*betaB+b2*betaC; betaC=0", "B": "betaA=1+a0*betaA+b1*betaB; betaB=1+a1*betaA+b0*betaB; betaC=0", "C": "betaA=1+a0*betaA+a1*betaB; betaB=1+b1*betaA+b0*betaB+b2*betaC; betaC=1+c0*betaC", "D": "betaA=1+a0*betaA+a1*betaB; betaB=1+b1*betaA+b0*betaB+b2*betaC; betaC=0" }, "correct": true, "kind": "recognition", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "17.2d" ], "exam": "final_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.92, "probabilities": { "C": 0.01, "B": 0.04, "D": 0.94, "A": 0.01 }, "batch_response_ms": 368.06929200247396, "repaired": false }, { "id": "2.1a", "section": 2, "context": "", "prompt": "Truth status of P OR Q OR R OR NOT R?", "response_mode": "single_choice", "options": { "A": "False", "B": "Could be either", "C": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.1a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.01, "A": 0.0, "C": 0.99 }, "batch_response_ms": 350.9275839969632, "repaired": false }, { "id": "2.1b", "section": 2, "context": "", "prompt": "Truth status of (NOT P implies (R AND NOT R)) implies P?", "response_mode": "single_choice", "options": { "A": "True", "B": "False", "C": "Could be either" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.1b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.92, "probabilities": { "B": 0.02, "A": 0.95, "C": 0.03 }, "batch_response_ms": 350.9275839969632, "repaired": false }, { "id": "2.1c", "section": 2, "context": "", "prompt": "Truth status of (R AND NOT R) OR (Q AND P)?", "response_mode": "single_choice", "options": { "A": "Could be either", "B": "False", "C": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.1c" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.88, "probabilities": { "B": 0.07, "A": 0.93, "C": 0.0 }, "batch_response_ms": 350.9275839969632, "repaired": false }, { "id": "2.2a", "section": 2, "context": "For all x in nonempty U, P(x) implies [Q(x) OR exists y R(x,y)].", "prompt": "P(x) is true and all R(x,y) false. Truth status of Q(x)?", "response_mode": "single_choice", "options": { "A": "Could be either", "B": "True", "C": "False" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.2a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.99, "A": 0.01, "C": 0.0 }, "batch_response_ms": 350.9275839969632, "repaired": false }, { "id": "2.2b", "section": 2, "context": "For all x in nonempty U, P(x) implies [Q(x) OR exists y R(x,y)].", "prompt": "Q(x) is false and all R(x,y) false. Truth status of P(x)?", "response_mode": "single_choice", "options": { "A": "True", "B": "Could be either", "C": "False" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.2b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.01, "A": 0.0, "C": 0.99 }, "batch_response_ms": 350.9275839969632, "repaired": false }, { "id": "2.2c", "section": 2, "context": "For all x in nonempty U, P(x) implies [Q(x) OR exists y R(x,y)].", "prompt": "P(x) is true and some R(x,y) true. Truth status of Q(x)?", "response_mode": "single_choice", "options": { "A": "False", "B": "True", "C": "Could be either" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.2c" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "A": 0.0, "C": 1.0 }, "batch_response_ms": 350.9275839969632, "repaired": false }, { "id": "2.3", "section": 2, "context": "", "prompt": "Express |a−b| using only min(a,b) and max(a,b).", "response_mode": "single_choice", "options": { "A": "max(a,b)−min(a,b)", "B": "max(a,b)*min(a,b)", "C": "min(a,b)−max(a,b)", "D": "max(a,b)+min(a,b)" }, "correct": true, "kind": "recognition", "topic": "algebra", "original_format": "written", "repairs": [], "source_parts": [ "2.3" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 1.0, "C": 0.0 }, "batch_response_ms": 350.9275839969632, "repaired": false }, { "id": "3.2", "section": 3, "context": "", "prompt": "Complete: forall n [P(n)=>P(n+1)] is equivalent to NOT exists n [P(n) AND __].", "response_mode": "single_choice", "options": { "A": "NOT P(n+1)", "B": "P(n+1)", "C": "P(n)", "D": "NOT P(n)" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "written", "repairs": [], "source_parts": [ "3.2" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 0.0, "A": 0.99, "B": 0.0, "D": 0.01 }, "batch_response_ms": 382.2543750029581, "repaired": false }, { "id": "3.3a", "section": 3, "context": "F0=F1=1 and Fi=F(i−1)+F(i−2).", "prompt": "Largest real c>=1 such that Fi>=c^(i−1) for every i>=1?", "response_mode": "single_choice", "options": { "A": "3/2", "B": "2", "C": "(1+sqrt(5))/2", "D": "sqrt(2)" }, "correct": true, "kind": "multi_step", "topic": "recurrences", "original_format": "written", "repairs": [], "source_parts": [ "3.3a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.75, "probabilities": { "C": 0.8099999999999999, "B": 0.0, "A": 0.04, "D": 0.15 }, "batch_response_ms": 382.2543750029581, "repaired": false }, { "id": "3.3b", "section": 3, "context": "F0=F1=1 and Fi=F(i−1)+F(i−2).", "prompt": "Which proves the optimal exponential lower bound?", "response_mode": "single_choice", "options": { "A": "Let phi=(1+sqrt(5))/2, so phi²=phi+1. Bases i=1,2 hold, and adding inductive bounds gives Fi>=phi^(i−1). Similarly Fi<=phi^i; any c>phi eventually has c^(i−1)>phi^i, so cannot work for all i.", "B": "A lower bound that works for one i must work for every later i.", "C": "Since F2=2, c=2 works for every i without checking the recurrence.", "D": "The recurrence implies Fi=phi^(i−1) exactly for all i." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "3.3b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 1.0, "D": 0.0 }, "batch_response_ms": 382.2543750029581, "repaired": false }, { "id": "4.1a", "section": 4, "context": "Jobs A:1>2,B:2>1; candidates 1:B>A,2:A>B.", "prompt": "Job-optimal matching?", "response_mode": "single_choice", "options": { "A": "A–2,B–1", "B": "A–1,B–2" }, "correct": false, "kind": "multi_step", "topic": "matching", "original_format": "written", "repairs": [], "source_parts": [ "4.1a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.63, "probabilities": { "A": 0.81, "B": 0.19 }, "batch_response_ms": 321.4624169995659, "repaired": false }, { "id": "4.1b", "section": 4, "context": "Jobs A:1>2,B:2>1; candidates 1:B>A,2:A>B.", "prompt": "Candidate-optimal matching?", "response_mode": "single_choice", "options": { "A": "A–1,B–2", "B": "A–2,B–1" }, "correct": true, "kind": "multi_step", "topic": "matching", "original_format": "written", "repairs": [], "source_parts": [ "4.1b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.35, "probabilities": { "A": 0.32, "B": 0.68 }, "batch_response_ms": 321.4624169995659, "repaired": false }, { "id": "4.2a", "section": 4, "context": "Jobs A:1>2>3>4; B:2>1>4>3; C:3>4>1>2; D:4>3>2>1. Candidates 1:B>A>C>D; 2:A>B>D>C; 3:D>C>A>B; 4:C>D>B>A.", "prompt": "Number of stable matchings?", "response_mode": "single_choice", "options": { "A": "1", "B": "2", "C": "8", "D": "4" }, "correct": true, "kind": "multi_step", "topic": "matching", "original_format": "written", "repairs": [], "source_parts": [ "4.2a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.17, "probabilities": { "D": 0.38, "B": 0.18, "A": 0.21, "C": 0.23 }, "batch_response_ms": 321.4624169995659, "repaired": false }, { "id": "4.2b", "section": 4, "context": "Jobs A:1>2>3>4; B:2>1>4>3; C:3>4>1>2; D:4>3>2>1. Candidates 1:B>A>C>D; 2:A>B>D>C; 3:D>C>A>B; 4:C>D>B>A.", "prompt": "Choose a stable matching optimal for neither entire side.", "response_mode": "single_choice", "options": { "A": "A–2,B–1,C–4,D–3", "B": "A–1,B–2,C–3,D–4", "C": "A–1,B–2,C–4,D–3", "D": "A–3,B–4,C–1,D–2" }, "correct": false, "kind": "multi_step", "topic": "matching", "original_format": "written", "repairs": [], "source_parts": [ "4.2b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.33, "probabilities": { "D": 0.49, "A": 0.24, "B": 0.16, "C": 0.11 }, "batch_response_ms": 321.4624169995659, "repaired": false }, { "id": "5.1a", "section": 5, "context": "", "prompt": "Graph G has k connected components. Sum of vertex degrees?", "response_mode": "single_choice", "options": { "A": "2|V|", "B": "|E|", "C": "2|E|", "D": "|V|−k" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "5.1a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 }, "batch_response_ms": 364.28287500166334, "repaired": false }, { "id": "5.1b", "section": 5, "context": "", "prompt": "Maximum edges in a simple graph with v vertices and k components,1<=k<=v?", "response_mode": "single_choice", "options": { "A": "C(v,2)−k", "B": "v−k", "C": "C(v−k+1,2)", "D": "k*C(v/k,2)" }, "correct": true, "kind": "multi_step", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "5.1b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 }, "batch_response_ms": 364.28287500166334, "repaired": false }, { "id": "5.1c", "section": 5, "context": "", "prompt": "Minimum edges with v vertices and k components?", "response_mode": "single_choice", "options": { "A": "v+k", "B": "k−1", "C": "v−1", "D": "v−k" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "5.1c" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 364.28287500166334, "repaired": false }, { "id": "5.2a", "section": 5, "context": "", "prompt": "Minimum vertex colors for a positive-dimensional hypercube?", "response_mode": "single_choice", "options": { "A": "log2(|V|)", "B": "log2(|V|)+1", "C": "2", "D": "1" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "5.2a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.96, "probabilities": { "B": 0.01, "D": 0.0, "A": 0.01, "C": 0.98 }, "batch_response_ms": 364.28287500166334, "repaired": false }, { "id": "5.2b", "section": 5, "context": "", "prompt": "Minimum edge colors for a hypercube on |V| vertices?", "response_mode": "single_choice", "options": { "A": "2", "B": "log2(|V|)+1", "C": "log2(|V|)", "D": "|V|/2" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "5.2b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.94, "probabilities": { "B": 0.03, "D": 0.0, "A": 0.01, "C": 0.96 }, "batch_response_ms": 364.28287500166334, "repaired": false }, { "id": "5.3", "section": 5, "context": "", "prompt": "Delete all edges of a simple k-cycle from a connected graph, retaining vertices. Sharp upper bound on component count?", "response_mode": "single_choice", "options": { "A": "k", "B": "2k", "C": "k−1", "D": "k+1" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "5.3" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.48, "probabilities": { "B": 0.04, "D": 0.29, "A": 0.62, "C": 0.05 }, "batch_response_ms": 364.28287500166334, "repaired": false }, { "id": "6.1", "section": 6, "context": "", "prompt": "2^63 modulo 77?", "response_mode": "single_choice", "options": { "A": "8", "B": "2", "C": "16", "D": "1" }, "correct": true, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "6.1" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.17, "probabilities": { "D": 0.13, "C": 0.35, "B": 0.14, "A": 0.38 }, "batch_response_ms": 399.7862910000549, "repaired": false }, { "id": "6.2", "section": 6, "context": "", "prompt": "1*2*3*4*5*6 modulo 7?", "response_mode": "single_choice", "options": { "A": "0", "B": "5", "C": "1", "D": "6" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "6.2" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.89, "probabilities": { "D": 0.91, "C": 0.03, "B": 0.03, "A": 0.03 }, "batch_response_ms": 399.7862910000549, "repaired": false }, { "id": "6.3", "section": 6, "context": "", "prompt": "Inverse of 63 modulo 71 in 0..70?", "response_mode": "single_choice", "options": { "A": "62", "B": "8", "C": "63", "D": "9" }, "correct": true, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "6.3" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.22, "probabilities": { "D": 0.18, "C": 0.02, "B": 0.38, "A": 0.42 }, "batch_response_ms": 399.7862910000549, "repaired": false }, { "id": "6.4a", "section": 6, "context": "p,q are distinct primes; a,b range independently over residues 0..pq−1.", "prompt": "Number of a coprime to p?", "response_mode": "single_choice", "options": { "A": "(p−1)q", "B": "pq−p", "C": "pq−1", "D": "(p−1)(q−1)" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "6.4a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.45, "probabilities": { "D": 0.0, "C": 0.0, "B": 0.41, "A": 0.59 }, "batch_response_ms": 399.7862910000549, "repaired": false }, { "id": "6.4b", "section": 6, "context": "p,q are distinct primes; a,b range independently over residues 0..pq−1.", "prompt": "Number of a coprime to both p and q?", "response_mode": "single_choice", "options": { "A": "(p−1)(q−1)", "B": "pq−p−q", "C": "pq−1", "D": "p+q−2" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "6.4b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "D": 0.0, "C": 0.0, "B": 0.01, "A": 0.99 }, "batch_response_ms": 399.7862910000549, "repaired": false }, { "id": "6.4c", "section": 6, "context": "p,q are distinct primes; a,b range independently over residues 0..pq−1.", "prompt": "Number of pairs (a,b) for which ax=b mod pq has exactly one solution?", "response_mode": "single_choice", "options": { "A": "pq−1", "B": "(pq)²", "C": "pq(p−1)(q−1)", "D": "(p−1)(q−1)" }, "correct": true, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "6.4c" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.8, "probabilities": { "D": 0.05, "C": 0.85, "B": 0.05, "A": 0.05 }, "batch_response_ms": 399.7862910000549, "repaired": false }, { "id": "6.4d", "section": 6, "context": "p,q are distinct primes; a,b range independently over residues 0..pq−1.", "prompt": "gcd(a,pq)=p and x is a solution. Smallest positive shift t for which x+t is another solution?", "response_mode": "single_choice", "options": { "A": "pq", "B": "1", "C": "q", "D": "p" }, "correct": false, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "6.4d" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.26, "probabilities": { "D": 0.17, "C": 0.38, "B": 0.01, "A": 0.44 }, "batch_response_ms": 399.7862910000549, "repaired": false }, { "id": "6.4e", "section": 6, "context": "p,q are distinct primes; a,b range independently over residues 0..pq−1.", "prompt": "Number of pairs (a,b) with exactly p solutions?", "response_mode": "single_choice", "options": { "A": "q(q−1)", "B": "pq", "C": "q(p−1)", "D": "p(p−1)" }, "correct": true, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "6.4e" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.3, "probabilities": { "D": 0.07, "C": 0.36, "B": 0.1, "A": 0.47 }, "batch_response_ms": 399.7862910000549, "repaired": false }, { "id": "6.4f", "section": 6, "context": "p,q are distinct primes; a,b range independently over residues 0..pq−1.", "prompt": "Number of pairs (a,b) with no solutions?", "response_mode": "single_choice", "options": { "A": "(q−1)(pq−q)+(p−1)(pq−p)+(pq−1)", "B": "(pq−1)²", "C": "(p−1)(q−1)pq", "D": "(q−1)(pq−q)+(p−1)(pq−p)" }, "correct": true, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "6.4f" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.33, "probabilities": { "D": 0.18, "C": 0.27, "B": 0.05, "A": 0.5 }, "batch_response_ms": 399.7862910000549, "repaired": false }, { "id": "6.5", "section": 6, "context": "", "prompt": "For every N>=2, N is prime if and only if no a,b in {1,...,N−1} have ab equal to __ mod N. Choose the residue that makes this equivalence valid.", "response_mode": "single_choice", "options": { "A": "1", "B": "2", "C": "0", "D": "N−1" }, "correct": true, "kind": "recognition", "topic": "number_theory", "original_format": "written", "repairs": [ "Exclude N=1 and strengthen to iff: the original one-way implication also permits vacuously true distractors." ], "source_parts": [ "6.5" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.9, "probabilities": { "D": 0.01, "C": 0.93, "B": 0.0, "A": 0.06 }, "batch_response_ms": 399.7862910000549, "repaired": true }, { "id": "6.6", "section": 6, "context": "", "prompt": "Use CRT to count units modulo pq for distinct primes p,q.", "response_mode": "single_choice", "options": { "A": "CRT bijects residues mod pq with pairs mod p and q; a unit corresponds exactly to a pair with both coordinates nonzero, giving (p−1)(q−1).", "B": "All nonzero residues modulo pq are units, giving pq−1.", "C": "CRT requires both coordinates to match as integers, giving min(p,q)−1.", "D": "Exactly one coordinate must be nonzero, giving p+q−2." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "6.6" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "C": 0.0, "B": 0.0, "A": 1.0 }, "batch_response_ms": 399.7862910000549, "repaired": false }, { "id": "7.1", "section": 7, "context": "", "prompt": "GF(5): normalized Lagrange Delta0 for x-coordinates 0,1,4?", "response_mode": "single_choice", "options": { "A": "−x²+4", "B": "−x²−4", "C": "x²−1", "D": "x²+4" }, "correct": false, "kind": "calculation", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "7.1" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.5, "probabilities": { "C": 0.05, "A": 0.63, "B": 0.13, "D": 0.19 }, "batch_response_ms": 315.1168339973083, "repaired": false }, { "id": "7.2a", "section": 7, "context": "GF(5), a constant message polynomial is encoded; at most one error. Received points (0,1),(2,1),(3,4).", "prompt": "Original polynomial?", "response_mode": "single_choice", "options": { "A": "3", "B": "1", "C": "x+1", "D": "4" }, "correct": true, "kind": "recognition", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "7.2a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.59, "probabilities": { "C": 0.01, "A": 0.09, "B": 0.69, "D": 0.21 }, "batch_response_ms": 315.1168339973083, "repaired": false }, { "id": "7.2b", "section": 7, "context": "GF(5), a constant message polynomial is encoded; at most one error. Received points (0,1),(2,1),(3,4).", "prompt": "With Q(x)=q1*x+q0 and E(x)=x+b0, write the BW equation for received (3,4).", "response_mode": "single_choice", "options": { "A": "q1+q0=4b0 mod 5", "B": "3q1+q0=4(3+b0) mod 5", "C": "3q1+q0=3+b0 mod 5", "D": "4q1+q0=3(4+b0) mod 5" }, "correct": true, "kind": "recognition", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "7.2b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.93, "probabilities": { "C": 0.01, "B": 0.94, "A": 0.02, "D": 0.03 }, "batch_response_ms": 315.1168339973083, "repaired": false }, { "id": "7.3a", "section": 7, "context": "", "prompt": "Normalized Lagrange basis on distinct field coordinates: Delta_i(i)?", "response_mode": "single_choice", "options": { "A": "0", "B": "i", "C": "1", "D": "yi" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "7.3a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 0.99, "A": 0.01, "B": 0.0, "D": 0.0 }, "batch_response_ms": 315.1168339973083, "repaired": false }, { "id": "7.3b", "section": 7, "context": "", "prompt": "Normalized Lagrange basis: Delta_i(j) for a different interpolation coordinate j?", "response_mode": "single_choice", "options": { "A": "i−j", "B": "yj", "C": "1", "D": "0" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "7.3b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.29, "probabilities": { "C": 0.46, "A": 0.06, "B": 0.02, "D": 0.45999999999999996 }, "batch_response_ms": 315.1168339973083, "repaired": false }, { "id": "7.3c", "section": 7, "context": "", "prompt": "Interpolate (0,y0),(1,y1),(2,y2) using normalized basis D0,D1,D2.", "response_mode": "single_choice", "options": { "A": "D0+D1+D2", "B": "y0*D1+y1*D2+y2*D0", "C": "y0+y1*x+y2*x²", "D": "y0*D0+y1*D1+y2*D2" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "7.3c" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 0.0, "A": 0.01, "B": 0.0, "D": 0.99 }, "batch_response_ms": 315.1168339973083, "repaired": false }, { "id": "8.1", "section": 8, "context": "", "prompt": "Distinct anagrams of BERKELEY?", "response_mode": "single_choice", "options": { "A": "7!/3!", "B": "8!/(2!2!)", "C": "8!", "D": "8!/3!" }, "correct": false, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "8.1" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.32, "probabilities": { "B": 0.48, "C": 0.01, "D": 0.43, "A": 0.08 }, "batch_response_ms": 322.78100000257837, "repaired": false }, { "id": "8.2", "section": 8, "context": "", "prompt": "Assign n labeled balls to m labeled bins without restrictions. Count?", "response_mode": "single_choice", "options": { "A": "n^m", "B": "m^n", "C": "C(n+m−1,m−1)", "D": "n!" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "8.2" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.99, "A": 0.01, "D": 0.0, "C": 0.0 }, "batch_response_ms": 322.78100000257837, "repaired": false }, { "id": "8.3a", "section": 8, "context": "", "prompt": "Number of unordered five-card hands all of one suit in a standard deck, including straight flushes?", "response_mode": "single_choice", "options": { "A": "4*13^5", "B": "C(52,5)", "C": "4*C(13,5)", "D": "C(13,5)" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "8.3a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "A": 0.0, "D": 0.0, "C": 1.0 }, "batch_response_ms": 322.78100000257837, "repaired": false }, { "id": "8.3b", "section": 8, "context": "", "prompt": "You hold two hearts. Choose five unordered cards from the remaining 50. Count selections with at least three additional hearts.", "response_mode": "single_choice", "options": { "A": "C(13,3)C(39,2)", "B": "C(11,3)C(47,2)", "C": "C(11,3)C(39,2)", "D": "C(11,3)C(39,2)+C(11,4)C(39,1)+C(11,5)" }, "correct": true, "kind": "multi_step", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "8.3b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.96, "probabilities": { "B": 0.01, "A": 0.0, "D": 0.97, "C": 0.02 }, "batch_response_ms": 322.78100000257837, "repaired": false }, { "id": "8.4", "section": 8, "context": "", "prompt": "Count integer solutions sum(xi,i=1..k)=n, each xi>=−1, n>=0.", "response_mode": "single_choice", "options": { "A": "k^n", "B": "C(n+2k−1,k−1)", "C": "C(n+k−1,k−1)", "D": "C(n−1,k−1)" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "8.4" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.96, "probabilities": { "B": 0.97, "C": 0.03, "D": 0.0, "A": 0.0 }, "batch_response_ms": 322.78100000257837, "repaired": false }, { "id": "9.1", "section": 9, "context": "Unit square grid from A=(0,0) to B=(7,5); only right and up steps. P=(3,2).", "prompt": "Number of paths A to B?", "response_mode": "single_choice", "options": { "A": "2^12", "B": "C(35,5)", "C": "C(12,5)", "D": "7!*5!" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "9.1" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.01, "A": 0.0, "C": 0.99, "D": 0.0 }, "batch_response_ms": 408.7955830000283, "repaired": false }, { "id": "9.2", "section": 9, "context": "Unit square grid from A=(0,0) to B=(7,5); only right and up steps. P=(3,2).", "prompt": "Number of paths A to B through P?", "response_mode": "single_choice", "options": { "A": "C(12,5)", "B": "C(7,3)C(5,3)/2", "C": "C(5,3)+C(7,4)", "D": "C(5,2)C(7,3)" }, "correct": true, "kind": "multi_step", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "9.2" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.89, "probabilities": { "D": 0.91, "B": 0.05, "C": 0.01, "A": 0.03 }, "batch_response_ms": 408.7955830000283, "repaired": false }, { "id": "9.3", "section": 9, "context": "Unit square grid from A=(0,0) to B=(7,5); only right and up steps. P=(3,2).", "prompt": "Combinatorially prove C(12,7)=sum(i=0..5)C(3+i,i)C(8−i,5−i).", "response_mode": "single_choice", "options": { "A": "Choose any i vertical steps twice; this counts each path once.", "B": "Classify by every vertex visited in column 3; each path visits exactly one such vertex.", "C": "Each of the six possible heights gives equally many paths, so the answer is six times any summand.", "D": "Classify each path by the unique right step from column 3 to 4 at height i. There are C(3+i,i) prefixes to (3,i), and C(8−i,5−i) suffixes from (4,i) to B." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "9.3" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.89, "probabilities": { "C": 0.0, "A": 0.0, "D": 0.92, "B": 0.08 }, "batch_response_ms": 408.7955830000283, "repaired": false }, { "id": "10.1", "section": 10, "context": "", "prompt": "Distinct real-valued functions represented by finite integer-coefficient polynomials?", "response_mode": "single_choice", "options": { "A": "Countably infinite", "B": "Uncountably infinite", "C": "Finite" }, "correct": true, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "10.1" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "A": 0.99, "C": 0.0, "B": 0.01 }, "batch_response_ms": 317.39345799724106, "repaired": false }, { "id": "10.2", "section": 10, "context": "", "prompt": "Distinct real-valued functions represented by finite real-coefficient polynomials?", "response_mode": "single_choice", "options": { "A": "Uncountably infinite", "B": "Countably infinite", "C": "Finite" }, "correct": false, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "10.2" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.78, "probabilities": { "A": 0.15, "C": 0.0, "B": 0.85 }, "batch_response_ms": 317.39345799724106, "repaired": false }, { "id": "10.3", "section": 10, "context": "", "prompt": "Distinct functions GF(p)->GF(p) represented by finite polynomials for fixed prime p?", "response_mode": "single_choice", "options": { "A": "Countably infinite", "B": "Finite", "C": "Uncountably infinite" }, "correct": true, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "10.3" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.61, "probabilities": { "A": 0.22, "C": 0.04, "B": 0.74 }, "batch_response_ms": 317.39345799724106, "repaired": false }, { "id": "11.1", "section": 11, "context": "", "prompt": "Fill four blanks: TestHalt(F,a) defines fresh dummy():pass and inner(b): [1];[2], then returns TestCalls([3],[4],a), where TestCalls(P,Q,x) detects whether P(x) ever calls Q.", "response_mode": "single_choice", "options": { "A": "(F(a); dummy(); dummy; inner)", "B": "(F(b); dummy(); inner; dummy)", "C": "(F(b); dummy(); F; dummy)", "D": "(dummy(); F(b); inner; dummy)" }, "correct": true, "kind": "multi_step", "topic": "computability", "original_format": "written", "repairs": [], "source_parts": [ "11.1", "11.2", "11.3", "11.4" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.25, "probabilities": { "D": 0.17, "A": 0.27, "B": 0.43, "C": 0.13 }, "batch_response_ms": 357.2974999988219, "repaired": false }, { "id": "12.1a", "section": 12, "context": "P(A)=1/2,P(B)=1/3,P(C)=1/6,P(A union B)=2/3,P(A given C)=2/3.", "prompt": "P(A intersection B)?", "response_mode": "single_choice", "options": { "A": "1/6", "B": "1/3", "C": "2/3", "D": "1/9" }, "correct": true, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "12.1a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.92, "probabilities": { "B": 0.05, "C": 0.01, "D": 0.01, "A": 0.9299999999999999 }, "batch_response_ms": 326.0857919995033, "repaired": false }, { "id": "12.1b", "section": 12, "context": "P(A)=1/2,P(B)=1/3,P(C)=1/6,P(A union B)=2/3,P(A given C)=2/3.", "prompt": "P(B given A)?", "response_mode": "single_choice", "options": { "A": "2/3", "B": "1/6", "C": "1/3", "D": "1/2" }, "correct": true, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "12.1b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.15, "probabilities": { "B": 0.29, "C": 0.37, "D": 0.22, "A": 0.12 }, "batch_response_ms": 326.0857919995033, "repaired": false }, { "id": "12.1c", "section": 12, "context": "P(A)=1/2,P(B)=1/3,P(C)=1/6,P(A union B)=2/3,P(A given C)=2/3.", "prompt": "P(C intersection A)?", "response_mode": "single_choice", "options": { "A": "2/9", "B": "1/3", "C": "1/9", "D": "1/6" }, "correct": true, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "12.1c" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.82, "probabilities": { "B": 0.01, "C": 0.86, "D": 0.01, "A": 0.12 }, "batch_response_ms": 326.0857919995033, "repaired": false }, { "id": "12.1d", "section": 12, "context": "P(A)=1/2,P(B)=1/3,P(C)=1/6,P(A union B)=2/3,P(A given C)=2/3.", "prompt": "P(C given A)?", "response_mode": "single_choice", "options": { "A": "2/9", "B": "1/9", "C": "1/3", "D": "2/3" }, "correct": true, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "12.1d" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.15, "probabilities": { "B": 0.26, "A": 0.36, "D": 0.04, "C": 0.34 }, "batch_response_ms": 326.0857919995033, "repaired": false }, { "id": "12.2a", "section": 12, "context": "ID,IE indicate events D,E with P(D)=1/3,P(E)=1/2,P(D intersection E)=1/5. The best affine estimate of ID from IE is a(IE−E[IE])+b.", "prompt": "Cov(ID,IE)?", "response_mode": "single_choice", "options": { "A": "1/6", "B": "1/5", "C": "−1/30", "D": "1/30" }, "correct": true, "kind": "calculation", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "12.2a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.72, "probabilities": { "B": 0.01, "C": 0.19, "D": 0.79, "A": 0.01 }, "batch_response_ms": 326.0857919995033, "repaired": false }, { "id": "12.2bi", "section": 12, "context": "ID,IE indicate events D,E with P(D)=1/3,P(E)=1/2,P(D intersection E)=1/5. The best affine estimate of ID from IE is a(IE−E[IE])+b.", "prompt": "Sign of a?", "response_mode": "single_choice", "options": { "A": "Positive", "B": "Negative", "C": "Zero" }, "correct": true, "kind": "recognition", "topic": "regression", "original_format": "choice", "repairs": [], "source_parts": [ "12.2bi" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.92, "probabilities": { "B": 0.05, "C": 0.01, "A": 0.94 }, "batch_response_ms": 326.0857919995033, "repaired": false }, { "id": "12.2bii", "section": 12, "context": "ID,IE indicate events D,E with P(D)=1/3,P(E)=1/2,P(D intersection E)=1/5. The best affine estimate of ID from IE is a(IE−E[IE])+b.", "prompt": "Value of a?", "response_mode": "single_choice", "options": { "A": "1/3", "B": "1/5", "C": "1/30", "D": "2/15" }, "correct": true, "kind": "multi_step", "topic": "regression", "original_format": "written", "repairs": [], "source_parts": [ "12.2bii" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.82, "probabilities": { "B": 0.02, "A": 0.02, "D": 0.86, "C": 0.1 }, "batch_response_ms": 326.0857919995033, "repaired": false }, { "id": "12.2biii", "section": 12, "context": "ID,IE indicate events D,E with P(D)=1/3,P(E)=1/2,P(D intersection E)=1/5. The best affine estimate of ID from IE is a(IE−E[IE])+b.", "prompt": "Value of b?", "response_mode": "single_choice", "options": { "A": "0", "B": "1/5", "C": "1/3", "D": "1/2" }, "correct": true, "kind": "recognition", "topic": "regression", "original_format": "written", "repairs": [], "source_parts": [ "12.2biii" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.77, "probabilities": { "B": 0.06, "C": 0.84, "D": 0.05, "A": 0.05 }, "batch_response_ms": 326.0857919995033, "repaired": false }, { "id": "12.3a", "section": 12, "context": "Independent X,Y have E[X]=2,E[Y]=3,Var(X)=1,Var(Y)=2.", "prompt": "E[XY]?", "response_mode": "single_choice", "options": { "A": "5", "B": "6", "C": "9", "D": "1" }, "correct": true, "kind": "calculation", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "12.3a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 1.0, "A": 0.0, "D": 0.0, "C": 0.0 }, "batch_response_ms": 326.0857919995033, "repaired": false }, { "id": "12.3b", "section": 12, "context": "Independent X,Y have E[X]=2,E[Y]=3,Var(X)=1,Var(Y)=2.", "prompt": "E[X²]?", "response_mode": "single_choice", "options": { "A": "1", "B": "4", "C": "5", "D": "3" }, "correct": true, "kind": "calculation", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "12.3b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "C": 1.0, "D": 0.0, "A": 0.0 }, "batch_response_ms": 326.0857919995033, "repaired": false }, { "id": "12.3c", "section": 12, "context": "Independent X,Y have E[X]=2,E[Y]=3,Var(X)=1,Var(Y)=2.", "prompt": "Var(XY)?", "response_mode": "single_choice", "options": { "A": "2", "B": "36", "C": "55", "D": "19" }, "correct": true, "kind": "multi_step", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "12.3c" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.91, "probabilities": { "B": 0.01, "C": 0.05, "D": 0.94, "A": 0.0 }, "batch_response_ms": 326.0857919995033, "repaired": false }, { "id": "12.4a", "section": 12, "context": "", "prompt": "Density f(x)=x on [0,c], zero otherwise. What is c>0?", "response_mode": "single_choice", "options": { "A": "1/2", "B": "2", "C": "sqrt(2)", "D": "1" }, "correct": true, "kind": "calculation", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "12.4a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.89, "probabilities": { "B": 0.06, "C": 0.92, "D": 0.01, "A": 0.01 }, "batch_response_ms": 326.0857919995033, "repaired": false }, { "id": "12.4b", "section": 12, "context": "", "prompt": "Density f(x)=x on [0,c], zero elsewhere, with c>0 chosen to normalize it. CDF on [0,c]?", "response_mode": "single_choice", "options": { "A": "x", "B": "x²", "C": "x²/2", "D": "2x" }, "correct": false, "kind": "calculation", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "12.4b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.46, "probabilities": { "B": 0.6, "C": 0.36, "D": 0.03, "A": 0.01 }, "batch_response_ms": 326.0857919995033, "repaired": false }, { "id": "13.1a", "section": 13, "context": "", "prompt": "Independent X~Bin(n,p),Y~Bin(n,q). Cov(X,Y)?", "response_mode": "single_choice", "options": { "A": "n(p+q)", "B": "np(1−q)", "C": "0", "D": "npq" }, "correct": true, "kind": "recognition", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "13.1a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "C": 1.0, "A": 0.0, "D": 0.0 }, "batch_response_ms": 349.60312499970314, "repaired": false }, { "id": "13.1b", "section": 13, "context": "", "prompt": "Independent X~Bin(n,p),Y~Bin(n,q). Var(X+Y)?", "response_mode": "single_choice", "options": { "A": "n²[p(1−p)+q(1−q)]", "B": "npq", "C": "n[p(1−p)+q(1−q)]", "D": "n(p+q)(1−p−q)" }, "correct": true, "kind": "calculation", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "13.1b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "C": 1.0, "A": 0.0, "D": 0.0 }, "batch_response_ms": 349.60312499970314, "repaired": false }, { "id": "13.2", "section": 13, "context": "", "prompt": "X~Geom(p) counts trials in {1,2,...}. For integer x>=1, compute E[X−x given X>=x].", "response_mode": "single_choice", "options": { "A": "x/p", "B": "0", "C": "(1−p)/p", "D": "1/p" }, "correct": true, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [ "Correct official 1/p key: >= retains a possible zero residual; with this support the answer is 1/p−1." ], "source_parts": [ "13.2" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.47, "probabilities": { "B": 0.02, "C": 0.61, "A": 0.01, "D": 0.36 }, "batch_response_ms": 349.60312499970314, "repaired": true }, { "id": "13.3", "section": 13, "context": "", "prompt": "X~Poisson(lambda). P(X>0)?", "response_mode": "single_choice", "options": { "A": "lambda*exp(−lambda)", "B": "exp(−lambda)", "C": "1/lambda", "D": "1−exp(−lambda)" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "13.3" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "A": 0.0, "C": 0.0, "D": 1.0 }, "batch_response_ms": 349.60312499970314, "repaired": false }, { "id": "13.4", "section": 13, "context": "", "prompt": "Independent Poisson(lambda),Poisson(mu). P(X+Y>0)?", "response_mode": "single_choice", "options": { "A": "exp(−lambda−mu)", "B": "1−exp(−lambda)*mu", "C": "(1−exp(−lambda))(1−exp(−mu))", "D": "1−exp(−lambda−mu)" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "13.4" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "A": 0.0, "D": 1.0 }, "batch_response_ms": 349.60312499970314, "repaired": false }, { "id": "13.5", "section": 13, "context": "", "prompt": "PDF of Exp(rate lambda>0) on x>=0?", "response_mode": "single_choice", "options": { "A": "exp(−x/lambda)/lambda²", "B": "exp(−lambda*x)", "C": "lambda*exp(−lambda*x)", "D": "lambda*exp(lambda*x)" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "13.5" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "C": 1.0, "A": 0.0, "D": 0.0 }, "batch_response_ms": 349.60312499970314, "repaired": false }, { "id": "13.6", "section": 13, "context": "", "prompt": "PDF of min(X,Y) for independent Exp(rate lambda) variables, z>=0?", "response_mode": "single_choice", "options": { "A": "2lambda*exp(−lambda*z)", "B": "lambda²*exp(−2lambda*z)", "C": "2lambda*exp(−2lambda*z)", "D": "lambda*exp(−lambda*z)" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "13.6" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.97, "probabilities": { "B": 0.0, "A": 0.01, "C": 0.98, "D": 0.01 }, "batch_response_ms": 349.60312499970314, "repaired": false }, { "id": "13.7", "section": 13, "context": "", "prompt": "A dart is uniform over a disk of radius r>0. Density of distance x to center?", "response_mode": "single_choice", "options": { "A": "1/r on [0,r], zero elsewhere", "B": "x²/r² on [0,r], zero elsewhere", "C": "2x/r² on [0,r], zero elsewhere", "D": "2pi*x on [0,r], zero elsewhere" }, "correct": true, "kind": "multi_step", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "13.7" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "C": 1.0, "A": 0.0, "D": 0.0 }, "batch_response_ms": 349.60312499970314, "repaired": false }, { "id": "14.1", "section": 14, "context": "Independent X,Y~Uniform[0,r],r>0.", "prompt": "CDF of max(X,Y)?", "response_mode": "single_choice", "options": { "A": "0 below 0; 1−(1−z/r)² on [0,r]; 1 above r", "B": "0 below 0; z/r on [0,r]; 1 above r", "C": "0 below 0; (z/r)² on [0,r]; 1 above r", "D": "0 below 0; 2z/r on [0,r]; 1 above r" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "14.1" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.97, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.02, "C": 0.98 }, "batch_response_ms": 297.1683329997177, "repaired": false }, { "id": "14.2", "section": 14, "context": "Independent X,Y~Uniform[0,r],r>0.", "prompt": "PDF of max(X,Y) on [0,r], zero elsewhere?", "response_mode": "single_choice", "options": { "A": "2(r−z)/r²", "B": "2z/r²", "C": "z²/r²", "D": "1/r" }, "correct": true, "kind": "calculation", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "14.2" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "B": 1.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 297.1683329997177, "repaired": false }, { "id": "14.3", "section": 14, "context": "Independent X,Y~Uniform[0,r],r>0.", "prompt": "E[max(X,Y)]?", "response_mode": "single_choice", "options": { "A": "2r/3", "B": "r", "C": "r/2", "D": "r/3" }, "correct": true, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "14.3" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 1.0, "C": 0.0 }, "batch_response_ms": 297.1683329997177, "repaired": false }, { "id": "14.4", "section": 14, "context": "Independent X,Y~Uniform[0,r],r>0.", "prompt": "E[min(X,Y)]? Hint: avoid integration.", "response_mode": "single_choice", "options": { "A": "r/4", "B": "2r/3", "C": "r/3", "D": "r/2" }, "correct": true, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "14.4" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.91, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.06, "C": 0.9400000000000001 }, "batch_response_ms": 297.1683329997177, "repaired": false }, { "id": "14.5", "section": 14, "context": "Independent X,Y~Uniform[0,r],r>0.", "prompt": "E[|X−Y|]? Hint: use min, max and linearity.", "response_mode": "single_choice", "options": { "A": "2r/3", "B": "0", "C": "r/3", "D": "r/2" }, "correct": true, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "14.5" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.88, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.09, "C": 0.91 }, "batch_response_ms": 297.1683329997177, "repaired": false }, { "id": "15.1", "section": 15, "context": "A uniform random permutation assigns balls 1..100 to bins 1..100, one each. Ii indicates ball i lands within distance one of bin i, without wraparound.", "prompt": "Expected number close to their original bin?", "response_mode": "single_choice", "options": { "A": "2", "B": "3", "C": "2.98", "D": "1" }, "correct": true, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "15.1" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.58, "probabilities": { "D": 0.02, "C": 0.6900000000000001, "A": 0.18, "B": 0.11 }, "batch_response_ms": 358.66100000203005, "repaired": false }, { "id": "15.2a", "section": 15, "context": "A uniform random permutation assigns balls 1..100 to bins 1..100, one each. Ii indicates ball i lands within distance one of bin i, without wraparound.", "prompt": "E[I2*I5]?", "response_mode": "single_choice", "options": { "A": "6/(100*99)", "B": "3/100", "C": "9/10000", "D": "9/(100*99)" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "15.2a" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.66, "probabilities": { "D": 0.74, "C": 0.11, "A": 0.13, "B": 0.02 }, "batch_response_ms": 358.66100000203005, "repaired": false }, { "id": "15.2b", "section": 15, "context": "A uniform random permutation assigns balls 1..100 to bins 1..100, one each. Ii indicates ball i lands within distance one of bin i, without wraparound.", "prompt": "E[I2*I4]?", "response_mode": "single_choice", "options": { "A": "8/(100*99)", "B": "9/10000", "C": "9/(100*99)", "D": "6/(100*99)" }, "correct": false, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "15.2b" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.4, "probabilities": { "D": 0.2, "C": 0.55, "A": 0.19, "B": 0.06 }, "batch_response_ms": 358.66100000203005, "repaired": false }, { "id": "15.2c", "section": 15, "context": "A uniform random permutation assigns balls 1..100 to bins 1..100, one each. Ii indicates ball i lands within distance one of bin i, without wraparound.", "prompt": "Do there exist distinct i,j for which Ii,Ij are independent?", "response_mode": "single_choice", "options": { "A": "Yes", "B": "No" }, "correct": false, "kind": "recognition", "topic": "probability", "original_format": "choice", "repairs": [], "source_parts": [ "15.2c" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.58, "probabilities": { "A": 0.79, "B": 0.21 }, "batch_response_ms": 358.66100000203005, "repaired": false }, { "id": "16.1", "section": 16, "context": "Choose one of two 52-card decks with equal probability. Loaded deck:39 red,13 black. Fair deck:26 red,26 black. Draw uniformly without replacement.", "prompt": "Given first card red, probability the chosen deck is fair?", "response_mode": "single_choice", "options": { "A": "1/2", "B": "3/5", "C": "2/5", "D": "1/4" }, "correct": true, "kind": "multi_step", "topic": "bayes", "original_format": "written", "repairs": [], "source_parts": [ "16.1" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.67, "probabilities": { "D": 0.04, "B": 0.18, "A": 0.02, "C": 0.76 }, "batch_response_ms": 355.7217090019549, "repaired": false }, { "id": "16.2", "section": 16, "context": "Choose one of two 52-card decks with equal probability. Loaded deck:39 red,13 black. Fair deck:26 red,26 black. Draw uniformly without replacement.", "prompt": "Given first card red, probability the second is red?", "response_mode": "single_choice", "options": { "A": "(1/2)(38/51)+(1/2)(25/51)", "B": "38/51", "C": "(3/5)(38/51)+(2/5)(25/51)", "D": "(3/5)(39/52)+(2/5)(26/52)" }, "correct": true, "kind": "multi_step", "topic": "bayes", "original_format": "written", "repairs": [], "source_parts": [ "16.2" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.45, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.41, "C": 0.59 }, "batch_response_ms": 355.7217090019549, "repaired": false }, { "id": "17.1", "section": 17, "context": "", "prompt": "X is a sum of indicators, and p=P(X>=1). Prove p<=E[X].", "response_mode": "single_choice", "options": { "A": "Every probability is at most one, and every expectation is at least one.", "B": "X is always either zero or one, so equality must hold.", "C": "Indicators are always independent, so multiply their probabilities.", "D": "The union bound gives P(any indicator is one)<=sum of their success probabilities=sum of their expectations=E[X]. No independence needed." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "17.1" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 0.0, "D": 1.0, "B": 0.0, "C": 0.0 }, "batch_response_ms": 296.01220800032024, "repaired": false }, { "id": "17.2", "section": 17, "context": "", "prompt": "W counts successes in n independent Bernoulli(p) trials. For k>0, prove P(W>=k)<=np/k.", "response_mode": "single_choice", "options": { "A": "Chebyshev gives P(W>=k)=Var(W)/k, which equals np/k.", "B": "W is nonnegative and E[W]=np, so apply Markov’s inequality.", "C": "W is at most np in every outcome.", "D": "Independence makes W deterministic with value np." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "17.2" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 0.0, "D": 0.0, "B": 1.0, "C": 0.0 }, "batch_response_ms": 296.01220800032024, "repaired": false }, { "id": "17.3", "section": 17, "context": "", "prompt": "Y~Poisson(lambda),lambda>0. Prove P(Y>=10lambda)<=1/(81lambda).", "response_mode": "single_choice", "options": { "A": "The event equals |Y−lambda|>=10lambda, giving the stated bound.", "B": "The event implies |Y−lambda|>=9lambda. Chebyshev and Var(Y)=lambda give lambda/(9lambda)²=1/(81lambda).", "C": "Markov gives lambda/(10lambda)=1/(81lambda).", "D": "Poisson variables cannot exceed ten times their means." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "17.3" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "A": 0.0, "D": 0.0, "B": 1.0, "C": 0.0 }, "batch_response_ms": 296.01220800032024, "repaired": false }, { "id": "18.1", "section": 18, "context": "", "prompt": "For arbitrary A,B,C with positive conditioning probabilities, P(C given A union B)=P(C given A)+P(C given B)−P(C given A intersection B).", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "probability", "original_format": "choice", "repairs": [], "source_parts": [ "18.1" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.89, "probabilities": { "A": 0.95, "B": 0.05 }, "batch_response_ms": 359.9019590001262, "repaired": false }, { "id": "18.2", "section": 18, "context": "", "prompt": "Choose a counterexample and its four probabilities in the order: C|A union B, C|A, C|B, C|A intersection B.", "response_mode": "single_choice", "options": { "A": "Two fair independent flips: A=first H,B=second H,C=HH. Values (0,1/2,1/2,1).", "B": "Two fair independent flips: A=first H,B=second H,C=HH. Values (1/3,1/2,1/2,1).", "C": "Set A=B=C to the same positive event; values (1,1,1,1).", "D": "Set C to the empty event; values (0,0,0,0)." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "18.2" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.68, "probabilities": { "C": 0.01, "A": 0.21, "D": 0.01, "B": 0.77 }, "batch_response_ms": 359.9019590001262, "repaired": false }, { "id": "19.1", "section": 19, "context": "Independent Xi~Poisson(lambda),lambda>0; An=(X1+...+Xn)/n.", "prompt": "E[An]?", "response_mode": "single_choice", "options": { "A": "lambda", "B": "n*lambda", "C": "lambda/n", "D": "1/lambda" }, "correct": true, "kind": "recognition", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "19.1" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 1.0, "C": 0.0, "B": 0.0 }, "batch_response_ms": 388.4730829995533, "repaired": false }, { "id": "19.2", "section": 19, "context": "Independent Xi~Poisson(lambda),lambda>0; An=(X1+...+Xn)/n.", "prompt": "Smallest integer sample size guaranteed by Chebyshev to make P(|An−E[An]|>=0.1E[An])<=0.05?", "response_mode": "single_choice", "options": { "A": "ceil(2000*lambda)", "B": "ceil(400/lambda)", "C": "ceil(2000/lambda)", "D": "ceil(200/lambda)" }, "correct": true, "kind": "multi_step", "topic": "bounds", "original_format": "written", "repairs": [], "source_parts": [ "19.2" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.33, "probabilities": { "D": 0.44, "A": 0.02, "C": 0.49, "B": 0.05 }, "batch_response_ms": 388.4730829995533, "repaired": false }, { "id": "19.3", "section": 19, "context": "Independent Xi~Poisson(lambda),lambda>0; An=(X1+...+Xn)/n.", "prompt": "Instead suppose An is exactly normal with its actual mean and variance, and use P(|Z|>=2)<=0.05. Sufficient integer n from this rule?", "response_mode": "single_choice", "options": { "A": "ceil(400*lambda)", "B": "ceil(40/lambda)", "C": "ceil(2000/lambda)", "D": "ceil(400/lambda)" }, "correct": true, "kind": "multi_step", "topic": "bounds", "original_format": "written", "repairs": [], "source_parts": [ "19.3" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.83, "probabilities": { "D": 0.87, "A": 0.08, "C": 0.01, "B": 0.04 }, "batch_response_ms": 388.4730829995533, "repaired": false }, { "id": "20.1", "section": 20, "context": "Repeated independent fair four-sided die rolls. State0:last roll not 1 (also start); state1:one trailing 1; state2:at least two trailing 1s.", "prompt": "Let b_i be expected future even rolls before first reaching state2 from state i. Choose the first-step equations.", "response_mode": "single_choice", "options": { "A": "b0=1/2+3b0/4+b1/4; b1=1/2+3b0/4+b2/4; b2=0", "B": "b0=1/4+3b0/4+b1/4; b1=1/4+3b0/4; b2=0", "C": "b0=1+3b0/4+b1/4; b1=1+3b0/4; b2=0", "D": "b0=1/2+b1/4; b1=1/2+b2/4; b2=0" }, "correct": true, "kind": "recognition", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "20.1" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.67, "probabilities": { "B": 0.2, "A": 0.75, "D": 0.01, "C": 0.04 }, "batch_response_ms": 310.453583002527, "repaired": false }, { "id": "20.2", "section": 20, "context": "Repeated independent fair four-sided die rolls. State0:last roll not 1 (also start); state1:one trailing 1; state2:at least two trailing 1s.", "prompt": "Continue rolling even after reaching state2. Is there a unique stationary distribution?", "response_mode": "single_choice", "options": { "A": "No", "B": "Yes" }, "correct": true, "kind": "recognition", "topic": "markov", "original_format": "choice", "repairs": [], "source_parts": [ "20.2" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.65, "probabilities": { "B": 0.83, "A": 0.17 }, "batch_response_ms": 310.453583002527, "repaired": false }, { "id": "20.3", "section": 20, "context": "Repeated independent fair four-sided die rolls. State0:last roll not 1 (also start); state1:one trailing 1; state2:at least two trailing 1s.", "prompt": "For the continuing chain, choose stationary equations plus normalization.", "response_mode": "single_choice", "options": { "A": "p0=3p0/4; p1=p0/4; p2=p1/4+p2; sum p=1", "B": "p0=3(p0+p1)/4; p1=(p0+p2)/4; p2=p1/4; sum p=1", "C": "p0=3(p0+p1+p2)/4; p1=p0/4; p2=(p1+p2)/4; sum p=1", "D": "p0=p1=p2=1/3" }, "correct": false, "kind": "recognition", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "20.3" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.63, "probabilities": { "B": 0.72, "A": 0.02, "D": 0.0, "C": 0.26 }, "batch_response_ms": 310.453583002527, "repaired": false }, { "id": "21.1", "section": 21, "context": "Two players start ready and shoot independently with success probability 2/3. Both make: win; both miss: loss. If exactly one misses, next turn that player retrieves while the other shoots. A lone shooter’s success makes both ready next turn; a miss leaves one retrieving. State0=both ready,1=one retrieving,2=win,3=loss. Arrows A:0->2,B:0->1,C:1->0,D:0->3,E:1->1.", "prompt": "Give (A,B,C,D,E).", "response_mode": "single_choice", "options": { "A": "(4/9,4/9,2/3,1/9,1/3)", "B": "(4/9,4/9,1/3,1/9,2/3)", "C": "(2/3,2/9,1/3,1/3,2/3)", "D": "(4/9,2/9,2/3,1/9,1/3)" }, "correct": true, "kind": "multi_step", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "21.1A", "21.1B", "21.1C", "21.1D", "21.1E" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.95, "probabilities": { "A": 0.96, "B": 0.02, "C": 0.0, "D": 0.02 }, "batch_response_ms": 378.19520800258033, "repaired": false }, { "id": "21.2", "section": 21, "context": "Two players start ready and shoot independently with success probability 2/3. Both make: win; both miss: loss. If exactly one misses, next turn that player retrieves while the other shoots. A lone shooter’s success makes both ready next turn; a miss leaves one retrieving. State0=both ready,1=one retrieving,2=win,3=loss. Arrows A:0->2,B:0->1,C:1->0,D:0->3,E:1->1.", "prompt": "Probability of winning from both ready?", "response_mode": "single_choice", "options": { "A": "8/9", "B": "2/3", "C": "4/9", "D": "4/5" }, "correct": false, "kind": "multi_step", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "21.2" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.55, "probabilities": { "A": 0.66, "B": 0.07, "C": 0.14, "D": 0.13 }, "batch_response_ms": 378.19520800258033, "repaired": false }, { "id": "21.3", "section": 21, "context": "Two players start ready and shoot independently with success probability 2/3. Both make: win; both miss: loss. If exactly one misses, next turn that player retrieves while the other shoots. A lone shooter’s success makes both ready next turn; a miss leaves one retrieving. State0=both ready,1=one retrieving,2=win,3=loss. Arrows A:0->2,B:0->1,C:1->0,D:0->3,E:1->1.", "prompt": "Expected turns until win or loss from both ready?", "response_mode": "single_choice", "options": { "A": "4", "B": "9/5", "C": "5", "D": "3" }, "correct": false, "kind": "multi_step", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "21.3" ], "exam": "final_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.66, "probabilities": { "A": 0.09, "B": 0.74, "C": 0.04, "D": 0.13 }, "batch_response_ms": 378.19520800258033, "repaired": false }, { "id": "2.1", "section": 2, "context": "", "prompt": "False implies False.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.1" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "A": 1.0 }, "batch_response_ms": 355.03979200075264, "repaired": false }, { "id": "2.2", "section": 2, "context": "", "prompt": "False implies True.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.2" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "A": 1.0 }, "batch_response_ms": 355.03979200075264, "repaired": false }, { "id": "2.3", "section": 2, "context": "", "prompt": "[NOT exists x P(x)] implies [forall x (P(x) implies Q(x))].", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.3" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.99, "A": 0.01 }, "batch_response_ms": 355.03979200075264, "repaired": false }, { "id": "2.4", "section": 2, "context": "", "prompt": "[forall x exists y (Q(x,y) implies P(x))] is equivalent to [forall x (NOT P(x) implies NOT forall y Q(x,y))].", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.4" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.48, "probabilities": { "B": 0.74, "A": 0.26 }, "batch_response_ms": 355.03979200075264, "repaired": false }, { "id": "2.5", "section": 2, "context": "", "prompt": "No stable matching can give every job its least-preferred candidate.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": false, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "2.5" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.41, "probabilities": { "B": 0.29, "A": 0.71 }, "batch_response_ms": 355.03979200075264, "repaired": false }, { "id": "2.6", "section": 2, "context": "", "prompt": "If job-proposing deferred acceptance finishes on day one, the result is optimal for both jobs and candidates.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "2.6" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.68, "probabilities": { "B": 0.16, "A": 0.84 }, "batch_response_ms": 355.03979200075264, "repaired": false }, { "id": "3.1", "section": 3, "context": "", "prompt": "For natural x<=y, choose a valid proof that x<=y/2 or y−x<=y/2.", "response_mode": "single_choice", "options": { "A": "Since x<=y, necessarily x<=y/2.", "B": "Every natural y is even, so one term equals y/2.", "C": "If both inequalities fail, adding x>y/2 and y−x>y/2 gives y>y, a contradiction.", "D": "Since y−x>=0, necessarily y−x<=y/2." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "3.1" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "C": 1.0, "A": 0.0 }, "batch_response_ms": 618.1105409996235, "repaired": false }, { "id": "3.2a", "section": 3, "context": "", "prompt": "How many ordered natural pairs (a,b) satisfy b²=3a²? Here zero is natural.", "response_mode": "single_choice", "options": { "A": "2", "B": "1", "C": "0", "D": "Infinitely many" }, "correct": true, "kind": "multi_step", "topic": "number_theory", "original_format": "written", "repairs": [], "source_parts": [ "3.2a" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.65, "probabilities": { "B": 0.74, "A": 0.21, "D": 0.03, "C": 0.02 }, "batch_response_ms": 618.1105409996235, "repaired": false }, { "id": "3.2b", "section": 3, "context": "", "prompt": "Choose a valid argument determining all natural solutions of b²=3a², allowing zero.", "response_mode": "single_choice", "options": { "A": "Squares are always even, whereas three times a square is always odd.", "B": "A nonzero square has even exponent of prime3, but 3a² has odd exponent; thus no nonzero solution exists and (0,0) is the only solution.", "C": "Dividing by three implies b=a, hence all pairs with a=b are solutions.", "D": "For a=3k, b=3k also satisfies the equation for every natural k." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "3.2b" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 1.0, "A": 0.0, "D": 0.0, "C": 0.0 }, "batch_response_ms": 618.1105409996235, "repaired": false }, { "id": "4.1", "section": 4, "context": "", "prompt": "Prime p and nonzero a modulo p: how many distinct residues ax arise as x ranges over 0..p−1?", "response_mode": "single_choice", "options": { "A": "p−1", "B": "p", "C": "1", "D": "gcd(a,p)" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "4.1" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.78, "probabilities": { "A": 0.16, "D": 0.0, "B": 0.84, "C": 0.0 }, "batch_response_ms": 293.14133300067624, "repaired": false }, { "id": "4.2", "section": 4, "context": "", "prompt": "Let gcd(a,N)=d. How many distinct residues ax mod N arise for x=0..N−1?", "response_mode": "single_choice", "options": { "A": "N", "B": "d", "C": "N/d", "D": "N−d" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "4.2" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.98, "probabilities": { "D": 0.0, "A": 0.0, "B": 0.01, "C": 0.99 }, "batch_response_ms": 293.14133300067624, "repaired": false }, { "id": "4.3", "section": 4, "context": "", "prompt": "For prime p, evaluate 1²·2²···(p−1)² modulo p.", "response_mode": "single_choice", "options": { "A": "1", "B": "2", "C": "0", "D": "p−1" }, "correct": true, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "4.3" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.77, "probabilities": { "A": 0.83, "D": 0.17, "B": 0.0, "C": 0.0 }, "batch_response_ms": 293.14133300067624, "repaired": false }, { "id": "5.1", "section": 5, "context": "", "prompt": "Choose a fair four-sided or fair six-sided die with equal probability. Roll and observe3. Probability it was the four-sided die?", "response_mode": "single_choice", "options": { "A": "3/5", "B": "1/4", "C": "1/2", "D": "2/5" }, "correct": false, "kind": "multi_step", "topic": "bayes", "original_format": "written", "repairs": [], "source_parts": [ "5.1" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.51, "probabilities": { "D": 0.64, "A": 0.25, "C": 0.07, "B": 0.04 }, "batch_response_ms": 364.61879200214753, "repaired": false }, { "id": "6.1", "section": 6, "context": "People indexed0,...,n−1 arrive in order, n>=8. Person i chooses min(i,7) distinct earlier people as friends. Friendship is undirected.", "prompt": "How many friendship edges result?", "response_mode": "single_choice", "options": { "A": "C(n,2)", "B": "7n−21", "C": "7n", "D": "7n−28" }, "correct": false, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [ "Use n>=8 rather than n>=7 consistently with Q6.3 intended answer." ], "source_parts": [ "6.1" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.86, "probabilities": { "C": 0.0, "B": 0.91, "A": 0.0, "D": 0.09 }, "batch_response_ms": 334.5862919995852, "repaired": true }, { "id": "6.2", "section": 6, "context": "People indexed0,...,n−1 arrive in order, n>=8. Person i chooses min(i,7) distinct earlier people as friends. Friendship is undirected.", "prompt": "Maximum possible degree of a person?", "response_mode": "single_choice", "options": { "A": "14", "B": "7", "C": "n−7", "D": "n−1" }, "correct": false, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "6.2" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.38, "probabilities": { "C": 0.18, "A": 0.16, "B": 0.54, "D": 0.12 }, "batch_response_ms": 334.5862919995852, "repaired": false }, { "id": "6.3a", "section": 6, "context": "People indexed0,...,n−1 arrive in order, n>=8. Person i chooses min(i,7) distinct earlier people as friends. Friendship is undirected.", "prompt": "Smallest number of colors sufficient for every possible resulting graph?", "response_mode": "single_choice", "options": { "A": "n", "B": "9", "C": "7", "D": "8" }, "correct": true, "kind": "multi_step", "topic": "graphs", "original_format": "written", "repairs": [ "At n=7 the source answer8 is not minimal; restrict n>=8." ], "source_parts": [ "6.3a" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.62, "probabilities": { "C": 0.17, "B": 0.09, "A": 0.02, "D": 0.72 }, "batch_response_ms": 334.5862919995852, "repaired": true }, { "id": "6.3b", "section": 6, "context": "People indexed0,...,n−1 arrive in order, n>=8. Person i chooses min(i,7) distinct earlier people as friends. Friendship is undirected.", "prompt": "Which proves the optimal universal coloring bound?", "response_mode": "single_choice", "options": { "A": "The first eight vertices form K8. Color in arrival order; each later vertex has seven earlier neighbors, so eight colors suffice.", "B": "Every new vertex requires a fresh color because it has earlier neighbors.", "C": "All vertices have degree seven, so seven colors always suffice.", "D": "The graph is a tree, so two colors suffice." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "6.3b" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 0.0, "A": 1.0, "B": 0.0, "D": 0.0 }, "batch_response_ms": 334.5862919995852, "repaired": false }, { "id": "6.4", "section": 6, "context": "People indexed0,...,n−1 arrive in order, n>=8. Person i chooses min(i,7) distinct earlier people as friends. Friendship is undirected.", "prompt": "If each person uniformly chooses the prescribed-size subset of earlier people, expected degree of person0?", "response_mode": "single_choice", "options": { "A": "sum(i=1 to n) min(i,7)/i", "B": "sum(i=1 to n−1) min(i,7)/i", "C": "7", "D": "7(n−1)" }, "correct": true, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [ "Correct source summation endpoint: n people indexed0..n−1, not0..n." ], "source_parts": [ "6.4" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.96, "probabilities": { "C": 0.01, "A": 0.02, "B": 0.97, "D": 0.0 }, "batch_response_ms": 334.5862919995852, "repaired": true }, { "id": "7.1", "section": 7, "context": "", "prompt": "The three-dimensional cube has more edges than K4.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "7.1" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.83, "probabilities": { "B": 0.91, "A": 0.09 }, "batch_response_ms": 480.2326249991893, "repaired": false }, { "id": "7.2", "section": 7, "context": "", "prompt": "Sharp maximum number of vertices in a graph with e edges and c connected components?", "response_mode": "single_choice", "options": { "A": "2e+c", "B": "e−c", "C": "e+c−1", "D": "e+c" }, "correct": false, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.2" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.41, "probabilities": { "B": 0.01, "C": 0.03, "D": 0.41, "A": 0.55 }, "batch_response_ms": 480.2326249991893, "repaired": false }, { "id": "7.3", "section": 7, "context": "", "prompt": "A finite undirected graph can have an odd number of odd-degree vertices.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "7.3" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.01, "A": 0.99 }, "batch_response_ms": 480.2326249991893, "repaired": false }, { "id": "7.4", "section": 7, "context": "", "prompt": "Average vertex degree in a tree with n vertices?", "response_mode": "single_choice", "options": { "A": "2", "B": "n−1", "C": "1−1/n", "D": "2−2/n" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.4" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "C": 0.0, "D": 1.0, "A": 0.0 }, "batch_response_ms": 480.2326249991893, "repaired": false }, { "id": "7.5", "section": 7, "context": "", "prompt": "Connected planar embedding has v vertices and average face boundary length s>2. Choose edge count with valid justification.", "response_mode": "single_choice", "options": { "A": "e=(s−2)(v−2)/s, by reversing the ratio in the face-count identity.", "B": "e=sv/2, since face length equals average vertex degree.", "C": "e=s(v−2)/(s−2), since sf=2e and v−e+f=2.", "D": "e=s(v−2)/2, since every face contributes two vertices." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "7.5" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "C": 1.0, "D": 0.0, "A": 0.0 }, "batch_response_ms": 480.2326249991893, "repaired": false }, { "id": "8.1a", "section": 8, "context": "Over GF(7), degree-at-most-two P satisfies P(0)=0 and P(1)=P(6)=2.", "prompt": "Remainder when P is divided by x−6?", "response_mode": "single_choice", "options": { "A": "2", "B": "6", "C": "1", "D": "0" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "8.1a" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "D": 0.0, "B": 0.0, "A": 1.0, "C": 0.0 }, "batch_response_ms": 288.80108399971505, "repaired": false }, { "id": "8.1b", "section": 8, "context": "Over GF(7), degree-at-most-two P satisfies P(0)=0 and P(1)=P(6)=2.", "prompt": "Find P(x).", "response_mode": "single_choice", "options": { "A": "x²+1", "B": "2x²+2", "C": "2x²", "D": "2x" }, "correct": false, "kind": "calculation", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "8.1b" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.22, "probabilities": { "D": 0.42, "B": 0.09, "A": 0.14, "C": 0.35 }, "batch_response_ms": 288.80108399971505, "repaired": false }, { "id": "8.1c", "section": 8, "context": "Over GF(7), degree-at-most-two P satisfies P(0)=0 and P(1)=P(6)=2.", "prompt": "Which constant polynomial disagrees with exactly one of these three given values?", "response_mode": "single_choice", "options": { "A": "0", "B": "1", "C": "2", "D": "6" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "8.1c" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.27, "probabilities": { "D": 0.06, "B": 0.2, "A": 0.28, "C": 0.46 }, "batch_response_ms": 288.80108399971505, "repaired": false }, { "id": "8.2", "section": 8, "context": "", "prompt": "Over the reals, degree-at-most-two P has P(0)=0 and P(1)=P(−1)=A. Find P.", "response_mode": "single_choice", "options": { "A": "Ax", "B": "Ax²", "C": "A(x²+1)", "D": "A" }, "correct": true, "kind": "calculation", "topic": "polynomials", "original_format": "written", "repairs": [ "Specify real coefficients to avoid characteristic-two coincidence of coordinates." ], "source_parts": [ "8.2" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.98, "probabilities": { "D": 0.01, "B": 0.99, "A": 0.0, "C": 0.0 }, "batch_response_ms": 288.80108399971505, "repaired": true }, { "id": "8.3", "section": 8, "context": "", "prompt": "Over GF(p), p>=d+1, fixed degree-at-most-d P. How many degree-at-most-d Q agree with P at coordinates1,...,k, where0<=k<=d?", "response_mode": "single_choice", "options": { "A": "p^(d−k)", "B": "p^(d+1−k)", "C": "1", "D": "p^k" }, "correct": false, "kind": "calculation", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "8.3" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.35, "probabilities": { "D": 0.0, "B": 0.49, "A": 0.51, "C": 0.0 }, "batch_response_ms": 288.80108399971505, "repaired": false }, { "id": "9.1", "section": 9, "context": "", "prompt": "Between any two distinct reals in [0,1] there is a rational.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "9.1" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], 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E[T]?", "response_mode": "single_choice", "options": { "A": "mu*lambda", "B": "lambda/mu", "C": "mu/lambda", "D": "1/lambda" }, "correct": true, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [ "Explicit independence between bulb count and lifetimes needed for the source solution." ], "source_parts": [ "14.1" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "C": 1.0, "A": 0.0, "D": 0.0 }, "batch_response_ms": 367.11920900052064, "repaired": true }, { "id": "15.1", "section": 15, "context": "X~Normal(mu,sigma²), where sigma² denotes the variance.", "prompt": "E[X]?", "response_mode": "single_choice", "options": { "A": "mu²", "B": "mu", "C": "0", "D": "sigma" }, "correct": true, "kind": "recognition", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "15.1" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 1.0, "C": 0.0 }, "batch_response_ms": 302.3862919981184, "repaired": false }, { "id": "15.2", "section": 15, "context": "X~Normal(mu,sigma²), where sigma² denotes the variance.", "prompt": "E[X²]?", "response_mode": "single_choice", "options": { "A": "sigma²", "B": "mu²+sigma²", "C": "mu²", "D": "mu+sigma²" }, "correct": true, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "15.2" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 1.0, "C": 0.0 }, "batch_response_ms": 302.3862919981184, "repaired": false }, { "id": "15.3", "section": 15, "context": "X~Normal(mu,sigma²), where sigma² denotes the variance.", "prompt": "E[X³]?", "response_mode": "single_choice", "options": { "A": "mu³+3mu²*sigma", "B": "mu³+sigma³", "C": "mu³+3mu*sigma²", "D": "mu³" }, "correct": true, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "15.3" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.0, "C": 1.0 }, "batch_response_ms": 302.3862919981184, "repaired": false }, { "id": "16.1", "section": 16, "context": "Joint probabilities, columns S=1,2,3: row P=1:(.05,.10,0); row P=2:(.10,.05,.15); row P=3:(.05,.10,.40).", "prompt": "Marginals (S=1,2,3) and (P=1,2,3)?", "response_mode": "single_choice", "options": { "A": "S:(1/3,1/3,1/3); P:(1/3,1/3,1/3)", "B": "S:(.25,.20,.55); P:(.30,.15,.55)", "C": "S:(.20,.25,.55); P:(.15,.30,.55)", "D": "S:(.15,.30,.55); P:(.20,.25,.55)" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "16.1" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.95, "probabilities": { "C": 0.97, "B": 0.0, "A": 0.0, "D": 0.03 }, "batch_response_ms": 320.69995799975004, "repaired": false }, { "id": "16.2", "section": 16, "context": "Joint probabilities, columns S=1,2,3: row P=1:(.05,.10,0); row P=2:(.10,.05,.15); row P=3:(.05,.10,.40).", "prompt": "(E[S],E[P])?", "response_mode": "single_choice", "options": { "A": "(2.40,2.35)", "B": "(.55,.55)", "C": "(2.35,2.40)", "D": "(2,2)" }, "correct": false, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "16.2" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.46, "probabilities": { "C": 0.36, "A": 0.59, "B": 0.0, "D": 0.05 }, "batch_response_ms": 320.69995799975004, "repaired": false }, { "id": "16.3", "section": 16, "context": "Joint probabilities, columns S=1,2,3: row P=1:(.05,.10,0); row P=2:(.10,.05,.15); row P=3:(.05,.10,.40).", "prompt": "P(S=P)?", "response_mode": "single_choice", "options": { "A": "1/3", "B": "2/5", "C": "1/2", "D": "3/4" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "16.3" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.36, "probabilities": { "C": 0.52, "B": 0.38, "A": 0.05, "D": 0.05 }, "batch_response_ms": 320.69995799975004, "repaired": false }, { "id": "16.4", "section": 16, "context": "Joint probabilities, columns S=1,2,3: row P=1:(.05,.10,0); row P=2:(.10,.05,.15); row P=3:(.05,.10,.40).", "prompt": "P(S=3 given S>=P)?", "response_mode": "single_choice", "options": { "A": "4/5", "B": "11/20", "C": "3/5", "D": "11/15" }, "correct": false, "kind": "multi_step", "topic": "conditional_probability", "original_format": "written", "repairs": [], "source_parts": [ "16.4" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.32, "probabilities": { "C": 0.21, "B": 0.11, "A": 0.49, "D": 0.19 }, "batch_response_ms": 320.69995799975004, "repaired": false }, { "id": "17.1", "section": 17, "context": "I_A is1 when event A occurs and0 otherwise. Events need not be independent.", "prompt": "Why does E[I_A]=P(A)?", "response_mode": "single_choice", "options": { "A": "The expectation is1*P(A)+0*P(not A).", "B": "All indicators are fair Bernoulli variables.", "C": "The expectation of every indicator equals its variance.", "D": "The expectation sums P(A) and P(not A)." }, "correct": true, "kind": "recognition", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "17.1" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "D": 0.0, "A": 1.0 }, "batch_response_ms": 380.83770900266245, "repaired": false }, { "id": "17.2a", "section": 17, "context": "I_A is1 when event A occurs and0 otherwise. Events need not be independent.", "prompt": "Choose the valid pointwise identity and justification.", "response_mode": "single_choice", "options": { "A": "I_A I_B=I_A+I_B, because indicators only take0 and1.", "B": "min(I_A,I_B)=I_(A union B), because minimum detects either event.", "C": "I_A I_B=I_(A intersection B)=min(I_A,I_B), since all three are1 exactly when both events occur.", "D": "I_A I_B=I_(A union B), since multiplication represents either event." }, "correct": true, "kind": "recognition", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "17.2a" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 1.0, "D": 0.0, "B": 0.0, "A": 0.0 }, "batch_response_ms": 380.83770900266245, "repaired": false }, { "id": "17.2b", "section": 17, "context": "I_A is1 when event A occurs and0 otherwise. Events need not be independent.", "prompt": "Choose the valid union identity and justification.", "response_mode": "single_choice", "options": { "A": "I_(A union B)=I_A I_B because the events are independent.", "B": "I_(A union B)=I_A+I_B without any subtraction because indicators are disjoint.", "C": "I_(A union B)=I_A+I_B−I_(A intersection B)=max(I_A,I_B); the overlap term corrects double counting.", "D": "I_(A union B)=min(I_A,I_B) because a union requires both events." }, "correct": true, "kind": "recognition", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "17.2b" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 1.0, "D": 0.0, "B": 0.0, "A": 0.0 }, "batch_response_ms": 380.83770900266245, "repaired": false }, { "id": "17.2c", "section": 17, "context": "I_A is1 when event A occurs and0 otherwise. Events need not be independent.", "prompt": "Why are I_A,I_B uncorrelated exactly when A,B are independent?", "response_mode": "single_choice", "options": { "A": "Every pair of indicators has zero covariance.", "B": "All uncorrelated random variables are independent, regardless of their distributions.", "C": "Zero covariance means A and B cannot both occur.", "D": "Cov(I_A,I_B)=P(A intersection B)−P(A)P(B); zero covariance is exactly the independence criterion for two events." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "17.2c" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "D": 1.0, "B": 0.0, "A": 0.0 }, "batch_response_ms": 380.83770900266245, "repaired": false }, { "id": "17.3", "section": 17, "context": "I_A is1 when event A occurs and0 otherwise. Events need not be independent.", "prompt": "For real alpha_i of either sign and arbitrary events A_i, why is sum(i,j) alpha_i alpha_j P(A_i intersection A_j)>=0? Hint: consider the square of sum_i alpha_i I_(A_i).", "response_mode": "single_choice", "options": { "A": "Each summand is nonnegative because every product alpha_i alpha_j is nonnegative.", "B": "The sum equals E[(sum_i alpha_i I_(A_i))²], an expectation of a nonnegative quantity.", "C": "The sum equals (sum_i alpha_i P(A_i))² for all events.", "D": "Distinct indicators are independent, so every cross term vanishes." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "17.3" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "D": 0.0, "B": 1.0, "A": 0.0 }, "batch_response_ms": 380.83770900266245, "repaired": false }, { "id": "18.1", "section": 18, "context": "P(M=m)=alpha*m for integers1<=m<=n, zero otherwise. Useful sums: sum(m=1..n)m=n(n+1)/2; sum(m=1..n)m²=n(n+1)(2n+1)/6.", "prompt": "Normalization alpha?", "response_mode": "single_choice", "options": { "A": "1/[n(n+1)]", "B": "1/n", "C": "2/[n(n+1)]", "D": "6/[n(n+1)(2n+1)]" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "18.1" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "C": 1.0, "A": 0.0 }, "batch_response_ms": 332.5494159980735, "repaired": false }, { "id": "18.2", "section": 18, "context": "P(M=m)=alpha*m for integers1<=m<=n, zero otherwise. Useful sums: sum(m=1..n)m=n(n+1)/2; sum(m=1..n)m²=n(n+1)(2n+1)/6.", "prompt": "E[M], expressed in terms of alpha?", "response_mode": "single_choice", "options": { "A": "alpha²*n(n+1)(2n+1)/6", "B": "alpha*n(n+1)(2n+1)/6", "C": "alpha*n(n+1)/2", "D": "alpha*n²" }, "correct": true, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "18.2" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.64, "probabilities": { "B": 0.73, "D": 0.0, "C": 0.02, "A": 0.25 }, "batch_response_ms": 332.5494159980735, "repaired": false }, { "id": "18.3", "section": 18, "context": "P(M=m)=alpha*m for integers1<=m<=n, zero otherwise. Useful sums: sum(m=1..n)m=n(n+1)/2; sum(m=1..n)m²=n(n+1)(2n+1)/6.", "prompt": "CDF at integer m?", "response_mode": "single_choice", "options": { "A": "0 if m<1; alpha*m² if1<=m<=n;1 if m>n", "B": "0 if m<1; alpha*m(m+1)/2 if1<=m<=n;1 if m>n", "C": "0 if m<1; alpha*m(m−1)/2 if1<=m<=n;1 if m>n", "D": "0 if m<1; alpha*m if1<=m<=n;1 if m>n" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "18.3" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.99, "D": 0.0, "C": 0.01, "A": 0.0 }, "batch_response_ms": 332.5494159980735, "repaired": false }, { "id": "19.1", "section": 19, "context": "A bidirectional laser is anchored at (−1,0). Its angle Theta from the positive x-axis is uniform on (−pi/2,pi/2). Y is its vertical coordinate where the beam intersects x=0.", "prompt": "Y as a function of Theta?", "response_mode": "single_choice", "options": { "A": "tan(Theta)", "B": "cos(Theta)", "C": "sin(Theta)", "D": "1/tan(Theta)" }, "correct": true, "kind": "recognition", "topic": "transformations", "original_format": "written", "repairs": [], "source_parts": [ "19.1" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 0.0, "D": 0.0, "A": 1.0, "B": 0.0 }, "batch_response_ms": 335.57733300040127, "repaired": false }, { "id": "19.2", "section": 19, "context": "A bidirectional laser is anchored at (−1,0). Its angle Theta from the positive x-axis is uniform on (−pi/2,pi/2). Y is its vertical coordinate where the beam intersects x=0.", "prompt": "CDF F_Y(y) for real y?", "response_mode": "single_choice", "options": { "A": "1/2+arctan(y)/pi", "B": "1/2+tan(y)/pi", "C": "arctan(y)/pi", "D": "1/[pi(1+y²)]" }, "correct": true, "kind": "multi_step", "topic": "transformations", "original_format": "written", "repairs": [], "source_parts": [ "19.2" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 0.0, "D": 0.0, "A": 1.0, "B": 0.0 }, "batch_response_ms": 335.57733300040127, "repaired": false }, { "id": "19.3", "section": 19, "context": "A bidirectional laser is anchored at (−1,0). Its angle Theta from the positive x-axis is uniform on (−pi/2,pi/2). Y is its vertical coordinate where the beam intersects x=0.", "prompt": "Density f_Y(y) for real y?", "response_mode": "single_choice", "options": { "A": "1/[pi(1+y²)]", "B": "1/2+arctan(y)/pi", "C": "exp(−y²/2)/sqrt(2pi)", "D": "1/[pi(1+y)]" }, "correct": true, "kind": "calculation", "topic": "transformations", "original_format": "written", "repairs": [], "source_parts": [ "19.3" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "D": 0.0, "A": 1.0, "B": 0.0 }, "batch_response_ms": 335.57733300040127, "repaired": false }, { "id": "20.1", "section": 20, "context": "Four guests receive their four distinct umbrellas in a uniformly random permutation. C_i means guest i receives their own umbrella.", "prompt": "P(C_1)?", "response_mode": "single_choice", "options": { "A": "1/24", "B": "1/4", "C": "3/4", "D": "1/2" }, "correct": true, "kind": "recognition", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "20.1" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "A": 0.01, "C": 0.0, "B": 0.99, "D": 0.0 }, "batch_response_ms": 433.86820800151327, "repaired": false }, { "id": "20.2", "section": 20, "context": "Four guests receive their four distinct umbrellas in a uniformly random permutation. C_i means guest i receives their own umbrella.", "prompt": "Why is P(C_i)=P(C_j) for every i,j?", "response_mode": "single_choice", "options": { "A": "Symmetry", "B": "Transitivity", "C": "Independence", "D": "Disjointness" }, "correct": true, "kind": "recognition", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "20.2" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 1.0, "C": 0.0, "B": 0.0, "D": 0.0 }, "batch_response_ms": 433.86820800151327, "repaired": false }, { "id": "20.3", "section": 20, "context": "Four guests receive their four distinct umbrellas in a uniformly random permutation. C_i means guest i receives their own umbrella.", "prompt": "Number of distinct pairwise-intersection terms in inclusion-exclusion?", "response_mode": "single_choice", "options": { "A": "16", "B": "12", "C": "6", "D": "4" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "20.3" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.98, "probabilities": { "A": 0.0, "C": 0.99, "B": 0.01, "D": 0.0 }, "batch_response_ms": 433.86820800151327, "repaired": false }, { "id": "20.4", "section": 20, "context": "Four guests receive their four distinct umbrellas in a uniformly random permutation. C_i means guest i receives their own umbrella.", "prompt": "Probability at least one guest receives their own umbrella?", "response_mode": "single_choice", "options": { "A": "9/24", "B": "1−(3/4)^4", "C": "15/24", "D": "1/4" }, "correct": true, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "20.4" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.9, "probabilities": { "A": 0.06, "C": 0.93, "B": 0.01, "D": 0.0 }, "batch_response_ms": 433.86820800151327, "repaired": false }, { "id": "21.1", "section": 21, "context": "n>=4 distinct balls independently choose uniformly among n labeled bins. X counts indices i=1,...,n−1 for which bins i and i+1 are both empty.", "prompt": "E[X]?", "response_mode": "single_choice", "options": { "A": "(n−1)((n−2)/n)^n", "B": "n((n−2)/n)^n", "C": "((n−2)/n)^n", "D": "(n−1)((n−1)/n)^(2n)" }, "correct": true, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [ "Specify independent uniform placements and n>=4 for the combined variance formula." ], "source_parts": [ "21.1" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.97, "probabilities": { "B": 0.0, "D": 0.01, "C": 0.01, "A": 0.98 }, "batch_response_ms": 463.3912919998693, "repaired": true }, { "id": "21.2", "section": 21, "context": "n>=4 distinct balls independently choose uniformly among n labeled bins. X counts indices i=1,...,n−1 for which bins i and i+1 are both empty.", "prompt": "Define a=((n−2)/n)^n, b=((n−3)/n)^n, c=((n−4)/n)^n. Var(X)?", "response_mode": "single_choice", "options": { "A": "(n−1)a+2(n−2)c+(n−2)(n−3)b−(n−1)²a²", "B": "(n−1)a(1−a)", "C": "(n−1)a+2(n−2)b+(n−2)(n−3)c−(n−1)²a²", "D": "(n−1)a+(n−2)b+(n−2)(n−3)c−(n−1)²a²" }, "correct": true, "kind": "multi_step", "topic": "variance", "original_format": "written", "repairs": [], "source_parts": [ "21.2" ], "exam": "final_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.75, "probabilities": { "B": 0.01, "D": 0.05, "C": 0.81, "A": 0.13 }, "batch_response_ms": 463.3912919998693, "repaired": false }, { "id": "1.a", "section": 1, "context": "", "prompt": "Suppose P(x) is false for every x>=0. Then for every x>=0, P(x) implies there exists y>=x with Q(y).", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "1.a" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.97, "probabilities": { "A": 0.01, "B": 0.99 }, "batch_response_ms": 319.60733300002175, "repaired": false }, { "id": "1.b", "section": 1, "context": "", "prompt": "Number of propositional forms on n variables up to logical equivalence, with AND, OR, NOT and no quantifiers?", "response_mode": "single_choice", "options": { "A": "n", "B": "2n", "C": "n²", "D": "2^n", "E": "2^(2^n)" }, "correct": true, "kind": "calculation", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "1.b" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "E" ], "expected": [ "E" ], "valid_response": true, "confidence": 1.0, "probabilities": { "E": 1.0, "D": 0.0, "A": 0.0, "C": 0.0, "B": 0.0 }, "batch_response_ms": 319.60733300002175, "repaired": false }, { "id": "1.c", "section": 1, "context": "", "prompt": "Odd prime p and integer n satisfy n=−n mod p. What residue must n have?", "response_mode": "single_choice", "options": { "A": "1", "B": "0", "C": "2", "D": "p/2", "E": "p−1" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "1.c" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.97, "probabilities": { "E": 0.0, "D": 0.02, "A": 0.0, "C": 0.0, "B": 0.98 }, "batch_response_ms": 319.60733300002175, "repaired": false }, { "id": "1.d", "section": 1, "context": "", "prompt": "A total program can decide for arbitrary P,input x and line number n whether that line is ever executed in P(x).", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "computability", "original_format": "choice", "repairs": [], "source_parts": [ "1.d" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.98, "probabilities": { "A": 0.01, "B": 0.99 }, "batch_response_ms": 319.60733300002175, "repaired": false }, { "id": "1.e", "section": 1, "context": "", "prompt": "For fixed nonnegative k, number of integer-coefficient polynomials of degree at most k?", "response_mode": "single_choice", "options": { "A": "Countably infinite", "B": "Uncountably infinite", "C": "Finite" }, "correct": true, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "1.e" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "A": 1.0, "C": 0.0, "B": 0.0 }, "batch_response_ms": 319.60733300002175, "repaired": false }, { "id": "1.f", "section": 1, "context": "", "prompt": "Ten message symbols in 0..15, correct up to six corrupted transmitted symbols with Reed–Solomon. Smallest prime field size?", "response_mode": "single_choice", "options": { "A": "11", "B": "17", "C": "23", "D": "29" }, "correct": false, "kind": "multi_step", "topic": "coding", "original_format": "choice", "repairs": [], "source_parts": [ "1.f" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.64, "probabilities": { "D": 0.08, "A": 0.07, "C": 0.12, "B": 0.73 }, "batch_response_ms": 319.60733300002175, "repaired": false }, { "id": "1.g", "section": 1, "context": "", "prompt": "A graph on n vertices is a disjoint union of k trees. Edge count?", "response_mode": "single_choice", "options": { "A": "k(n−1)", "B": "n−k+1", "C": "n−1", "D": "n−k", "E": "Not determined" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "1.g" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 1.0, "E": 0.0, "A": 0.0, "C": 0.0, "B": 0.0 }, "batch_response_ms": 319.60733300002175, "repaired": false }, { "id": "1.h", "section": 1, "context": "", "prompt": "Positive n,m,k with k<=n,m. Evaluate sum(i=0..k) C(n,i)C(m,k−i).", "response_mode": "single_choice", "options": { "A": "C(n+m,k)", "B": "C(nm,k)", "C": "n!m!/(k!)²", "D": "2^(n+m)", "E": "None of these" }, "correct": true, "kind": "recognition", "topic": "counting", "original_format": "choice", "repairs": [], "source_parts": [ "1.h" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "E": 0.0, "D": 0.0, "A": 1.0, "C": 0.0, "B": 0.0 }, "batch_response_ms": 319.60733300002175, "repaired": false }, { "id": "1.i", "section": 1, "context": "", "prompt": "Shortest paths in the 7-cube from 0000000 to 1011001?", "response_mode": "single_choice", "options": { "A": "128", "B": "1", "C": "16", "D": "24", "E": "4" }, "correct": false, "kind": "multi_step", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "1.i" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "E" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.98, "probabilities": { "D": 0.0, "E": 0.98, "A": 0.01, "C": 0.01, "B": 0.0 }, "batch_response_ms": 319.60733300002175, "repaired": false }, { "id": "1.j1", "section": 1, "context": "", "prompt": "There exists a simple undirected graph with 7 vertices and 7 edges containing an Eulerian tour.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "1.j1" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.35, "probabilities": { "A": 0.67, "B": 0.33 }, "batch_response_ms": 319.60733300002175, "repaired": false }, { "id": "1.j2", "section": 1, "context": "", "prompt": "There exists a simple undirected graph with 4 vertices and 5 edges containing an Eulerian tour.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "1.j2" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.52, "probabilities": { "A": 0.76, "B": 0.24 }, "batch_response_ms": 319.60733300002175, "repaired": false }, { "id": "1.j3", "section": 1, "context": "", "prompt": "There exists a simple undirected graph with 5 vertices and 7 edges containing an Eulerian tour.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": false, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "1.j3" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.46, "probabilities": { "A": 0.27, "B": 0.73 }, "batch_response_ms": 319.60733300002175, "repaired": false }, { "id": "2.a", "section": 2, "context": "", "prompt": "3^42 modulo 55?", "response_mode": "single_choice", "options": { "A": "27", "B": "1", "C": "9", "D": "3" }, "correct": false, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "2.a" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.53, "probabilities": { "D": 0.09, "A": 0.65, "C": 0.14, "B": 0.12 }, "batch_response_ms": 396.38312500028405, "repaired": false }, { "id": "2.b", "section": 2, "context": "", "prompt": "Solve 17x=7 mod 42, x in 0..41.", "response_mode": "single_choice", "options": { "A": "7", "B": "35", "C": "17", "D": "5" }, "correct": true, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "2.b" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.23, "probabilities": { "D": 0.36, "A": 0.15, "C": 0.07, "B": 0.42 }, "batch_response_ms": 396.38312500028405, "repaired": false }, { "id": "2.c", "section": 2, "context": "", "prompt": "GF(5), threshold three shares. P(1)=1,P(2)=1,P(4)=2; degree at most two. Secret P(0)?", "response_mode": "single_choice", "options": { "A": "1", "B": "2", "C": "4", "D": "3" }, "correct": false, "kind": "multi_step", "topic": "secret_sharing", "original_format": "written", "repairs": [], "source_parts": [ "2.c" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.06, "probabilities": { "D": 0.28, "B": 0.3, "C": 0.28, "A": 0.14 }, "batch_response_ms": 396.38312500028405, "repaired": false }, { "id": "2.d", "section": 2, "context": "", "prompt": "Is f(x)=x² mod 7 a bijection on 0..6? Give a witness or inverse.", "response_mode": "single_choice", "options": { "A": "Yes: its inverse is x^4 mod 7.", "B": "Yes: its inverse is itself.", "C": "No: f(1)=f(6)=1.", "D": "No: f(0)=f(1)=0." }, "correct": true, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "2.d" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "D": 0.0, "A": 0.0, "C": 1.0, "B": 0.0 }, "batch_response_ms": 396.38312500028405, "repaired": false }, { "id": "3.a", "section": 3, "context": "", "prompt": "For x≠1, prove (x^n−1)/(x−1)=sum(i=0..n−1)x^i by induction on n>=1.", "response_mode": "single_choice", "options": { "A": "Assume x=1, divide both sides by x−1, and then extend to all x.", "B": "Base n=1 is 1=1. If the identity holds at n=k, then x^(k+1)−1=x(x^k−1)+(x−1)=(x−1)[x sum(i=0..k−1)x^i+1]=(x−1)sum(i=0..k)x^i; divide by nonzero x−1.", "C": "Base n=1 holds. Replace n by n+1 in the desired identity, which proves the induction step without using the hypothesis.", "D": "The geometric series formula proves itself, so no base or induction step is needed." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "3.a" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 1.0, "C": 0.0 }, "batch_response_ms": 329.00541700291797, "repaired": false }, { "id": "3.b", "section": 3, "context": "", "prompt": "Prove 2^m−1 prime implies m prime for natural m. Hint: factor a geometric sum.", "response_mode": "single_choice", "options": { "A": "m=0,1 do not yield primes. If m=ab with a,b>1, then 2^m−1=(2^a−1)[1+2^a+...+(2^a)^(b−1)], a product of two integers greater than one.", "B": "If m=ab, then 2^m−1=(2^a−1)(2^b−1).", "C": "A prime exponent always makes a Mersenne prime, and reversing that statement proves the result.", "D": "Every odd number is prime, so m must be odd and hence prime." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "3.b" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 1.0, "B": 0.0, "C": 0.0 }, "batch_response_ms": 329.00541700291797, "repaired": false }, { "id": "4.a", "section": 4, "context": "Jobs 1:A>B>C>D,2:B>C>D>A,3:C>D>A>B,4:B>A>C>D. Candidates A:2>3>1>4,B:3>1>2>4,C:1>2>4>3,D:2>4>3>1.", "prompt": "Classify (1,B),(2,C),(3,A),(4,D).", "response_mode": "single_choice", "options": { "A": "Stable and job-optimal", "B": "Unstable; blocking pair (1,A)", "C": "Unstable; blocking pair (4,C)", "D": "Stable and candidate-optimal" }, "correct": false, "kind": "multi_step", "topic": "matching", "original_format": "written", "repairs": [], "source_parts": [ "4.a" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.16, "probabilities": { "B": 0.22, "C": 0.15, "A": 0.37, "D": 0.26 }, "batch_response_ms": 423.2227499996952, "repaired": false }, { "id": "4.b", "section": 4, "context": "Jobs 1:A>B>C>D,2:B>C>D>A,3:C>D>A>B,4:B>A>C>D. Candidates A:2>3>1>4,B:3>1>2>4,C:1>2>4>3,D:2>4>3>1.", "prompt": "Classify (1,A),(2,B),(3,C),(4,D).", "response_mode": "single_choice", "options": { "A": "Unstable; blocking pair (4,C)", "B": "Stable and job-optimal", "C": "Stable and candidate-optimal", "D": "Unstable; blocking pair (1,B)" }, "correct": false, "kind": "multi_step", "topic": "matching", "original_format": "written", "repairs": [], "source_parts": [ "4.b" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.38, "probabilities": { "B": 0.54, "C": 0.32, "A": 0.07, "D": 0.07 }, "batch_response_ms": 423.2227499996952, "repaired": false }, { "id": "5.a", "section": 5, "context": "Uniformly choose one fair four-sided die: faces {1,2,3,4} or faces {1,2,2,4}. Roll the chosen die twice independently conditional on identity.", "prompt": "Given first roll 3, probability second roll 2?", "response_mode": "single_choice", "options": { "A": "5/12", "B": "3/8", "C": "1/4", "D": "1/2" }, "correct": false, "kind": "multi_step", "topic": "bayes", "original_format": "written", "repairs": [], "source_parts": [ "5.a" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.13, "probabilities": { "D": 0.3, "C": 0.21, "A": 0.33999999999999997, "B": 0.15 }, "batch_response_ms": 413.6264589978964, "repaired": false }, { "id": "5.b", "section": 5, "context": "Uniformly choose one fair four-sided die: faces {1,2,3,4} or faces {1,2,2,4}. Roll the chosen die twice independently conditional on identity.", "prompt": "Given first roll 2, probability second roll 2?", "response_mode": "single_choice", "options": { "A": "1/4", "B": "3/8", "C": "5/12", "D": "1/2" }, "correct": true, "kind": "multi_step", "topic": "bayes", "original_format": "written", "repairs": [], "source_parts": [ "5.b" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.21, "probabilities": { "D": 0.28, "A": 0.03, "C": 0.41, "B": 0.28 }, "batch_response_ms": 413.6264589978964, "repaired": false }, { "id": "6.a", "section": 6, "context": "P,Q have degree at most d with every coefficient independently uniform in GF(p), prime p>d.", "prompt": "Probability P(x)=Q(x) for every field element?", "response_mode": "single_choice", "options": { "A": "1/p^d", "B": "1/p^(d+1)", "C": "1/(d+1)", "D": "1/p" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "6.a" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.95, "probabilities": { "C": 0.0, "A": 0.03, "D": 0.01, "B": 0.96 }, "batch_response_ms": 368.765582999913, "repaired": false }, { "id": "6.b", "section": 6, "context": "P,Q have degree at most d with every coefficient independently uniform in GF(p), prime p>d.", "prompt": "Probability P(0)=Q(0)?", "response_mode": "single_choice", "options": { "A": "1/p^(d+1)", "B": "1/p²", "C": "1/p", "D": "1/2" }, "correct": true, "kind": "recognition", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "6.b" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.98, "probabilities": { "C": 0.98, "A": 0.01, "D": 0.0, "B": 0.01 }, "batch_response_ms": 368.765582999913, "repaired": false }, { "id": "6.c", "section": 6, "context": "P,Q have degree at most d with every coefficient independently uniform in GF(p), prime p>d.", "prompt": "For 0<=k<=d−1, probability P(i)=Q(i) for every i=0,...,k?", "response_mode": "single_choice", "options": { "A": "(k+1)/p", "B": "1/p^k", "C": "1/p^(k+1)", "D": "1/p^(d−k)" }, "correct": true, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "6.c" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.86, "probabilities": { "C": 0.89, "A": 0.06, "D": 0.02, "B": 0.03 }, "batch_response_ms": 368.765582999913, "repaired": false }, { "id": "7.1", "section": 7, "context": "", "prompt": "N independent node lifetimes are Exp(rate lambda). Divide into ell groups of k=N/ell nodes. A group fails when any node fails; system fails when all groups fail. P(system fails by t>=0)?", "response_mode": "single_choice", "options": { "A": "(1−exp(−lambda*t))^ell", "B": "(1−exp(−k*lambda*t))^ell", "C": "1−exp(−N*lambda*t)", "D": "1−(1−exp(−lambda*t))^N" }, "correct": true, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "7.1" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.67, "probabilities": { "B": 0.76, "D": 0.01, "C": 0.01, "A": 0.22 }, "batch_response_ms": 364.40666599810356, "repaired": false }, { "id": "8.a", "section": 8, "context": "Density f(x)=alpha*x on [0,1], zero elsewhere; mu=E[X].", "prompt": "Normalizing alpha?", "response_mode": "single_choice", "options": { "A": "1", "B": "1/2", "C": "2", "D": "3" }, "correct": true, "kind": "calculation", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "8.a" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 0.0, "B": 0.0, "C": 1.0, "D": 0.0 }, "batch_response_ms": 304.48183399857953, "repaired": false }, { "id": "8.b", "section": 8, "context": "Density f(x)=alpha*x on [0,1], zero elsewhere; mu=E[X].", "prompt": "P(X<=1/2), expressed in alpha?", "response_mode": "single_choice", "options": { "A": "alpha/8", "B": "alpha/4", "C": "alpha/2", "D": "alpha/16" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "8.b" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.29, "probabilities": { "A": 0.47, "B": 0.46, "C": 0.01, "D": 0.06 }, "batch_response_ms": 304.48183399857953, "repaired": false }, { "id": "8.c", "section": 8, "context": "Density f(x)=alpha*x on [0,1], zero elsewhere; mu=E[X].", "prompt": "E[X], expressed in alpha?", "response_mode": "single_choice", "options": { "A": "alpha/4", "B": "alpha/3", "C": "alpha/2", "D": "alpha²/3" }, "correct": true, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "8.c" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.64, "probabilities": { "A": 0.08, "B": 0.73, "C": 0.13, "D": 0.06 }, "batch_response_ms": 304.48183399857953, "repaired": false }, { "id": "8.d", "section": 8, "context": "Density f(x)=alpha*x on [0,1], zero elsewhere; mu=E[X].", "prompt": "Var(X), expressed in alpha and mu?", "response_mode": "single_choice", "options": { "A": "alpha/3−mu²", "B": "alpha/4−mu", "C": "alpha/4+mu²", "D": "alpha/4−mu²" }, "correct": true, "kind": "calculation", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "8.d" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.7, "probabilities": { "A": 0.21, "B": 0.01, "C": 0.0, "D": 0.78 }, "batch_response_ms": 304.48183399857953, "repaired": false }, { "id": "9.1", "section": 9, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "prompt": "Under the given assumptions, X and Z are independent.", "response_mode": "single_choice", "options": { "A": "Could be either", "B": "True", "C": "False" }, "correct": true, "kind": "recognition", "topic": "moments", "original_format": "choice", "repairs": [], "source_parts": [ "9.1" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.51, "probabilities": { "C": 0.32, "B": 0.01, "A": 0.67 }, "batch_response_ms": 398.80608299790765, "repaired": false }, { "id": "9.2", "section": 9, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "prompt": "Under the given assumptions, E[X+2Y−Z]=0.", "response_mode": "single_choice", "options": { "A": "Could be either", "B": "True", "C": "False" }, "correct": true, "kind": "recognition", "topic": "moments", "original_format": "choice", "repairs": [], "source_parts": [ "9.2" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.27, "probabilities": { "C": 0.46, "B": 0.51, "A": 0.03 }, "batch_response_ms": 398.80608299790765, "repaired": false }, { "id": "9.3", "section": 9, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "prompt": "Under the given assumptions, P(Y>=10)<=1/10.", "response_mode": "single_choice", "options": { "A": "True", "B": "False", "C": "Could be either" }, "correct": false, "kind": "recognition", "topic": "moments", "original_format": "choice", "repairs": [], "source_parts": [ "9.3" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.43, "probabilities": { "C": 0.22, "B": 0.16, "A": 0.62 }, "batch_response_ms": 398.80608299790765, "repaired": false }, { "id": "9.4", "section": 9, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "prompt": "Under the given assumptions, E[XY]=2.", "response_mode": "single_choice", "options": { "A": "Could be either", "B": "False", "C": "True" }, "correct": true, "kind": "recognition", "topic": "moments", "original_format": "choice", "repairs": [], "source_parts": [ "9.4" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.93, "probabilities": { "C": 0.03, "B": 0.95, "A": 0.02 }, "batch_response_ms": 398.80608299790765, "repaired": false }, { "id": "9.5", "section": 9, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "prompt": "Under the given assumptions, Var(X−Y)=20.", "response_mode": "single_choice", "options": { "A": "Could be either", "B": "True", "C": "False" }, "correct": true, "kind": "recognition", "topic": "moments", "original_format": "choice", "repairs": [], "source_parts": [ "9.5" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.87, "probabilities": { "C": 0.04, "B": 0.91, "A": 0.05 }, "batch_response_ms": 398.80608299790765, "repaired": false }, { "id": "9.6", "section": 9, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "prompt": "Under the given assumptions, Var(2X)=20.", "response_mode": "single_choice", "options": { "A": "True", "B": "Could be either", "C": "False" }, "correct": false, "kind": "recognition", "topic": "moments", "original_format": "choice", "repairs": [], "source_parts": [ "9.6" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.97, "probabilities": { "C": 0.01, "B": 0.01, "A": 0.98 }, "batch_response_ms": 398.80608299790765, "repaired": false }, { "id": "9.7", "section": 9, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "prompt": "Under the given assumptions, P(|X|>=5)<=2/5.", "response_mode": "single_choice", "options": { "A": "True", "B": "False", "C": "Could be either" }, "correct": true, "kind": "recognition", "topic": "moments", "original_format": "choice", "repairs": [], "source_parts": [ "9.7" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.69, "probabilities": { "C": 0.1, "B": 0.1, "A": 0.8 }, "batch_response_ms": 398.80608299790765, "repaired": false }, { "id": "9.8", "section": 9, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "prompt": "Under the given assumptions, P(|X|>=sqrt(10))>0.", "response_mode": "single_choice", "options": { "A": "True", "B": "Could be either", "C": "False" }, "correct": false, "kind": "recognition", "topic": "moments", "original_format": "choice", "repairs": [], "source_parts": [ "9.8" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.28, "probabilities": { "C": 0.05, "B": 0.52, "A": 0.43 }, "batch_response_ms": 398.80608299790765, "repaired": false }, { "id": "10.1", "section": 10, "context": "", "prompt": "Xi are iid uniform on {1,2,3}; Sn=sum Xi. Choose a,b so (Sn−a)/b converges to N(0,1).", "response_mode": "single_choice", "options": { "A": "a=n,b=sqrt(n)", "B": "a=2n,b=2n/3", "C": "a=2,b=sqrt(2/3)", "D": "a=2n,b=sqrt(2n/3)" }, "correct": true, "kind": "multi_step", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "10.a", "10.b" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "A": 0.0, "D": 1.0, "B": 0.0 }, "batch_response_ms": 334.2922499978158, "repaired": false }, { "id": "11.a", "section": 11, "context": "Piano has 52 white and 36 black keys. Chords are unordered sets of 2..10 distinct keys. Melodies are ordered sequences, repetition allowed unless constrained.", "prompt": "Five-key chords with exactly two white and three black?", "response_mode": "single_choice", "options": { "A": "C(88,5)", "B": "C(52,3)C(36,2)", "C": "52²*36³", "D": "C(52,2)C(36,3)" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "11.a" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "D": 1.0, "A": 0.0 }, "batch_response_ms": 356.3250830011384, "repaired": false }, { "id": "11.b", "section": 11, "context": "Piano has 52 white and 36 black keys. Chords are unordered sets of 2..10 distinct keys. Melodies are ordered sequences, repetition allowed unless constrained.", "prompt": "Chords from a fixed set of ten keys?", "response_mode": "single_choice", "options": { "A": "C(10,2)", "B": "2^10−11", "C": "2^10", "D": "2^10−1" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "11.b" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.95, "probabilities": { "B": 0.96, "C": 0.0, "A": 0.01, "D": 0.03 }, "batch_response_ms": 356.3250830011384, "repaired": false }, { "id": "11.c", "section": 11, "context": "Piano has 52 white and 36 black keys. Chords are unordered sets of 2..10 distinct keys. Melodies are ordered sequences, repetition allowed unless constrained.", "prompt": "Melodies of exactly twelve keys?", "response_mode": "single_choice", "options": { "A": "12^88", "B": "88!/76!", "C": "C(88,12)", "D": "88^12" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "11.c" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.97, "probabilities": { "B": 0.0, "C": 0.0, "D": 0.98, "A": 0.02 }, "batch_response_ms": 356.3250830011384, "repaired": false }, { "id": "11.d", "section": 11, "context": "Piano has 52 white and 36 black keys. Chords are unordered sets of 2..10 distinct keys. Melodies are ordered sequences, repetition allowed unless constrained.", "prompt": "Melodies with 40 white and 10 black notes, no adjacent black notes?", "response_mode": "single_choice", "options": { "A": "52^40*36^10*C(41,10)", "B": "C(52,40)C(36,10)", "C": "52^40*36^10*C(39,10)", "D": "52^40*36^10*C(50,10)" }, "correct": true, "kind": "multi_step", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "11.d" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.94, "probabilities": { "B": 0.0, "C": 0.03, "D": 0.02, "A": 0.95 }, "batch_response_ms": 356.3250830011384, "repaired": false }, { "id": "11.e", "section": 11, "context": "Piano has 52 white and 36 black keys. Chords are unordered sets of 2..10 distinct keys. Melodies are ordered sequences, repetition allowed unless constrained.", "prompt": "Choose 200 keys as a multiset, unlimited repetition?", "response_mode": "single_choice", "options": { "A": "C(200,88)", "B": "C(287,87)", "C": "C(288,88)", "D": "88^200" }, "correct": false, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "11.e" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.19, "probabilities": { "B": 0.32, "C": 0.39, "D": 0.25, "A": 0.04 }, "batch_response_ms": 356.3250830011384, "repaired": false }, { "id": "11.f", "section": 11, "context": "Piano has 52 white and 36 black keys. Chords are unordered sets of 2..10 distinct keys. Melodies are ordered sequences, repetition allowed unless constrained.", "prompt": "Split a fixed 200-note melody into four consecutive parts, at least twenty notes each. Count?", "response_mode": "single_choice", "options": { "A": "C(199,3)", "B": "C(120,3)", "C": "C(123,3)", "D": "C(124,4)" }, "correct": false, "kind": "multi_step", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "11.f" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.52, "probabilities": { "B": 0.14, "C": 0.2, "A": 0.64, "D": 0.02 }, "batch_response_ms": 356.3250830011384, "repaired": false }, { "id": "12.a", "section": 12, "context": "Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.", "prompt": "Probability a fixed vertex is isolated?", "response_mode": "single_choice", "options": { "A": "1/8", "B": "1/4", "C": "1/n", "D": "1/2" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "12.a" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.96, "probabilities": { "B": 0.97, "C": 0.0, "A": 0.02, "D": 0.01 }, "batch_response_ms": 408.9225840034487, "repaired": false }, { "id": "12.b", "section": 12, "context": "Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.", "prompt": "E[X]?", "response_mode": "single_choice", "options": { "A": "n/2", "B": "1/4", "C": "n/8", "D": "n/4" }, "correct": true, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "12.b" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.7, "probabilities": { "B": 0.0, "C": 0.03, "A": 0.19, "D": 0.78 }, "batch_response_ms": 408.9225840034487, "repaired": false }, { "id": "12.c", "section": 12, "context": "Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.", "prompt": "Probability adjacent vertices are both isolated?", "response_mode": "single_choice", "options": { "A": "1/4", "B": "1/2", "C": "1/16", "D": "1/8" }, "correct": false, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "12.c" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.34, "probabilities": { "B": 0.01, "C": 0.51, "A": 0.18, "D": 0.3 }, "batch_response_ms": 408.9225840034487, "repaired": false }, { "id": "12.d", "section": 12, "context": "Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.", "prompt": "For two nonadjacent vertices (when present), probability both isolated?", "response_mode": "single_choice", "options": { "A": "1/4", "B": "1/16", "C": "1/32", "D": "1/8" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "12.d" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.64, "probabilities": { "B": 0.74, "C": 0.09, "A": 0.07, "D": 0.1 }, "batch_response_ms": 408.9225840034487, "repaired": false }, { "id": "12.e", "section": 12, "context": "Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.", "prompt": "Var(X)?", "response_mode": "single_choice", "options": { "A": "5n/16", "B": "3n/16", "C": "n/4", "D": "n²/16" }, "correct": false, "kind": "multi_step", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "12.e" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.46, "probabilities": { "B": 0.6, "C": 0.11, "A": 0.28, "D": 0.01 }, "batch_response_ms": 408.9225840034487, "repaired": false }, { "id": "12.f", "section": 12, "context": "Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.", "prompt": "Chebyshev upper bound on P(X>=n/2), clipped to probability range?", "response_mode": "single_choice", "options": { "A": "min(1,5/(4n))", "B": "min(1,5/n)", "C": "min(1,20/n)", "D": "min(1,3/n)" }, "correct": false, "kind": "multi_step", "topic": "bounds", "original_format": "written", "repairs": [], "source_parts": [ "12.f" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.22, "probabilities": { "B": 0.3, "C": 0.42, "A": 0.17, "D": 0.11 }, "batch_response_ms": 408.9225840034487, "repaired": false }, { "id": "13.a", "section": 13, "context": "Dolphins arrive as a homogeneous Poisson process with mean spacing 20 minutes. Each dolphin performs a trick independently with probability p each minute, then leaves at that minute’s end independently with probability q>0. All behaviors independent across dolphins and time.", "prompt": "Distribution of arrivals in one hour?", "response_mode": "single_choice", "options": { "A": "Poisson(3)", "B": "Poisson(20)", "C": "Bin(60,1/20)", "D": "Geom(1/20)" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [ "Specify homogeneous Poisson arrivals; average spacing alone is insufficient." ], "source_parts": [ "13.a" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "A": 1.0, "D": 0.0, "B": 0.0 }, "batch_response_ms": 310.4866249996121, "repaired": true }, { "id": "13.b", "section": 13, "context": "Dolphins arrive as a homogeneous Poisson process with mean spacing 20 minutes. Each dolphin performs a trick independently with probability p each minute, then leaves at that minute’s end independently with probability q>0. All behaviors independent across dolphins and time.", "prompt": "Distribution of minutes a dolphin stays, counting its first minute?", "response_mode": "single_choice", "options": { "A": "Geom(p) on 1,2,...", "B": "Geom(q) on 1,2,...", "C": "Exp(rate q)", "D": "Poisson(1/q)" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "13.b" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.89, "probabilities": { "C": 0.0, "A": 0.08, "D": 0.0, "B": 0.92 }, "batch_response_ms": 310.4866249996121, "repaired": false }, { "id": "13.c", "section": 13, "context": "Dolphins arrive as a homogeneous Poisson process with mean spacing 20 minutes. Each dolphin performs a trick independently with probability p each minute, then leaves at that minute’s end independently with probability q>0. All behaviors independent across dolphins and time.", "prompt": "Probability a dolphin stays exactly five minutes?", "response_mode": "single_choice", "options": { "A": "(1−q)^4*q", "B": "(1−q)^4", "C": "(1−q)^5*q", "D": "q^4*(1−q)" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "13.c" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.93, "probabilities": { "D": 0.0, "A": 0.95, "C": 0.04, "B": 0.01 }, "batch_response_ms": 310.4866249996121, "repaired": false }, { "id": "13.d", "section": 13, "context": "Dolphins arrive as a homogeneous Poisson process with mean spacing 20 minutes. Each dolphin performs a trick independently with probability p each minute, then leaves at that minute’s end independently with probability q>0. All behaviors independent across dolphins and time.", "prompt": "Expected minutes a dolphin stays?", "response_mode": "single_choice", "options": { "A": "q", "B": "1/p", "C": "1/q", "D": "(1−q)/q" }, "correct": true, "kind": "recognition", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "13.d" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.89, "probabilities": { "C": 0.91, "A": 0.01, "D": 0.06, "B": 0.02 }, "batch_response_ms": 310.4866249996121, "repaired": false }, { "id": "13.e", "section": 13, "context": "Dolphins arrive as a homogeneous Poisson process with mean spacing 20 minutes. Each dolphin performs a trick independently with probability p each minute, then leaves at that minute’s end independently with probability q>0. All behaviors independent across dolphins and time.", "prompt": "Given a twenty-minute stay, probability of exactly k tricks?", "response_mode": "single_choice", "options": { "A": "C(20,k)q^k(1−q)^(20−k)", "B": "p^k(1−p)^(20−k)", "C": "C(20,k)p^k*q", "D": "C(20,k)p^k(1−p)^(20−k)" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "13.e" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.78, "probabilities": { "D": 0.84, "A": 0.1, "C": 0.02, "B": 0.04 }, "batch_response_ms": 310.4866249996121, "repaired": false }, { "id": "13.f", "section": 13, "context": "Dolphins arrive as a homogeneous Poisson process with mean spacing 20 minutes. Each dolphin performs a trick independently with probability p each minute, then leaves at that minute’s end independently with probability q>0. All behaviors independent across dolphins and time.", "prompt": "Given a twenty-minute stay, expected tricks?", "response_mode": "single_choice", "options": { "A": "20q", "B": "20p", "C": "20/p", "D": "p/q" }, "correct": true, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "13.f" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.64, "probabilities": { "C": 0.01, "A": 0.05, "D": 0.2, "B": 0.74 }, "batch_response_ms": 310.4866249996121, "repaired": false }, { "id": "13.g", "section": 13, "context": "Dolphins arrive as a homogeneous Poisson process with mean spacing 20 minutes. Each dolphin performs a trick independently with probability p each minute, then leaves at that minute’s end independently with probability q>0. All behaviors independent across dolphins and time.", "prompt": "Unconditional expected total tricks per dolphin?", "response_mode": "single_choice", "options": { "A": "1/(pq)", "B": "p/(1−q)", "C": "q/p", "D": "p/q" }, "correct": true, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "13.g" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.5, "probabilities": { "C": 0.0, "A": 0.02, "D": 0.62, "B": 0.36 }, "batch_response_ms": 310.4866249996121, "repaired": false }, { "id": "14.a", "section": 14, "context": "Markov chain: 1->2 with probability1; 2->3 with1; 3->1 and 3->4 each1/2; 4->5 with1; 5->2 with1.", "prompt": "Is it irreducible? Justify.", "response_mode": "single_choice", "options": { "A": "No; no self-loops exist.", "B": "No; transitions from states1,2,4,5 are deterministic.", "C": "Yes; every state can reach state2, and state2 can reach every state.", "D": "Yes; every state has the same number of outgoing edges." }, "correct": true, "kind": "multi_step", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "14.a" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.95, "probabilities": { "B": 0.04, "C": 0.96, "A": 0.0, "D": 0.0 }, "batch_response_ms": 488.92054200041457, "repaired": false }, { "id": "14.b", "section": 14, "context": "Markov chain: 1->2 with probability1; 2->3 with1; 3->1 and 3->4 each1/2; 4->5 with1; 5->2 with1.", "prompt": "Is it aperiodic? Justify.", "response_mode": "single_choice", "options": { "A": "Yes; all states have period0.", "B": "No; no self-loop exists, so period is2.", "C": "No; a five-state chain always has period5.", "D": "Yes; state2 has return paths of lengths3 and4, whose gcd is1." }, "correct": true, "kind": "multi_step", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "14.b" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "C": 0.0, "A": 0.0, "D": 1.0 }, "batch_response_ms": 488.92054200041457, "repaired": false }, { "id": "14.c", "section": 14, "context": "Markov chain: 1->2 with probability1; 2->3 with1; 3->1 and 3->4 each1/2; 4->5 with1; 5->2 with1.", "prompt": "Stationary pi1=pi5=1/7. Find pi2.", "response_mode": "single_choice", "options": { "A": "3/7", "B": "1/7", "C": "2/7", "D": "1/5" }, "correct": true, "kind": "calculation", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "14.c" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.58, "probabilities": { "B": 0.09, "C": 0.6900000000000001, "D": 0.0, "A": 0.22 }, "batch_response_ms": 488.92054200041457, "repaired": false }, { "id": "14.d", "section": 14, "context": "Markov chain: 1->2 with probability1; 2->3 with1; 3->1 and 3->4 each1/2; 4->5 with1; 5->2 with1.", "prompt": "bi is expected hitting time of state1 from i. Given b3=5, find b5.", "response_mode": "single_choice", "options": { "A": "5", "B": "10", "C": "7", "D": "6" }, "correct": false, "kind": "calculation", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "14.d" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.21, "probabilities": { "B": 0.1, "C": 0.38, "A": 0.11, "D": 0.41 }, "batch_response_ms": 488.92054200041457, "repaired": false }, { "id": "14.e", "section": 14, "context": "Markov chain: 1->2 with probability1; 2->3 with1; 3->1 and 3->4 each1/2; 4->5 with1; 5->2 with1.", "prompt": "Replace 4->5 by a probability-one self-loop at4. Expected hitting time of state1 from5?", "response_mode": "single_choice", "options": { "A": "14", "B": "Infinity", "C": "5", "D": "7" }, "correct": true, "kind": "multi_step", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "14.e" ], "exam": "final_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.9, "probabilities": { "B": 0.93, "C": 0.02, "A": 0.02, "D": 0.03 }, "batch_response_ms": 488.92054200041457, "repaired": false }, { "id": "2.1", "section": 2, "context": "The last question of this same exam states: it is Jay’s birthday today (May 16).", "prompt": "Which TA has a birthday on May 16?", "response_mode": "single_choice", "options": { "A": "Andy", "B": "Malia", "C": "Evelyn", "D": "Jay" }, "correct": true, "kind": "recognition", "topic": "course_context", "original_format": "written", "repairs": [], "source_parts": [ "2.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.99, "probabilities": { "D": 1.0, "A": 0.0, "B": 0.0, "C": 0.0 }, "batch_response_ms": 300.96354100169265, "repaired": false }, { "id": "3.1a", "section": 3, "context": "", "prompt": "Complete: (P implies NOT Q) is equivalent to (NOT P OR __).", "response_mode": "single_choice", "options": { "A": "Q", "B": "P", "C": "NOT Q", "D": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "written", "repairs": [], "source_parts": [ "3.1a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "D": 0.0, "A": 0.0, "C": 1.0, "B": 0.0 }, "batch_response_ms": 429.8258330018143, "repaired": false }, { "id": "3.1b", "section": 3, "context": "", "prompt": "P AND (Q OR R) is equivalent to (P OR Q) AND (P OR R).", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.1b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.52, "probabilities": { "A": 0.76, "B": 0.24 }, "batch_response_ms": 429.8258330018143, "repaired": false }, { "id": "3.1c", "section": 3, "context": "", "prompt": "Truth status of P AND (NOT P OR Q) AND (NOT P OR NOT Q)?", "response_mode": "single_choice", "options": { "A": "Could be either", "B": "False", "C": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.1c" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.89, "probabilities": { "A": 0.07, "C": 0.01, "B": 0.92 }, "batch_response_ms": 429.8258330018143, "repaired": false }, { "id": "3.2ai", "section": 3, "context": "N includes zero. For every n, P(n)=>Q(n+1) and Q(n)=>P(n+1). Also P(0) holds.", "prompt": "P(n) holds for every even n.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.2ai" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.92, "probabilities": { "A": 0.96, "B": 0.04 }, "batch_response_ms": 429.8258330018143, "repaired": false }, { "id": "3.2aii", "section": 3, "context": "N includes zero. For every n, P(n)=>Q(n+1) and Q(n)=>P(n+1). Also P(0) holds.", "prompt": "For every n, NOT Q(n+1) implies NOT P(n).", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.2aii" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.95, "probabilities": { "A": 0.03, "B": 0.97 }, "batch_response_ms": 429.8258330018143, "repaired": false }, { "id": "3.2aiii", "section": 3, "context": "N includes zero. For every n, P(n)=>Q(n+1) and Q(n)=>P(n+1). Also P(0) holds.", "prompt": "Q(0) implies Q(n) for every n.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.2aiii" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.43, "probabilities": { "A": 0.29, "B": 0.71 }, "batch_response_ms": 429.8258330018143, "repaired": false }, { "id": "3.2b", "section": 3, "context": "", "prompt": "NOT forall n [P(n) OR Q(n)] implies exists n NOT P(n).", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.2b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "A": 0.99, "B": 0.01 }, "batch_response_ms": 429.8258330018143, "repaired": false }, { "id": "4.1", "section": 4, "context": "", "prompt": "Distinct primes p,q>2 and positive e,d satisfy ed=1 mod (p−1)(q−1). Prove x^(ed)=x mod pq for every integer x, without merely citing RSA correctness.", "response_mode": "single_choice", "options": { "A": "Cancel x from the congruence for all x, including multiples of p and q, then use its inverse.", "B": "Fermat applies modulo pq directly with exponent pq−1, and ed=pq−1.", "C": "Modulo each prime, if x is zero the equality is immediate; otherwise Fermat applies because ed−1 is a multiple of that prime minus one. CRT then combines the equalities modulo p and q.", "D": "Since ed=1 modulo the totient, ed=1 as integers, so the equality is exact." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "4.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 0.0, "B": 0.0, "C": 1.0, "D": 0.0 }, "batch_response_ms": 314.1008749989851, "repaired": false }, { "id": "4.2a", "section": 4, "context": "", "prompt": "Up to isomorphism, give all trees on four vertices as edge sets on {1,2,3,4}.", "response_mode": "single_choice", "options": { "A": "{12,23,34} and {12,23,34,41}", "B": "Only {12,23,34}", "C": "{12,23,31} and {12,23,34}", "D": "{12,23,34} and {12,13,14}" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "4.2a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.97, "probabilities": { "A": 0.0, "B": 0.02, "C": 0.0, "D": 0.98 }, "batch_response_ms": 314.1008749989851, "repaired": false }, { "id": "4.2b", "section": 4, "context": "", "prompt": "Every three-vertex tree is a subgraph of K3 (not necessarily induced).", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "4.2b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.51, "probabilities": { "A": 0.75, "B": 0.25 }, "batch_response_ms": 314.1008749989851, "repaired": false }, { "id": "4.2c", "section": 4, "context": "", "prompt": "Prove a simple graph of minimum degree d contains every k-vertex tree for 1<=k<=d.", "response_mode": "single_choice", "options": { "A": "Any k vertices of the graph form a tree when k<=d.", "B": "Induct on tree size. Remove a leaf, embed the smaller tree, and map the leaf to an unused neighbor of its parent’s image. At most k−2 other embedded vertices can occupy neighbors, fewer than d. The single-vertex base is immediate.", "C": "Every degree-d vertex has all its neighbors mutually adjacent, so it contains every tree directly.", "D": "Induct by adding arbitrary edges to the target tree; every missing edge must already appear in the host graph." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "4.2c" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 0.0, "B": 1.0, "C": 0.0, "D": 0.0 }, "batch_response_ms": 314.1008749989851, "repaired": false }, { "id": "5.1", "section": 5, "context": "", "prompt": "All n jobs have the same ranking. In synchronous job-proposing deferred acceptance, counting the initial proposal day as day 1, how many days until all are held?", "response_mode": "single_choice", "options": { "A": "n−1", "B": "n", "C": "1", "D": "n²" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "written", "repairs": [ "Fix day-count convention; original grading accepted n and n−1." ], "source_parts": [ "5.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.22, "probabilities": { "D": 0.04, "A": 0.36, "C": 0.19, "B": 0.41 }, "batch_response_ms": 312.6039579983626, "repaired": true }, { "id": "5.2", "section": 5, "context": "", "prompt": "If all jobs have the same strict ranking, the stable matching is unique.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "5.2" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.77, "probabilities": { "A": 0.88, "B": 0.12 }, "batch_response_ms": 312.6039579983626, "repaired": false }, { "id": "5.3", "section": 5, "context": "", "prompt": "If all candidates have the same strict ranking, the stable matching is unique.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "5.3" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.86, "probabilities": { "A": 0.93, "B": 0.07 }, "batch_response_ms": 312.6039579983626, "repaired": false }, { "id": "5.4a", "section": 5, "context": "", "prompt": "All preference lists are independently uniform permutations, n jobs and n candidates. Expected number of candidates receiving exactly i first-day proposals?", "response_mode": "single_choice", "options": { "A": "n*(1−1/n)^i", "B": "n/i", "C": "n*C(n,i)*(1/n)^i*(1−1/n)^(n−i)", "D": "C(n,i)*(1/n)^i*(1−1/n)^(n−i)" }, "correct": true, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "5.4a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.68, "probabilities": { "D": 0.19, "A": 0.04, "C": 0.76, "B": 0.01 }, "batch_response_ms": 312.6039579983626, "repaired": false }, { "id": "5.4b", "section": 5, "context": "", "prompt": "For n=2 with independently uniform strict preference lists, probability both possible matchings are stable?", "response_mode": "single_choice", "options": { "A": "1/16", "B": "1/4", "C": "1/8", "D": "1/2" }, "correct": false, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "5.4b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.17, "probabilities": { "D": 0.39, "A": 0.05, "C": 0.24, "B": 0.32 }, "batch_response_ms": 312.6039579983626, "repaired": false }, { "id": "6.1", "section": 6, "context": "", "prompt": "A connected simple planar embedding has n vertices and every face boundary has four edge-sides. Edge count?", "response_mode": "single_choice", "options": { "A": "2n−2", "B": "3n−6", "C": "n−1", "D": "2n−4" }, "correct": false, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [ "State connectedness required by the Euler formula used in the key." ], "source_parts": [ "6.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.42, "probabilities": { "C": 0.0, "B": 0.0, "A": 0.57, "D": 0.43 }, "batch_response_ms": 414.6086250002554, "repaired": true }, { "id": "6.2a", "section": 6, "context": "G has a connected simple planar embedding with v vertices,e edges,f faces, all face boundaries simple cycles. F keeps G, adds one vertex per face and joins it to every vertex on that face, including the outer face. F is planar.", "prompt": "Number of vertices of F?", "response_mode": "single_choice", "options": { "A": "v+f+e", "B": "2v", "C": "v+f", "D": "v+e" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "6.2a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 1.0, "B": 0.0, "A": 0.0, "D": 0.0 }, "batch_response_ms": 414.6086250002554, "repaired": false }, { "id": "6.2b", "section": 6, "context": "G has a connected simple planar embedding with v vertices,e edges,f faces, all face boundaries simple cycles. F keeps G, adds one vertex per face and joins it to every vertex on that face, including the outer face. F is planar.", "prompt": "Number of edges of F?", "response_mode": "single_choice", "options": { "A": "3(v+f)−6", "B": "3v−6", "C": "2e", "D": "e+f" }, "correct": true, "kind": "multi_step", "topic": "graphs", "original_format": "written", "repairs": [ "Specify connected embedding and simple facial cycles, required for the source triangulation argument." ], "source_parts": [ "6.2b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.61, "probabilities": { "C": 0.2, "B": 0.01, "A": 0.7, "D": 0.09 }, "batch_response_ms": 414.6086250002554, "repaired": true }, { "id": "6.3", "section": 6, "context": "", "prompt": "Remove one edge from a tree. How many connected components result?", "response_mode": "single_choice", "options": { "A": "2", "B": "1", "C": "3", "D": "0" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "6.3" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 1.0, "D": 0.0 }, "batch_response_ms": 414.6086250002554, "repaired": false }, { "id": "6.4", "section": 6, "context": "", "prompt": "Minimum edges removed from K(2n) to make it bipartite?", "response_mode": "single_choice", "options": { "A": "n−1", "B": "n²", "C": "n(n−1)", "D": "2n(n−1)" }, "correct": false, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "6.4" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.54, "probabilities": { "C": 0.33, "B": 0.65, "A": 0.01, "D": 0.01 }, "batch_response_ms": 414.6086250002554, "repaired": false }, { "id": "6.5", "section": 6, "context": "", "prompt": "Minimum vertex colors for a tree with at least one edge?", "response_mode": "single_choice", "options": { "A": "1", "B": "d", "C": "2", "D": "d+1" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [ "Exclude the one-vertex tree." ], "source_parts": [ "6.5" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 1.0, "B": 0.0, "A": 0.0, "D": 0.0 }, "batch_response_ms": 414.6086250002554, "repaired": true }, { "id": "6.6", "section": 6, "context": "", "prompt": "Minimum edge colors for a tree of maximum degree d?", "response_mode": "single_choice", "options": { "A": "2d", "B": "d", "C": "d+1", "D": "2" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "6.6" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.92, "probabilities": { "C": 0.04, "B": 0.94, "A": 0.0, "D": 0.02 }, "batch_response_ms": 414.6086250002554, "repaired": false }, { "id": "6.7", "section": 6, "context": "", "prompt": "Maximum edges in a simple n-vertex graph, n>=3, where every cycle has length at least n?", "response_mode": "single_choice", "options": { "A": "2n−2", "B": "n", "C": "n−1", "D": "n+1" }, "correct": false, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "6.7" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.15, "probabilities": { "C": 0.24, "B": 0.34, "A": 0.37, "D": 0.05 }, "batch_response_ms": 414.6086250002554, "repaired": false }, { "id": "6.8", "section": 6, "context": "", "prompt": "Length in edges of an Eulerian tour in a graph with n vertices,m edges?", "response_mode": "single_choice", "options": { "A": "n", "B": "n−1", "C": "m", "D": "2m" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "6.8" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.86, "probabilities": { "C": 0.89, "B": 0.0, "A": 0.0, "D": 0.11 }, "batch_response_ms": 414.6086250002554, "repaired": false }, { "id": "6.9a", "section": 6, "context": "", "prompt": "Delete the edges of a simple k-cycle from a connected graph, retaining all vertices. What universal lower bound on the remaining component count is attainable for some such graphs?", "response_mode": "single_choice", "options": { "A": "k", "B": "1", "C": "k−1", "D": "2" }, "correct": false, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [ "Ask universal attainable bound; for particular tiny n,k, one component need not be attainable." ], "source_parts": [ "6.9a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.22, "probabilities": { "C": 0.11, "B": 0.26, "A": 0.42, "D": 0.21 }, "batch_response_ms": 414.6086250002554, "repaired": true }, { "id": "6.9b", "section": 6, "context": "", "prompt": "Delete the edges of a simple k-cycle from a connected graph, retaining all vertices. Maximum possible remaining component count?", "response_mode": "single_choice", "options": { "A": "k+1", "B": "2k", "C": "k", "D": "k−1" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "6.9b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.52, "probabilities": { "C": 0.63, "B": 0.04, "A": 0.25, "D": 0.08 }, "batch_response_ms": 414.6086250002554, "repaired": false }, { "id": "7.1", "section": 7, "context": "", "prompt": "2^36 mod 7?", "response_mode": "single_choice", "options": { "A": "1", "B": "2", "C": "4", "D": "0" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "7.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.21, "probabilities": { "B": 0.33, "D": 0.0, "C": 0.25, "A": 0.42 }, "batch_response_ms": 435.76137499985634, "repaired": false }, { "id": "7.2", "section": 7, "context": "", "prompt": "2^36 mod 35?", "response_mode": "single_choice", "options": { "A": "4", "B": "16", "C": "1", "D": "8" }, "correct": false, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "7.2" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.37, "probabilities": { "B": 0.53, "D": 0.07, "C": 0.18, "A": 0.22 }, "batch_response_ms": 435.76137499985634, "repaired": false }, { "id": "7.5", "section": 7, "context": "", "prompt": "gcd(385,70)?", "response_mode": "single_choice", "options": { "A": "5", "B": "35", "C": "70", "D": "7" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "7.5" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.93, "probabilities": { "B": 0.95, "D": 0.04, "C": 0.0, "A": 0.01 }, "batch_response_ms": 435.76137499985634, "repaired": false }, { "id": "7.7a", "section": 7, "context": "", "prompt": "Number of distinct square residues modulo 3, including zero?", "response_mode": "single_choice", "options": { "A": "2", "B": "3", "C": "1", "D": "0" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "7.7a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.93, "probabilities": { "B": 0.04, "D": 0.0, "C": 0.01, "A": 0.95 }, "batch_response_ms": 435.76137499985634, "repaired": false }, { "id": "7.7c", "section": 7, "context": "", "prompt": "Number of distinct square residues modulo 15, including zero?", "response_mode": "single_choice", "options": { "A": "6", "B": "8", "C": "4", "D": "7" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "7.7c" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.56, "probabilities": { "B": 0.05, "D": 0.12, "C": 0.16, "A": 0.67 }, "batch_response_ms": 435.76137499985634, "repaired": false }, { "id": "7.3", "section": 7, "context": "", "prompt": "If x=0 mod d, then bx−kd=0 mod d for all integers b,k.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "7.3" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "A": 1.0 }, "batch_response_ms": 435.76137499985634, "repaired": false }, { "id": "7.4", "section": 7, "context": "", "prompt": "If x=1 mod d, then bx−kd=1 mod d for all integers b,k.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": false, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "7.4" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.49, "probabilities": { "B": 0.74, "A": 0.26 }, "batch_response_ms": 435.76137499985634, "repaired": false }, { "id": "7.6a", "section": 7, "context": "gcd(a,m)=d and 2a=b mod m, with positive modulus m.", "prompt": "Assume d>1. Give the least positive t, nonzero modulo m, with (2+t)a=b mod m.", "response_mode": "single_choice", "options": { "A": "m/d", "B": "d", "C": "a/d", "D": "m" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [ "Require d>1; if d=1 the source requests an impossible nonzero shift modulo m." ], "source_parts": [ "7.6a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.79, "probabilities": { "B": 0.13, "D": 0.02, "C": 0.01, "A": 0.84 }, "batch_response_ms": 435.76137499985634, "repaired": true }, { "id": "7.6b", "section": 7, "context": "gcd(a,m)=d and 2a=b mod m, with positive modulus m.", "prompt": "b must be even.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "7.6b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.46, "probabilities": { "B": 0.73, "A": 0.27 }, "batch_response_ms": 435.76137499985634, "repaired": false }, { "id": "7.6c", "section": 7, "context": "gcd(a,m)=d and 2a=b mod m, with positive modulus m.", "prompt": "d divides b.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "7.6c" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.91, "probabilities": { "B": 0.05, "A": 0.95 }, "batch_response_ms": 435.76137499985634, "repaired": false }, { "id": "7.6d", "section": 7, "context": "gcd(a,m)=d and 2a=b mod m, with positive modulus m.", "prompt": "How many distinct solutions modulo m does ax=b mod m have, given it is solvable?", "response_mode": "single_choice", "options": { "A": "d", "B": "m/d", "C": "1", "D": "m" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "7.6d" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.75, "probabilities": { "B": 0.16, "D": 0.0, "C": 0.03, "A": 0.81 }, "batch_response_ms": 435.76137499985634, "repaired": false }, { "id": "7.7b", "section": 7, "context": "", "prompt": "Prime p>2: count square residues including zero.", "response_mode": "single_choice", "options": { "A": "(p−1)/2", "B": "p", "C": "(p+1)/2", "D": "p−1" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "7.7b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "D": 0.0, "C": 1.0, "A": 0.0 }, "batch_response_ms": 435.76137499985634, "repaired": false }, { "id": "7.7d", "section": 7, "context": "", "prompt": "Distinct primes p,q>2: count square residues modulo pq, including zero. Hint: CRT.", "response_mode": "single_choice", "options": { "A": "(p−1)(q−1)/4", "B": "(p+q)/2", "C": "(pq+1)/2", "D": "(p+1)(q+1)/4" }, "correct": true, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "7.7d" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "D": 1.0, "C": 0.0, "A": 0.0 }, "batch_response_ms": 435.76137499985634, "repaired": false }, { "id": "8.1", "section": 8, "context": "", "prompt": "GF(5) interpolation yields P=x²+4x+1. One normalized Lagrange basis polynomial is Delta=x²+3x+2. Recover the three interpolation points.", "response_mode": "single_choice", "options": { "A": "{(1,2),(3,0),(4,0)}", "B": "{(1,1),(3,2),(4,3)}", "C": "{(0,1),(3,2),(4,3)}", "D": "{(1,1),(2,3),(4,3)}" }, "correct": true, "kind": "multi_step", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "8.1.point1", "8.1.point2", "8.1.point3" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.45, "probabilities": { "B": 0.58, "C": 0.12, "D": 0.07, "A": 0.23 }, "batch_response_ms": 399.2643340025097, "repaired": false }, { "id": "8.2", "section": 8, "context": "", "prompt": "Every polynomial of degree exactly two is a bijection over every prime field.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "choice", "repairs": [], "source_parts": [ "8.2" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.99, "A": 0.01 }, "batch_response_ms": 399.2643340025097, "repaired": false }, { "id": "8.3", "section": 8, "context": "", "prompt": "Every polynomial of degree exactly one is a bijection over a prime field.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "choice", "repairs": [], "source_parts": [ "8.3" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.89, "probabilities": { "B": 0.05, "A": 0.95 }, "batch_response_ms": 399.2643340025097, "repaired": false }, { "id": "8.4a", "section": 8, "context": "", "prompt": "Where do 2x+3 and 3x+2 intersect over GF(5)?", "response_mode": "single_choice", "options": { "A": "4", "B": "0", "C": "1", "D": "2" }, "correct": false, "kind": "calculation", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "8.4a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.18, "probabilities": { "B": 0.07, "A": 0.39, "D": 0.16, "C": 0.38 }, "batch_response_ms": 399.2643340025097, "repaired": false }, { "id": "8.4b", "section": 8, "context": "", "prompt": "Distinct formal polynomials of degrees dP,dQ over GF(p), p>max(dP,dQ): tight general upper bound on intersection coordinates?", "response_mode": "single_choice", "options": { "A": "dP+dQ", "B": "p", "C": "min(dP,dQ)", "D": "max(dP,dQ)" }, "correct": false, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "8.4b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.32, "probabilities": { "B": 0.0, "C": 0.49, "D": 0.41, "A": 0.1 }, "batch_response_ms": 399.2643340025097, "repaired": false }, { "id": "8.4c", "section": 8, "context": "", "prompt": "Over GF(p),p>3, fixed P. Count polynomials Q of degree at most three with Q(1)=P(1),Q(2)=P(2).", "response_mode": "single_choice", "options": { "A": "(p−1)p", "B": "p²", "C": "1", "D": "p" }, "correct": true, "kind": "calculation", "topic": "polynomials", "original_format": "written", "repairs": [ "Explicit degree-at-most convention matching source reference." ], "source_parts": [ "8.4c" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.81, "probabilities": { "B": 0.86, "A": 0.07, "D": 0.07, "C": 0.0 }, "batch_response_ms": 399.2643340025097, "repaired": true }, { "id": "8.5a", "section": 8, "context": "", "prompt": "Encode n message symbols as polynomial coefficients. Degree bound of message P?", "response_mode": "single_choice", "options": { "A": "At most n+2k", "B": "At most n−1", "C": "Exactly n", "D": "Exactly k" }, "correct": true, "kind": "recognition", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "8.5a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 1.0, "C": 0.0, "D": 0.0, "A": 0.0 }, "batch_response_ms": 399.2643340025097, "repaired": false }, { "id": "8.5b", "section": 8, "context": "", "prompt": "How many distinct evaluations suffice to correct k corruptions in n message symbols?", "response_mode": "single_choice", "options": { "A": "n+2k−1", "B": "2n+k", "C": "n+k", "D": "n+2k" }, "correct": true, "kind": "recognition", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "8.5b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.82, "probabilities": { "B": 0.0, "A": 0.05, "D": 0.87, "C": 0.08 }, "batch_response_ms": 399.2643340025097, "repaired": false }, { "id": "8.5c", "section": 8, "context": "", "prompt": "Why does E(i)(P(i)−ri)=0 at each transmitted coordinate, where E vanishes at error locations?", "response_mode": "single_choice", "options": { "A": "At a corrupted coordinate E(i)=0; at an uncorrupted one P(i)−ri=0. Thus at least one factor vanishes everywhere.", "B": "Both factors vanish at every coordinate.", "C": "Every field product with an error polynomial is zero.", "D": "The product has degree zero, so it is identically zero." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "8.5c" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "D": 0.0, "A": 1.0 }, "batch_response_ms": 399.2643340025097, "repaired": false }, { "id": "9.1", "section": 9, "context": "", "prompt": "Distinct arrangements of BAA?", "response_mode": "single_choice", "options": { "A": "1", "B": "3", "C": "6", "D": "2" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "9.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.84, "probabilities": { "B": 0.88, "D": 0.11, "A": 0.01, "C": 0.0 }, "batch_response_ms": 311.1654579988681, "repaired": false }, { "id": "9.2", "section": 9, "context": "", "prompt": "Distinct arrangements of 126?", "response_mode": "single_choice", "options": { "A": "1", "B": "3", "C": "9", "D": "6" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "9.2" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.31, "probabilities": { "D": 0.48, "B": 0.11, "A": 0.34, "C": 0.07 }, "batch_response_ms": 311.1654579988681, "repaired": false }, { "id": "9.3", "section": 9, "context": "", "prompt": "n distinct bears, k distinct children,n>k. Assign one bear each, without reuse. Count?", "response_mode": "single_choice", "options": { "A": "n^k", "B": "k^n", "C": "C(n,k)", "D": "n!/(n−k)!" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "9.3" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 311.1654579988681, "repaired": false }, { "id": "9.4", "section": 9, "context": "", "prompt": "Bipartite labeled graph with fixed parts each of size n and exactly m edges. Count?", "response_mode": "single_choice", "options": { "A": "2^(n²)", "B": "C(n²,m)", "C": "n^(2m)", "D": "C(2n,m)" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "9.4" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "B": 1.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 311.1654579988681, "repaired": false }, { "id": "9.5", "section": 9, "context": "", "prompt": "Number of functions from p residues to p residues?", "response_mode": "single_choice", "options": { "A": "p^p", "B": "p!", "C": "2^p", "D": "p²" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "9.5" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 1.0, "C": 0.0 }, "batch_response_ms": 311.1654579988681, "repaired": false }, { "id": "9.6", "section": 9, "context": "", "prompt": "Number of bijections from p residues to themselves?", "response_mode": "single_choice", "options": { "A": "p²", "B": "p^p", "C": "p", "D": "p!" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "9.6" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 1.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 311.1654579988681, "repaired": false }, { "id": "9.7", "section": 9, "context": "", "prompt": "Form three named teams of four distinct players from 15 people, no overlap. Count?", "response_mode": "single_choice", "options": { "A": "15!/(4!4!4!3!3!)", "B": "C(15,12)", "C": "C(15,4)^3", "D": "C(15,4)C(11,4)C(7,4)" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "9.7" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.8, "probabilities": { "D": 0.86, "B": 0.0, "A": 0.14, "C": 0.0 }, "batch_response_ms": 311.1654579988681, "repaired": false }, { "id": "10.1", "section": 10, "context": "", "prompt": "Give a combinatorial proof of sum(i=0..n) C(n,i)C(n,n−i)=C(2n,n).", "response_mode": "single_choice", "options": { "A": "Choose n objects from two disjoint n-object groups. Classify each choice by how many i come from the first group; exactly n−i come from the second.", "B": "Choose the same subset twice, which counts every n-subset of 2n once.", "C": "Choose i from both groups, always obtaining n total objects regardless of i.", "D": "Every binomial coefficient equals the number of permutations of its top argument." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "10.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 1.0, "C": 0.0 }, "batch_response_ms": 733.361208000133, "repaired": false }, { "id": "10.2", "section": 10, "context": "", "prompt": "Given sum(i=0..n) C(n,i)C(n,n−i)=C(2n,n), why does sum(i=0..n) C(n,i)²=C(2n,n) follow?", "response_mode": "single_choice", "options": { "A": "All summands are equal to C(2n,n)/n.", "B": "C(n,n−i)=C(n,i) by taking complements of subsets.", "C": "Every binomial coefficient equals its own square.", "D": "C(n,n−i)=n−i." }, "correct": true, "kind": "recognition", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "10.2" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "B": 1.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 733.361208000133, "repaired": false }, { "id": "11.1", "section": 11, "context": "", "prompt": "The power set of N has the same cardinality as [0,1].", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "11.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.9, "probabilities": { "B": 0.95, "A": 0.05 }, "batch_response_ms": 717.5805830011086, "repaired": false }, { "id": "11.2", "section": 11, "context": "", "prompt": "A finite Cartesian product of countable sets is countable.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "11.2" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.97, "probabilities": { "B": 0.99, "A": 0.01 }, "batch_response_ms": 717.5805830011086, "repaired": false }, { "id": "11.3a", "section": 11, "context": "A=Z×Z; f:A->C is onto and g:C->B is injective.", "prompt": "A is countable.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "11.3a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "A": 1.0 }, "batch_response_ms": 717.5805830011086, "repaired": false }, { "id": "11.3b", "section": 11, "context": "A=Z×Z; f:A->C is onto and g:C->B is injective.", "prompt": "These assumptions imply |A| is strictly smaller than |B|.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "11.3b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.01, "A": 0.99 }, "batch_response_ms": 717.5805830011086, "repaired": false }, { "id": "11.4", "section": 11, "context": "", "prompt": "Give a bijection (0,1]->[1,infinity).", "response_mode": "single_choice", "options": { "A": "f(x)=1/x", "B": "f(x)=1/(1−x)", "C": "f(x)=x+1", "D": "f(x)=−ln(x)" }, "correct": true, "kind": "recognition", "topic": "functions", "original_format": "written", "repairs": [], "source_parts": [ "11.4" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "C": 0.0, "D": 0.0, "A": 1.0 }, "batch_response_ms": 717.5805830011086, "repaired": false }, { "id": "12.1", "section": 12, "context": "", "prompt": "There is an algorithm deciding whether P(x) halts within |x|^k steps, for arbitrary P, nonempty finite x and nonnegative integer k.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "computability", "original_format": "choice", "repairs": [], "source_parts": [ "12.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.44, "probabilities": { "B": 0.72, "A": 0.28 }, "batch_response_ms": 326.3699160015676, "repaired": false }, { "id": "12.2", "section": 12, "context": "", "prompt": "Hypothetical HaltsOnEmpty decides whether a program halts on empty input. Fill: Halt(P,x) defines inner(y): __1__; then returns HaltsOnEmpty(__2__).", "response_mode": "single_choice", "options": { "A": "(return True; inner)", "B": "(P(x); inner)", "C": "(P(x); x)", "D": "(P(y); P)" }, "correct": true, "kind": "multi_step", "topic": "computability", "original_format": "written", "repairs": [], "source_parts": [ "12.2.1", "12.2.2" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.59, "probabilities": { "B": 0.7, "D": 0.12, "A": 0.02, "C": 0.16 }, "batch_response_ms": 326.3699160015676, "repaired": false }, { "id": "13.1", "section": 13, "context": "", "prompt": "Sharp upper bound on P(A union B) using only a=P(A),b=P(B)?", "response_mode": "single_choice", "options": { "A": "a+b", "B": "a*b", "C": "max(a,b)", "D": "min(1,a+b)" }, "correct": true, "kind": "recognition", "topic": "bounds", "original_format": "written", "repairs": [ "Clip at one; source unqualified union bound is not the lowest upper bound when a+b>1." ], "source_parts": [ "13.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.0, "D": 0.98, "C": 0.0, "A": 0.02 }, "batch_response_ms": 297.2415420008474, "repaired": true }, { "id": "13.2", "section": 13, "context": "", "prompt": "Sharp lower bound on P(A union B) using only a=P(A),b=P(B)?", "response_mode": "single_choice", "options": { "A": "a+b", "B": "max(a,b)", "C": "min(a,b)", "D": "a*b" }, "correct": true, "kind": "recognition", "topic": "bounds", "original_format": "written", "repairs": [], "source_parts": [ "13.2" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 1.0, "D": 0.0, "C": 0.0, "A": 0.0 }, "batch_response_ms": 297.2415420008474, "repaired": false }, { "id": "13.3", "section": 13, "context": "", "prompt": "A is a subset of B and both have positive probability. P(A given B)?", "response_mode": "single_choice", "options": { "A": "1", "B": "P(B)/P(A)", "C": "P(A)/P(B)", "D": "P(A)P(B)" }, "correct": true, "kind": "recognition", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "13.3" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "D": 0.0, "C": 1.0, "A": 0.0 }, "batch_response_ms": 297.2415420008474, "repaired": false }, { "id": "13.4", "section": 13, "context": "", "prompt": "A is a subset of B and P(A)>0. P(B given A)?", "response_mode": "single_choice", "options": { "A": "0", "B": "P(B)", "C": "P(A)/P(B)", "D": "1" }, "correct": true, "kind": "recognition", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "13.4" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 1.0, "C": 0.0, "A": 0.0 }, "batch_response_ms": 297.2415420008474, "repaired": false }, { "id": "13.5a", "section": 13, "context": "Events have probabilities a=P(A),b=P(B),c=P(C), with P(A intersection B)=P(B intersection C)=alpha. Assume these quantities are feasible.", "prompt": "Sharp upper bound on P(A intersection C)?", "response_mode": "single_choice", "options": { "A": "a+c", "B": "min(a,b,c)", "C": "alpha", "D": "min(a,c)" }, "correct": true, "kind": "recognition", "topic": "bounds", "original_format": "written", "repairs": [], "source_parts": [ "13.5a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.66, "probabilities": { "B": 0.08, "D": 0.75, "C": 0.16, "A": 0.01 }, "batch_response_ms": 297.2415420008474, "repaired": false }, { "id": "13.5b", "section": 13, "context": "Events have probabilities a=P(A),b=P(B),c=P(C), with P(A intersection B)=P(B intersection C)=alpha. Assume these quantities are feasible.", "prompt": "Sharp lower bound on P(A intersection C) using all the given quantities?", "response_mode": "single_choice", "options": { "A": "max(0,2alpha−b)+max(0,a+c−2alpha+b−1)", "B": "max(0,a+c−1)", "C": "max(0,2alpha−b)", "D": "alpha" }, "correct": true, "kind": "multi_step", "topic": "bounds", "original_format": "written", "repairs": [ "Correct incomplete source lower bound: intersections forced outside B must also be counted." ], "source_parts": [ "13.5b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.9, "probabilities": { "B": 0.02, "D": 0.0, "C": 0.05, "A": 0.93 }, "batch_response_ms": 297.2415420008474, "repaired": true }, { "id": "13.5c", "section": 13, "context": "Events have probabilities a=P(A),b=P(B),c=P(C), with P(A intersection B)=P(B intersection C)=alpha. Assume these quantities are feasible.", "prompt": "Justify a sharp lower bound using the partition into B and its complement.", "response_mode": "single_choice", "options": { "A": "The overlap is always exactly alpha, regardless of a,b,c.", "B": "The two known intersections are equal, so A and C must be identical.", "C": "Ignore the complement of B, because neither A nor C can extend beyond B.", "D": "Inside B, overlap is at least max(0,2alpha−b). Outside B, the masses a−alpha and c−alpha share a region of mass 1−b, forcing max(0,a+c−2alpha+b−1). Add the independently attainable minima." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [ "Extend source argument to cover the omitted complement region." ], "source_parts": [ "13.5c" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 1.0, "C": 0.0, "A": 0.0 }, "batch_response_ms": 297.2415420008474, "repaired": true }, { "id": "14.1", "section": 14, "context": "Choose a fair four-sided or fair six-sided die with equal probability, then roll the chosen die twice independently conditional on its identity. A:first roll 1; B:second roll 1; C:four-sided die selected.", "prompt": "P(A given B)?", "response_mode": "single_choice", "options": { "A": "5/24", "B": "13/60", "C": "1/6", "D": "1/4" }, "correct": false, "kind": "multi_step", "topic": "bayes", "original_format": "written", "repairs": [], "source_parts": [ "14.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.46, "probabilities": { "B": 0.08, "C": 0.59, "A": 0.1, "D": 0.23 }, "batch_response_ms": 372.60020800022176, "repaired": false }, { "id": "14.2", "section": 14, "context": "Choose a fair four-sided or fair six-sided die with equal probability, then roll the chosen die twice independently conditional on its identity. A:first roll 1; B:second roll 1; C:four-sided die selected.", "prompt": "P(C given A)?", "response_mode": "single_choice", "options": { "A": "2/5", "B": "1/4", "C": "1/2", "D": "3/5" }, "correct": false, "kind": "multi_step", "topic": "bayes", "original_format": "written", "repairs": [], "source_parts": [ "14.2" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.34, "probabilities": { "C": 0.14, "B": 0.02, "A": 0.51, "D": 0.33 }, "batch_response_ms": 372.60020800022176, "repaired": false }, { "id": "15.1", "section": 15, "context": "", "prompt": "Independent Bernoulli X,Y each have mean 1/2. P(X=Y)?", "response_mode": "single_choice", "options": { "A": "1", "B": "1/4", "C": "1/2", "D": "3/4" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "15.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.98, "probabilities": { "D": 0.01, "A": 0.0, "B": 0.0, "C": 0.99 }, "batch_response_ms": 371.270874999027, "repaired": false }, { "id": "15.2a", "section": 15, "context": "Bernoulli X,Y each have mean 1/2 and correlation 1/2.", "prompt": "E[XY]?", "response_mode": "single_choice", "options": { "A": "1/4", "B": "3/8", "C": "1/2", "D": "1/8" }, "correct": true, "kind": "multi_step", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "15.2a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.88, "probabilities": { "D": 0.02, "A": 0.01, "B": 0.91, "C": 0.06 }, "batch_response_ms": 371.270874999027, "repaired": false }, { "id": "15.2b", "section": 15, "context": "Bernoulli X,Y each have mean 1/2 and correlation 1/2.", "prompt": "P(X=Y)?", "response_mode": "single_choice", "options": { "A": "1/2", "B": "3/4", "C": "3/8", "D": "1/4" }, "correct": true, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "15.2b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.53, "probabilities": { "D": 0.02, "A": 0.27, "B": 0.66, "C": 0.05 }, "batch_response_ms": 371.270874999027, "repaired": false }, { "id": "16.1", "section": 16, "context": "Sample n people independently from a population with positive-test probability p. Xi indicates a positive test.", "prompt": "E[Xi]?", "response_mode": "single_choice", "options": { "A": "p(1−p)", "B": "1−p", "C": "p", "D": "np" }, "correct": true, "kind": "recognition", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "16.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 1.0, "A": 0.0, "B": 0.0, "D": 0.0 }, "batch_response_ms": 288.18266700181994, "repaired": false }, { "id": "16.2", "section": 16, "context": "Sample n people independently from a population with positive-test probability p. Xi indicates a positive test.", "prompt": "Sharp p-independent upper bound on Var(Xi)?", "response_mode": "single_choice", "options": { "A": "1", "B": "p", "C": "1/4", "D": "1/2" }, "correct": true, "kind": "recognition", "topic": "bounds", "original_format": "written", "repairs": [], "source_parts": [ "16.2" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.87, "probabilities": { "C": 0.91, "A": 0.02, "B": 0.07, "D": 0.0 }, "batch_response_ms": 288.18266700181994, "repaired": false }, { "id": "16.3", "section": 16, "context": "Sample n people independently from a population with positive-test probability p. Xi indicates a positive test.", "prompt": "50 of 100 independently sampled people test positive. Using Chebyshev and the worst-case Bernoulli variance, give a 95% confidence interval for p.", "response_mode": "single_choice", "options": { "A": "[1/2−1/20,1/2+1/20]", "B": "[1/2−1/sqrt(20),1/2+1/sqrt(20)]", "C": "[1/2−1/10,1/2+1/10]", "D": "[1/2−1/sqrt(100),1/2+1/sqrt(100)]" }, "correct": true, "kind": "multi_step", "topic": "bounds", "original_format": "written", "repairs": [ "State independent sampling assumed in source solution." ], "source_parts": [ "16.3" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.53, "probabilities": { "C": 0.26, "A": 0.06, "B": 0.65, "D": 0.03 }, "batch_response_ms": 288.18266700181994, "repaired": true }, { "id": "17.1", "section": 17, "context": "", "prompt": "Prove every tournament (one directed edge per unordered vertex pair) has a Hamiltonian path.", "response_mode": "single_choice", "options": { "A": "Every tournament has a vertex pointing to all others, so choose it first recursively.", "B": "All tournaments are acyclic, so a topological sort always gives a path.", "C": "Follow outgoing edges greedily; a visited vertex can never be encountered again.", "D": "Induct including zero/one-vertex bases. Pick v; split remaining vertices into those pointing to v and those v points to. Each induced tournament has a Hamiltonian path; concatenate the first path, v, then the second." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "17.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 1.0, "A": 0.0, "C": 0.0, "B": 0.0 }, "batch_response_ms": 435.24579100267147, "repaired": false }, { "id": "17.2a", "section": 17, "context": "", "prompt": "Orient every edge of a complete n-vertex graph independently with a fair coin. Number of possible tournaments?", "response_mode": "single_choice", "options": { "A": "C(n,2)", "B": "2^C(n,2)", "C": "2^(n²)", "D": "n!" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "17.2a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.0, "B": 1.0 }, "batch_response_ms": 435.24579100267147, "repaired": false }, { "id": "17.2b", "section": 17, "context": "", "prompt": "Orient every edge of a complete n-vertex graph independently with a fair coin. Expected Hamiltonian paths in the resulting tournament?", "response_mode": "single_choice", "options": { "A": "n!/2^(n−1)", "B": "n!/2^C(n,2)", "C": "2^(n−1)", "D": "n/2" }, "correct": true, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "17.2b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.88, "probabilities": { "D": 0.0, "A": 0.91, "C": 0.0, "B": 0.09 }, "batch_response_ms": 435.24579100267147, "repaired": false }, { "id": "18.1", "section": 18, "context": "Random variables have finite second moments; Var(X)>0.", "prompt": "The best affine least-squares estimator of Y from X passes through the origin iff E[Y] equals what?", "response_mode": "single_choice", "options": { "A": "Cov(X,Y)/E[X]", "B": "0", "C": "Cov(X,Y)*E[X]/Var(X)", "D": "E[X]" }, "correct": false, "kind": "recognition", "topic": "regression", "original_format": "written", "repairs": [], "source_parts": [ "18.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.84, "probabilities": { "B": 0.88, "D": 0.0, "A": 0.04, "C": 0.08 }, "batch_response_ms": 337.2207080028602, "repaired": false }, { "id": "18.2", "section": 18, "context": "Random variables have finite second moments; Var(X)>0.", "prompt": "E[X²] in terms of variance and mean?", "response_mode": "single_choice", "options": { "A": "Var(X)−E[X]²", "B": "E[X]²", "C": "Var(X)+E[X]²", "D": "Var(X)²+E[X]" }, "correct": true, "kind": "recognition", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "18.2" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.0, "C": 1.0 }, "batch_response_ms": 337.2207080028602, "repaired": false }, { "id": "18.3", "section": 18, "context": "Random variables have finite second moments; Var(X)>0.", "prompt": "If E[X]=E[Y]=t, what t makes Cov(X,Y)=E[XY]?", "response_mode": "single_choice", "options": { "A": "0", "B": "1/2", "C": "−1", "D": "1" }, "correct": true, "kind": "recognition", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "18.3" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 1.0, "C": 0.0 }, "batch_response_ms": 337.2207080028602, "repaired": false }, { "id": "18.4", "section": 18, "context": "Random variables have finite second moments; Var(X)>0.", "prompt": "If X,Y are independent, best affine least-squares estimator of Y given X?", "response_mode": "single_choice", "options": { "A": "E[XY]", "B": "E[X]", "C": "E[Y]", "D": "X" }, "correct": true, "kind": "recognition", "topic": "regression", "original_format": "written", "repairs": [], "source_parts": [ "18.4" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.0, "C": 1.0 }, "batch_response_ms": 337.2207080028602, "repaired": false }, { "id": "19.1", "section": 19, "context": "", "prompt": "Independent X,Y~Bin(n,p). Distribution of X+Y?", "response_mode": "single_choice", "options": { "A": "Poisson(2np)", "B": "Bin(n,2p)", "C": "Bin(2n,p)", "D": "Bin(2n,p²)" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "19.1" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "C": 1.0, "A": 0.0, "D": 0.0 }, "batch_response_ms": 289.44937500273227, "repaired": false }, { "id": "19.2", "section": 19, "context": "", "prompt": "Independent X,Y~Geom(p), counting trials from 1. Distribution of min(X,Y)?", "response_mode": "single_choice", "options": { "A": "Geom(2p−p²)", "B": "Geom(p²)", "C": "Geom(p)", "D": "Geom(2p)" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "19.2" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.97, "probabilities": { "B": 0.0, "A": 0.98, "C": 0.01, "D": 0.01 }, "batch_response_ms": 289.44937500273227, "repaired": false }, { "id": "19.3", "section": 19, "context": "", "prompt": "Fixed lambda>0 and nonnegative integer i. Limit as n->infinity of C(n,i)(lambda/n)^i(1−lambda/n)^(n−i)?", "response_mode": "single_choice", "options": { "A": "exp(−lambda)", "B": "exp(−i)*i^lambda/lambda!", "C": "lambda^i/i!", "D": "exp(−lambda)*lambda^i/i!" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "19.3" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "A": 0.0, "C": 0.0, "D": 1.0 }, "batch_response_ms": 289.44937500273227, "repaired": false }, { "id": "19.4a", "section": 19, "context": "First X~Uniform(0,1]; conditional on X=x, draw Y~Uniform[0,x].", "prompt": "Marginal density of X?", "response_mode": "single_choice", "options": { "A": "−ln(x) on (0,1], zero elsewhere", "B": "1 on (0,1], zero elsewhere", "C": "1/x on (0,1], zero elsewhere", "D": "2x on (0,1], zero elsewhere" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "19.4a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.96, "probabilities": { "B": 0.98, "A": 0.02, "C": 0.0, "D": 0.0 }, "batch_response_ms": 289.44937500273227, "repaired": false }, { "id": "19.4b", "section": 19, "context": "First X~Uniform(0,1]; conditional on X=x, draw Y~Uniform[0,x].", "prompt": "Describe the joint support region, replacing the requested shading.", "response_mode": "single_choice", "options": { "A": "0<=x,y<=1, the full square", "B": "0=0", "D": "0=a)?", "response_mode": "single_choice", "options": { "A": "[F(b)−F(a)]/F(a) for all a,b", "B": "F(b)/(1−F(a)) for all a,b", "C": "0 if b=s)?", "response_mode": "single_choice", "options": { "A": "(t−s)/(b−s)", "B": "(t−s)/(t−a)", "C": "(t−s)/(b−a)", "D": "(t−a)/(b−a)" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "20.2a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "D": 0.0, "A": 1.0, "C": 0.0 }, "batch_response_ms": 393.2557079970138, "repaired": false }, { "id": "20.2b", "section": 20, "context": "", "prompt": "A nondegenerate bounded uniform distribution is memoryless.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "choice", "repairs": [], "source_parts": [ "20.2b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 1.0, "A": 0.0 }, "batch_response_ms": 393.2557079970138, "repaired": false }, { "id": "20.3a", "section": 20, "context": "", "prompt": "X is the minimum of m independent Uniform[0,1] draws. Its density?", "response_mode": "single_choice", "options": { "A": "1/m on [0,1], zero elsewhere", "B": "(1−x)^m on [0,1], zero elsewhere", "C": "m*x^(m−1) on [0,1], zero elsewhere", "D": "m(1−x)^(m−1) on [0,1], zero elsewhere" }, "correct": true, "kind": "multi_step", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "20.3a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.0, "D": 0.99, "A": 0.0, "C": 0.01 }, "batch_response_ms": 393.2557079970138, "repaired": false }, { "id": "20.3b", "section": 20, "context": "", "prompt": "Expected minimum of m independent Uniform[0,1] draws?", "response_mode": "single_choice", "options": { "A": "1/(m+1)", "B": "m/(m+1)", "C": "1/m", "D": "1/2" }, "correct": true, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "20.3b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 1.0, "C": 0.0 }, "batch_response_ms": 393.2557079970138, "repaired": false }, { "id": "21.1a", "section": 21, "context": "States 0..29. From i, move to i+3 or i+5 modulo 30, each with probability 1/2.", "prompt": "Is this chain irreducible?", "response_mode": "single_choice", "options": { "A": "Yes", "B": "No" }, "correct": true, "kind": "recognition", "topic": "markov", "original_format": "choice", "repairs": [], "source_parts": [ "21.1a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.73, "probabilities": { "B": 0.14, "A": 0.86 }, "batch_response_ms": 344.0762909995101, "repaired": false }, { "id": "21.1b", "section": 21, "context": "States 0..29. From i, move to i+3 or i+5 modulo 30, each with probability 1/2.", "prompt": "Period of state 0?", "response_mode": "single_choice", "options": { "A": "10", "B": "1", "C": "2", "D": "6" }, "correct": false, "kind": "multi_step", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "21.1b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.36, "probabilities": { "B": 0.52, "D": 0.06, "C": 0.19, "A": 0.23 }, "batch_response_ms": 344.0762909995101, "repaired": false }, { "id": "21.1c", "section": 21, "context": "States 0..29. From i, move to i+3 or i+5 modulo 30, each with probability 1/2.", "prompt": "The uniform distribution on these states is invariant.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "markov", "original_format": "choice", "repairs": [], "source_parts": [ "21.1c" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.83, "probabilities": { "B": 0.08, "A": 0.92 }, "batch_response_ms": 344.0762909995101, "repaired": false }, { "id": "21.2a", "section": 21, "context": "Start at Evelyn. Each person chooses uniformly among outgoing neighbors: Evelyn->{Andy,Jay}; Andy->{Malia,Jay}; Jay->{Malia}; Malia->{Gavin}; Gavin->{Evelyn}. One transition per minute.", "prompt": "Expected minutes until first reaching Andy?", "response_mode": "single_choice", "options": { "A": "4", "B": "8", "C": "5", "D": "2" }, "correct": false, "kind": "multi_step", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "21.2a" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.27, "probabilities": { "B": 0.1, "D": 0.45, "C": 0.13, "A": 0.32 }, "batch_response_ms": 344.0762909995101, "repaired": false }, { "id": "21.2b", "section": 21, "context": "Start at Evelyn. Each person chooses uniformly among outgoing neighbors: Evelyn->{Andy,Jay}; Andy->{Malia,Jay}; Jay->{Malia}; Malia->{Gavin}; Gavin->{Evelyn}. One transition per minute.", "prompt": "Probability of reaching Jay before Gavin?", "response_mode": "single_choice", "options": { "A": "2/3", "B": "1/4", "C": "3/4", "D": "1/2" }, "correct": false, "kind": "multi_step", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "21.2b" ], "exam": "final_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.36, "probabilities": { "B": 0.04, "D": 0.12, "C": 0.32, "A": 0.52 }, "batch_response_ms": 344.0762909995101, "repaired": false }, { "id": "3.1", "section": 3, "context": "", "prompt": "(NOT P implies P) is equivalent to P.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.1" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.87, "probabilities": { "A": 0.07, "B": 0.93 }, "batch_response_ms": 397.77895800216356, "repaired": false }, { "id": "3.2", "section": 3, "context": "", "prompt": "[(P OR Q) AND (P implies R) AND (Q implies R)] is equivalent to R.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": false, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.2" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.16, "probabilities": { "A": 0.58, "B": 0.42 }, "batch_response_ms": 397.77895800216356, "repaired": false }, { "id": "3.3", "section": 3, "context": "", "prompt": "exists x [P(x) AND NOT Q(x)] is equivalent to forall x [NOT P(x) implies Q(x)].", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.3" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.57, "probabilities": { "A": 0.78, "B": 0.22 }, "batch_response_ms": 397.77895800216356, "repaired": false }, { "id": "3.4", "section": 3, "context": "", "prompt": "exists x [P(x) AND NOT Q(x)] is equivalent to NOT forall x [P(x) implies Q(x)].", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.4" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "A": 0.99, "B": 0.01 }, "batch_response_ms": 397.77895800216356, "repaired": false }, { "id": "3.5", "section": 3, "context": "", "prompt": "Simplify (P AND Q) OR (P AND NOT Q).", "response_mode": "single_choice", "options": { "A": "Q", "B": "P OR Q", "C": "False", "D": "P" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "written", "repairs": [], "source_parts": [ "3.5" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 1.0, "C": 0.0, "A": 0.0, "B": 0.0 }, "batch_response_ms": 397.77895800216356, "repaired": false }, { "id": "3.6", "section": 3, "context": "", "prompt": "Simplify P AND (P implies Q).", "response_mode": "single_choice", "options": { "A": "P AND Q", "B": "Q", "C": "P OR Q", "D": "P" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "written", "repairs": [], "source_parts": [ "3.6" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.97, "probabilities": { "D": 0.02, "C": 0.0, "A": 0.97, "B": 0.01 }, "batch_response_ms": 397.77895800216356, "repaired": false }, { "id": "4.1", "section": 4, "context": "", "prompt": "Which proof technique uses [P(1) AND ... AND P(k)] implies P(k+1)?", "response_mode": "single_choice", "options": { "A": "Contraposition", "B": "Strong induction", "C": "Proof by cases", "D": "Diagonalization" }, "correct": true, "kind": "recognition", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "4.1" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 1.0, "C": 0.0, "D": 0.0, "A": 0.0 }, "batch_response_ms": 351.2803330013412, "repaired": false }, { "id": "4.2", "section": 4, "context": "", "prompt": "Real x,y have (x+y)/2=a. Prove x<=a or y<=a.", "response_mode": "single_choice", "options": { "A": "If both x>a and y>a, their average would exceed a, a contradiction.", "B": "The average is always less than each input.", "C": "Since the average is a, both x and y must equal a.", "D": "Assume x<=a; this assumption proves the desired statement for every x,y." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "4.2" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "D": 0.0, "A": 1.0 }, "batch_response_ms": 351.2803330013412, "repaired": false }, { "id": "5.1", "section": 5, "context": "", "prompt": "Every graph with maximum degree at most two is planar.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "5.1" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.93, "probabilities": { "B": 0.97, "A": 0.03 }, "batch_response_ms": 301.9209169979149, "repaired": false }, { "id": "5.2", "section": 5, "context": "", "prompt": "A graph containing a vertex of degree six cannot be planar.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "5.2" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.97, "probabilities": { "B": 0.98, "A": 0.02 }, "batch_response_ms": 301.9209169979149, "repaired": false }, { "id": "5.3", "section": 5, "context": "", "prompt": "A simple cycle on n vertices is bipartite iff n is odd.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "5.3" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.01, "A": 0.99 }, "batch_response_ms": 301.9209169979149, "repaired": false }, { "id": "5.4a", "section": 5, "context": "", "prompt": "Minimum vertex colors for a d-dimensional hypercube, d>=1?", "response_mode": "single_choice", "options": { "A": "d+1", "B": "2", "C": "d", "D": "2^d" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [ "Require positive dimension to exclude a single vertex." ], "source_parts": [ "5.4a" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 1.0, "C": 0.0, "D": 0.0, "A": 0.0 }, "batch_response_ms": 301.9209169979149, "repaired": true }, { "id": "5.4b", "section": 5, "context": "", "prompt": "A connected simple planar graph with at least two edges has no triangles. Give and justify an edge bound in terms of n vertices.", "response_mode": "single_choice", "options": { "A": "e<=2n−4: every face has at least four edge-sides, so 4f<=2e; combine with n−e+f=2.", "B": "e<=2n: Euler’s formula is n+e=f+2.", "C": "e<=n−1: no triangles means there are no cycles.", "D": "e<=3n−6 and this is tight even without triangles; every face remains triangular." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "5.4b" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "D": 0.0, "A": 1.0 }, "batch_response_ms": 301.9209169979149, "repaired": false }, { "id": "5.4c", "section": 5, "context": "", "prompt": "Prove the four-dimensional hypercube is nonplanar.", "response_mode": "single_choice", "options": { "A": "It has degree four at every vertex, and any such graph is nonplanar.", "B": "It has an odd cycle, which all planar graphs forbid.", "C": "It has 16 vertices, exceeding the maximum number permitted in a planar graph.", "D": "It has 16 vertices and 32 edges and is bipartite; a triangle-free planar graph on 16 vertices has at most 28 edges." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "5.4c" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "D": 1.0, "A": 0.0 }, "batch_response_ms": 301.9209169979149, "repaired": false }, { "id": "6.1", "section": 6, "context": "", "prompt": "If all candidates rank the same job last, job-proposing deferred acceptance cannot finish in one day.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.1" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.72, "probabilities": { "A": 0.14, "B": 0.86 }, "batch_response_ms": 298.6532079994504, "repaired": false }, { "id": "6.2", "section": 6, "context": "", "prompt": "If all jobs have the same strict ranking of candidates, the stable matching is unique.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.2" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.85, "probabilities": { "A": 0.93, "B": 0.07 }, "batch_response_ms": 298.6532079994504, "repaired": false }, { "id": "6.3", "section": 6, "context": "", "prompt": "It is impossible for every candidate to get its favorite job in a job-optimal stable matching.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.3" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.3, "probabilities": { "A": 0.65, "B": 0.35 }, "batch_response_ms": 298.6532079994504, "repaired": false }, { "id": "6.4", "section": 6, "context": "", "prompt": "A job and candidate who rank each other first are paired in every stable matching.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.4" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.88, "probabilities": { "A": 0.06, "B": 0.94 }, "batch_response_ms": 298.6532079994504, "repaired": false }, { "id": "7.1a", "section": 7, "context": "", "prompt": "There exists integer x with x=1 mod 10 and x=2 mod 15.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": false, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "7.1a" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.22, "probabilities": { "A": 0.61, "B": 0.39 }, "batch_response_ms": 391.2162079977861, "repaired": false }, { "id": "7.1b", "section": 7, "context": "", "prompt": "Why are x=1 mod 10 and x=2 mod 15 inconsistent?", "response_mode": "single_choice", "options": { "A": "Their product is even, which prevents integer solutions.", "B": "Neither 10 nor 15 is prime, so no system using them has a solution.", "C": "The residues 1 and 2 must be identical as integers in any CRT system.", "D": "Reducing both modulo 5 requires x=1 and x=2 mod 5 simultaneously." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "7.1b" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "A": 0.0, "B": 0.0, "D": 1.0 }, "batch_response_ms": 391.2162079977861, "repaired": false }, { "id": "7.2a", "section": 7, "context": "", "prompt": "There exists integer x with x=6 mod 10 and x=1 mod 15.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": false, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "7.2a" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.19, "probabilities": { "A": 0.6, "B": 0.4 }, "batch_response_ms": 391.2162079977861, "repaired": false }, { "id": "7.2b", "section": 7, "context": "", "prompt": "Give a witness satisfying x=6 mod 10 and x=1 mod 15.", "response_mode": "single_choice", "options": { "A": "31", "B": "16", "C": "46+1", "D": "6" }, "correct": false, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "7.2b" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.54, "probabilities": { "D": 0.01, "A": 0.66, "B": 0.29, "C": 0.04 }, "batch_response_ms": 391.2162079977861, "repaired": false }, { "id": "7.3a", "section": 7, "context": "", "prompt": "Prove x²=(m−x)² modulo positive m.", "response_mode": "single_choice", "options": { "A": "The equality holds as integers for all x.", "B": "Both squares always have residue one.", "C": "Their difference is m²−2mx=m(m−2x), a multiple of m.", "D": "Squaring changes every residue to zero." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "7.3a" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 1.0, "A": 0.0, "B": 0.0, "D": 0.0 }, "batch_response_ms": 391.2162079977861, "repaired": false }, { "id": "7.3b", "section": 7, "context": "", "prompt": "For positive even n, prove sum(k=1..n) (−1)^k k² is divisible by n+1.", "response_mode": "single_choice", "options": { "A": "There are an even number of terms, so every such sum is divisible by n+1.", "B": "Pair adjacent terms, whose differences are all exactly n+1.", "C": "Every individual squared term is divisible by n+1.", "D": "Pair k with n+1−k. Their squares agree modulo n+1 and their signs are opposite since n is even, so every pair cancels." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "7.3b" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.95, "probabilities": { "C": 0.0, "A": 0.01, "D": 0.96, "B": 0.03 }, "batch_response_ms": 391.2162079977861, "repaired": false }, { "id": "8.1", "section": 8, "context": "", "prompt": "Over GF(5), how many formal polynomials of degree at most 3 pass through one specified point?", "response_mode": "single_choice", "options": { "A": "25", "B": "625", "C": "125", "D": "100" }, "correct": true, "kind": "calculation", "topic": "polynomials", "original_format": "written", "repairs": [ "Resolve source degree ambiguity as at most three; original grading accepted both strict and non-strict interpretations." ], "source_parts": [ "8.1" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.78, "probabilities": { "D": 0.03, "A": 0.03, "B": 0.1, "C": 0.84 }, "batch_response_ms": 340.02358299767366, "repaired": true }, { "id": "8.2a", "section": 8, "context": "A Shamir polynomial of degree at most two over GF(5) gives shares (1,3),(2,4),(4,2). Secret is P(0).", "prompt": "Find the secret.", "response_mode": "single_choice", "options": { "A": "1", "B": "4", "C": "2", "D": "3" }, "correct": false, "kind": "multi_step", "topic": "secret_sharing", "original_format": "written", "repairs": [], "source_parts": [ "8.2a" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.13, "probabilities": { "D": 0.27, "A": 0.33999999999999997, "B": 0.24, "C": 0.15 }, "batch_response_ms": 340.02358299767366, "repaired": false }, { "id": "8.2b", "section": 8, "context": "A Shamir polynomial of degree at most two over GF(5) gives shares (1,3),(2,4),(4,2). Secret is P(0).", "prompt": "What value belongs in a share at x=3?", "response_mode": "single_choice", "options": { "A": "1", "B": "2", "C": "4", "D": "3" }, "correct": false, "kind": "multi_step", "topic": "secret_sharing", "original_format": "written", "repairs": [], "source_parts": [ "8.2b" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.26, "probabilities": { "D": 0.27, "A": 0.44, "B": 0.12, "C": 0.17 }, "batch_response_ms": 340.02358299767366, "repaired": false }, { "id": "8.3a", "section": 8, "context": "", "prompt": "Eight Reed–Solomon evaluations are transmitted over GF(11), correcting up to two corrupted values. Maximum message length in field symbols?", "response_mode": "single_choice", "options": { "A": "6", "B": "2", "C": "4", "D": "8" }, "correct": true, "kind": "calculation", "topic": "coding", "original_format": "written", "repairs": [ "Ask maximum length; shorter messages also meet the stated guarantee." ], "source_parts": [ "8.3a" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.62, "probabilities": { "D": 0.0, "A": 0.27, "B": 0.01, "C": 0.72 }, "batch_response_ms": 340.02358299767366, "repaired": true }, { "id": "8.3b", "section": 8, "context": "", "prompt": "Minimal error locator is E(x)=x²−6x+8 over GF(11). Which evaluation coordinates are corrupted?", "response_mode": "single_choice", "options": { "A": "6 and 8", "B": "1 and 8", "C": "2 and 4", "D": "3 and 5" }, "correct": true, "kind": "calculation", "topic": "coding", "original_format": "written", "repairs": [ "Specify minimal locator to exclude padding roots." ], "source_parts": [ "8.3b" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.96, "probabilities": { "D": 0.01, "A": 0.01, "B": 0.01, "C": 0.97 }, "batch_response_ms": 340.02358299767366, "repaired": true }, { "id": "9.1", "section": 9, "context": "", "prompt": "Ten rounds give +30 for a win, −30 for a loss, starting at zero. Number of ordered outcomes ending at +60?", "response_mode": "single_choice", "options": { "A": "C(10,4)", "B": "2^10", "C": "C(10,2)", "D": "C(10,5)" }, "correct": true, "kind": "multi_step", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "9.1" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.32, "probabilities": { "C": 0.32, "B": 0.01, "A": 0.49, "D": 0.18 }, "batch_response_ms": 287.7190840008552, "repaired": false }, { "id": "9.2", "section": 9, "context": "", "prompt": "Ten-digit IDs allow 0..9 in every position, with no equal adjacent digits. Count?", "response_mode": "single_choice", "options": { "A": "9*10^9", "B": "10*9^9", "C": "10^10", "D": "10!" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "9.2" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 0.0, "B": 0.99, "A": 0.01, "D": 0.0 }, "batch_response_ms": 287.7190840008552, "repaired": false }, { "id": "9.3", "section": 9, "context": "", "prompt": "Distribute eight integer points into three named stats; first nonnegative, others strictly positive. Count?", "response_mode": "single_choice", "options": { "A": "3^8", "B": "C(8,2)", "C": "C(7,2)", "D": "C(10,2)" }, "correct": false, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "9.3" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.54, "probabilities": { "C": 0.18, "B": 0.05, "A": 0.11, "D": 0.66 }, "batch_response_ms": 287.7190840008552, "repaired": false }, { "id": "9.4", "section": 9, "context": "", "prompt": "Choose distinct captain and lieutenant from eleven people. Count?", "response_mode": "single_choice", "options": { "A": "121", "B": "110", "C": "55", "D": "22" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "9.4" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.97, "probabilities": { "C": 0.0, "B": 0.98, "A": 0.02, "D": 0.0 }, "batch_response_ms": 287.7190840008552, "repaired": false }, { "id": "9.5", "section": 9, "context": "", "prompt": "Seven distinct collectible items; choose a subset with at least two. Count?", "response_mode": "single_choice", "options": { "A": "120", "B": "21", "C": "127", "D": "128" }, "correct": false, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "9.5" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 1.0, "B": 0.0, "A": 0.0, "D": 0.0 }, "batch_response_ms": 287.7190840008552, "repaired": false }, { "id": "9.6", "section": 9, "context": "", "prompt": "Two islands each have n named checkpoints, n>=2. Travel only between islands. Starting at one fixed checkpoint, count directed tours visiting every checkpoint exactly once before returning.", "response_mode": "single_choice", "options": { "A": "n!(n−1)!/2", "B": "n!(n−1)!", "C": "(2n−1)!", "D": "(n!)²" }, "correct": true, "kind": "multi_step", "topic": "counting", "original_format": "written", "repairs": [ "Fix the starting checkpoint and distinguish reverse traversal, matching the source’s intended count." ], "source_parts": [ "9.6" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.57, "probabilities": { "C": 0.02, "B": 0.68, "A": 0.17, "D": 0.13 }, "batch_response_ms": 287.7190840008552, "repaired": true }, { "id": "9.7", "section": 9, "context": "", "prompt": "Number of functions from an n-element set to itself that are not bijections?", "response_mode": "single_choice", "options": { "A": "n^n−n!", "B": "n^n−n", "C": "n!−n", "D": "2^n−n!" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "9.7" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "C": 0.0, "A": 1.0, "B": 0.0 }, "batch_response_ms": 287.7190840008552, "repaired": false }, { "id": "10.1", "section": 10, "context": "", "prompt": "Functions N->{True,False} form a countable set.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "10.1" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.92, "probabilities": { "B": 0.96, "A": 0.04 }, "batch_response_ms": 396.4985419988807, "repaired": false }, { "id": "10.2", "section": 10, "context": "", "prompt": "Finite binary strings form a countable set.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "10.2" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.99, "A": 0.01 }, "batch_response_ms": 396.4985419988807, "repaired": false }, { "id": "10.3", "section": 10, "context": "", "prompt": "It is impossible to write a halting analyzer that correctly handles even some programs.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "computability", "original_format": "choice", "repairs": [], "source_parts": [ "10.3" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.63, "probabilities": { "B": 0.19, "A": 0.81 }, "batch_response_ms": 396.4985419988807, "repaired": false }, { "id": "10.4", "section": 10, "context": "", "prompt": "No always-terminating analyzer can correctly decide for all programs whether they ever print a particular fixed string.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "computability", "original_format": "choice", "repairs": [], "source_parts": [ "10.4" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.95, "probabilities": { "B": 0.97, "A": 0.03 }, "batch_response_ms": 396.4985419988807, "repaired": false }, { "id": "10.5", "section": 10, "context": "", "prompt": "It is impossible to decide for arbitrary program P and finite N whether P hits a runtime error within its first N execution steps.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "computability", "original_format": "choice", "repairs": [], "source_parts": [ "10.5" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.67, "probabilities": { "B": 0.83, "A": 0.17 }, "batch_response_ms": 396.4985419988807, "repaired": false }, { "id": "10.6", "section": 10, "context": "", "prompt": "Why can no total analyzer decide whether an arbitrary Python program ever assigns a string to a given variable? Assume unbounded theoretical computation.", "response_mode": "single_choice", "options": { "A": "A variable has no value before assignment, so no analyzer can read any program.", "B": "Finite programs contain uncountably many variable names.", "C": "Python never exposes variable types, even during execution.", "D": "Given P,input x, construct a program with a fresh variable initially 0, simulate P(x), then assign a string to that variable if simulation halts. The analyzer would decide halting." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "10.6" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 0.0, "D": 1.0 }, "batch_response_ms": 396.4985419988807, "repaired": false }, { "id": "11.1", "section": 11, "context": "A five-sided die has P(1)=1/2; values 2,3,4,5 share the remaining probability equally. A is an odd result; B is a perfect-square result.", "prompt": "P(A)?", "response_mode": "single_choice", "options": { "A": "1/2", "B": "3/5", "C": "5/8", "D": "3/4" }, "correct": false, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "11.1" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.74, "probabilities": { "A": 0.02, "C": 0.8, "B": 0.06, "D": 0.12 }, "batch_response_ms": 352.26962500019, "repaired": false }, { "id": "11.2", "section": 11, "context": "A five-sided die has P(1)=1/2; values 2,3,4,5 share the remaining probability equally. A is an odd result; B is a perfect-square result.", "prompt": "P(NOT B)?", "response_mode": "single_choice", "options": { "A": "2/5", "B": "1/2", "C": "5/8", "D": "3/8" }, "correct": false, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "11.2" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.76, "probabilities": { "A": 0.04, "C": 0.82, "B": 0.05, "D": 0.09 }, "batch_response_ms": 352.26962500019, "repaired": false }, { "id": "11.3", "section": 11, "context": "A five-sided die has P(1)=1/2; values 2,3,4,5 share the remaining probability equally. A is an odd result; B is a perfect-square result.", "prompt": "P(A union B)?", "response_mode": "single_choice", "options": { "A": "3/4", "B": "1", "C": "5/8", "D": "7/8" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "11.3" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.31, "probabilities": { "A": 0.3, "C": 0.1, "B": 0.12, "D": 0.48 }, "batch_response_ms": 352.26962500019, "repaired": false }, { "id": "11.4", "section": 11, "context": "A five-sided die has P(1)=1/2; values 2,3,4,5 share the remaining probability equally. A is an odd result; B is a perfect-square result.", "prompt": "P(A given B)?", "response_mode": "single_choice", "options": { "A": "4/5", "B": "3/4", "C": "1/2", "D": "5/8" }, "correct": false, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "11.4" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.12, "probabilities": { "A": 0.29, "C": 0.28, "B": 0.33, "D": 0.1 }, "batch_response_ms": 352.26962500019, "repaired": false }, { "id": "11.5", "section": 11, "context": "A five-sided die has P(1)=1/2; values 2,3,4,5 share the remaining probability equally. A is an odd result; B is a perfect-square result.", "prompt": "P(A intersection B given A union B)?", "response_mode": "single_choice", "options": { "A": "4/7", "B": "4/5", "C": "1/2", "D": "7/8" }, "correct": true, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "11.5" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.51, "probabilities": { "A": 0.63, "C": 0.13, "B": 0.21, "D": 0.03 }, "batch_response_ms": 352.26962500019, "repaired": false }, { "id": "11.6", "section": 11, "context": "A five-sided die has P(1)=1/2; values 2,3,4,5 share the remaining probability equally. A is an odd result; B is a perfect-square result.", "prompt": "P(A intersection NOT B given B)?", "response_mode": "single_choice", "options": { "A": "0", "B": "3/8", "C": "1", "D": "4/5" }, "correct": true, "kind": "recognition", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "11.6" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.92, "probabilities": { "A": 0.95, "C": 0.01, "B": 0.02, "D": 0.02 }, "batch_response_ms": 352.26962500019, "repaired": false }, { "id": "11.7", "section": 11, "context": "A five-sided die has P(1)=1/2; values 2,3,4,5 share the remaining probability equally. A is an odd result; B is a perfect-square result.", "prompt": "Now vary P(1)=p<1 and keep the other four outcomes equally likely. Which p makes A and B independent?", "response_mode": "single_choice", "options": { "A": "1/5", "B": "1/2", "C": "1/3", "D": "0" }, "correct": false, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "11.7" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.1, "probabilities": { "A": 0.31, "C": 0.3, "B": 0.33, "D": 0.06 }, "batch_response_ms": 352.26962500019, "repaired": false }, { "id": "12.1", "section": 12, "context": "A detector flags plagiarized submissions with probability p1 and original submissions with probability p2. Class CS1 has 10% plagiarized submissions; CS100 has 2%.", "prompt": "Must p1+p2 equal one?", "response_mode": "single_choice", "options": { "A": "No, it cannot.", "B": "Yes, always.", "C": "Depends; it can but need not." }, "correct": true, "kind": "recognition", "topic": "probability", "original_format": "choice", "repairs": [], "source_parts": [ "12.1" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.38, "probabilities": { "B": 0.01, "A": 0.4, "C": 0.59 }, "batch_response_ms": 298.2502080012637, "repaired": false }, { "id": "12.2a", "section": 12, "context": "A detector flags plagiarized submissions with probability p1 and original submissions with probability p2. Class CS1 has 10% plagiarized submissions; CS100 has 2%.", "prompt": "p1=0.8,p2=0.2. Given a CS1 submission was flagged, probability it was plagiarized?", "response_mode": "single_choice", "options": { "A": "4/5", "B": "4/13", "C": "1/4", "D": "1/10" }, "correct": true, "kind": "multi_step", "topic": "bayes", "original_format": "written", "repairs": [], "source_parts": [ "12.2a" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.7, "probabilities": { "B": 0.78, "D": 0.0, "C": 0.01, "A": 0.21 }, "batch_response_ms": 298.2502080012637, "repaired": false }, { "id": "12.2b", "section": 12, "context": "A detector flags plagiarized submissions with probability p1 and original submissions with probability p2. Class CS1 has 10% plagiarized submissions; CS100 has 2%.", "prompt": "p1=0.8,p2=0.2. Given a CS100 submission was flagged, probability it was plagiarized?", "response_mode": "single_choice", "options": { "A": "4/13", "B": "4/5", "C": "1/50", "D": "4/53" }, "correct": false, "kind": "multi_step", "topic": "bayes", "original_format": "written", "repairs": [], "source_parts": [ "12.2b" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.47, "probabilities": { "B": 0.04, "D": 0.34, "C": 0.01, "A": 0.61 }, "batch_response_ms": 298.2502080012637, "repaired": false }, { "id": "12.3", "section": 12, "context": "A detector flags plagiarized submissions with probability p1 and original submissions with probability p2. Class CS1 has 10% plagiarized submissions; CS100 has 2%.", "prompt": "Choose either class with equal probability, then a uniform submission within it. An auditor establishes plagiarism without the detector. Given plagiarism, probability the class was CS1?", "response_mode": "single_choice", "options": { "A": "1/10", "B": "1/2", "C": "5/6", "D": "4/13" }, "correct": true, "kind": "multi_step", "topic": "bayes", "original_format": "written", "repairs": [], "source_parts": [ "12.3" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.59, "probabilities": { "B": 0.11, "D": 0.18, "C": 0.69, "A": 0.02 }, "batch_response_ms": 298.2502080012637, "repaired": false }, { "id": "13.1", "section": 13, "context": "There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.", "prompt": "Probability the two California senators sit adjacent?", "response_mode": "single_choice", "options": { "A": "2/(2n)", "B": "2/(2n−1)", "C": "1/(2n−1)", "D": "1/n" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "13.1" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.46, "probabilities": { "C": 0.38, "B": 0.6, "A": 0.01, "D": 0.01 }, "batch_response_ms": 396.4044999993348, "repaired": false }, { "id": "13.2", "section": 13, "context": "There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.", "prompt": "Probability the California pair is adjacent given the New Mexico pair is adjacent?", "response_mode": "single_choice", "options": { "A": "2/(2n−1)", "B": "2/(2n−2)", "C": "1/(2n−2)", "D": "4/((2n−1)(2n−2))" }, "correct": false, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "13.2" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.25, "probabilities": { "C": 0.17, "A": 0.43, "B": 0.33, "D": 0.07 }, "batch_response_ms": 396.4044999993348, "repaired": false }, { "id": "13.3", "section": 13, "context": "There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.", "prompt": "One California senator is happy if adjacent to either of two specified other senators. Probability of happiness?", "response_mode": "single_choice", "options": { "A": "4/(2n−1)−2/((2n−1)(2n−2))", "B": "2/(2n−1)", "C": "4/(2n−1)−4/((2n−1)(2n−2))", "D": "4/(2n−1)" }, "correct": false, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "13.3" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.4, "probabilities": { "C": 0.55, "B": 0.01, "A": 0.36, "D": 0.08 }, "batch_response_ms": 396.4044999993348, "repaired": false }, { "id": "13.4a", "section": 13, "context": "There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.", "prompt": "Ni means senator i is adjacent to their state partner. P(Ni)?", "response_mode": "single_choice", "options": { "A": "1/(2n−1)", "B": "2/(2n−2)", "C": "1/n", "D": "2/(2n−1)" }, "correct": true, "kind": "calculation", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "13.4a" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.7, "probabilities": { "C": 0.01, "A": 0.21, "B": 0.01, "D": 0.77 }, "batch_response_ms": 396.4044999993348, "repaired": false }, { "id": "13.4b", "section": 13, "context": "There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.", "prompt": "Are the events Ni pairwise independent over all senators?", "response_mode": "single_choice", "options": { "A": "Yes", "B": "No" }, "correct": true, "kind": "recognition", "topic": "probability", "original_format": "choice", "repairs": [], "source_parts": [ "13.4b" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.63, "probabilities": { "A": 0.18, "B": 0.82 }, "batch_response_ms": 396.4044999993348, "repaired": false }, { "id": "13.4c", "section": 13, "context": "There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.", "prompt": "Why are the Ni not pairwise independent?", "response_mode": "single_choice", "options": { "A": "Every Ni has probability one.", "B": "For the two senators of one state, their events are identical with probability strictly between zero and one; conditioning on one makes the other certain.", "C": "Events on a finite sample space can never be independent.", "D": "All adjacency events are mutually exclusive." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "13.4c" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 0.0, "A": 0.0, "B": 0.99, "D": 0.01 }, "batch_response_ms": 396.4044999993348, "repaired": false }, { "id": "13.4d", "section": 13, "context": "There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.", "prompt": "Expected number of senators adjacent to their state partner?", "response_mode": "single_choice", "options": { "A": "4n/(2n−2)", "B": "4n/(2n−1)", "C": "2", "D": "2n/(2n−1)" }, "correct": false, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "13.4d" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.37, "probabilities": { "C": 0.05, "A": 0.03, "B": 0.39, "D": 0.53 }, "batch_response_ms": 396.4044999993348, "repaired": false }, { "id": "14.1", "section": 14, "context": "", "prompt": "Pairwise independent Xi have finite second moments. An attempted induction sets Y=X1+...+X(n−1), then applies a lemma requiring independent Y and Xn to add their variances. What is wrong?", "response_mode": "single_choice", "options": { "A": "Pairwise independence of the Xi does not ensure independence of Y and Xn.", "B": "Variance never distributes over independent sums.", "C": "The sum Y is not a random variable.", "D": "Pairwise independence forbids defining a sum." }, "correct": true, "kind": "recognition", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "14.1" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 1.0, "C": 0.0, "B": 0.0, "D": 0.0 }, "batch_response_ms": 347.5739169989538, "repaired": false }, { "id": "14.2", "section": 14, "context": "", "prompt": "Prove variance of a sum of pairwise independent finite-variance variables equals the sum of variances.", "response_mode": "single_choice", "options": { "A": "Pairwise independence implies mutual independence, so group arbitrary sums and use independence recursively.", "B": "Expand Var(sum Xi)=sum Var(Xi)+2sum(i=0. Give (E[X],E[Y],Var(Y),Cov(X,Y)).", "response_mode": "single_choice", "options": { "A": "(9/7,9/7,52/49,−52/49)", "B": "(0,9/7,19/7,0)", "C": "(0,3/2,5/4,0)", "D": "(0,9/7,52/49,0)" }, "correct": true, "kind": "multi_step", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "15.1d.EX", "15.1d.EY", "15.1d.VarY", "15.1d.Cov" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.79, "probabilities": { "D": 0.85, "C": 0.01, "B": 0.12, "A": 0.02 }, "batch_response_ms": 328.6960830009775, "repaired": false }, { "id": "15.1e", "section": 15, "context": "Choose uniformly from S={(x,y) in Z²: |x|+|y|=3}, the twelve integer points on the perimeter of the diamond with vertices (0,3),(3,0),(0,−3),(−3,0).", "prompt": "Instead choose uniformly from the four points in S with x,y>=0. Give (E[X],E[Y],Var(Y),Cov(X,Y)).", "response_mode": "single_choice", "options": { "A": "(1,1,2,−2)", "B": "(3/2,3/2,5/4,−5/4)", "C": "(3/2,3/2,5/4,0)", "D": "(3/2,3/2,5/4,5/4)" }, "correct": true, "kind": "multi_step", "topic": "moments", "original_format": "written", "repairs": [], "source_parts": [ "15.1e.EX", "15.1e.EY", "15.1e.VarY", "15.1e.Cov" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.96, "probabilities": { "D": 0.01, "C": 0.01, "B": 0.97, "A": 0.01 }, "batch_response_ms": 328.6960830009775, "repaired": false }, { "id": "16.1a", "section": 16, "context": "Independent X~Bin(n,p),Y~Bin(m,p).", "prompt": "P(X+Y=k)?", "response_mode": "single_choice", "options": { "A": "C(n+m,k)p^k(1−p)^(n+m−k)", "B": "C(n,k)C(m,k)p^(2k)(1−p)^(n+m−2k)", "C": "C(n+m,k)p^(n+m−k)(1−p)^k", "D": "p^k(1−p)^(n+m−k)" }, "correct": true, "kind": "calculation", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "16.1a" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 1.0, "B": 0.0, "C": 0.0, "D": 0.0 }, "batch_response_ms": 379.76145799984806, "repaired": false }, { "id": "16.1b", "section": 16, "context": "Independent X~Bin(n,p),Y~Bin(m,p).", "prompt": "Distribution of X+Y?", "response_mode": "single_choice", "options": { "A": "Bin(n+m,p)", "B": "Poisson((n+m)p)", "C": "Bin(nm,p)", "D": "Bin(n+m,2p)" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [], "source_parts": [ "16.1b" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 1.0, "B": 0.0, "C": 0.0, "D": 0.0 }, "batch_response_ms": 379.76145799984806, "repaired": false }, { "id": "16.1c", "section": 16, "context": "Independent X~Bin(n,p),Y~Bin(m,p).", "prompt": "Explain the distribution of X+Y without a probability formula.", "response_mode": "single_choice", "options": { "A": "The sum of any two distributions in the same family stays in that family.", "B": "Combine n and m independent Bernoulli trials with the same success probability p; their total counts successes in n+m such trials.", "C": "Independent random variables always sum to a Poisson variable.", "D": "Adding the trial counts also doubles the success probability." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "16.1c" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 0.0, "B": 1.0, "C": 0.0, "D": 0.0 }, "batch_response_ms": 379.76145799984806, "repaired": false }, { "id": "16.2a", "section": 16, "context": "", "prompt": "X counts heads in 100 independent fair flips. Chebyshev bound for P(X<=10 or X>=90)?", "response_mode": "single_choice", "options": { "A": "1/16", "B": "1/8", "C": "1/64", "D": "1/40" }, "correct": false, "kind": "multi_step", "topic": "bounds", "original_format": "written", "repairs": [], "source_parts": [ "16.2a" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.29, "probabilities": { "A": 0.47, "B": 0.13, "C": 0.32, "D": 0.08 }, "batch_response_ms": 379.76145799984806, "repaired": false }, { "id": "16.2b", "section": 16, "context": "", "prompt": "For X counting heads in 100 independent fair flips, P(X<=10)=P(X>=90).", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "probability", "original_format": "choice", "repairs": [], "source_parts": [ "16.2b" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.94, "probabilities": { "A": 0.97, "B": 0.03 }, "batch_response_ms": 379.76145799984806, "repaired": false }, { "id": "16.2c", "section": 16, "context": "", "prompt": "X counts heads in 100 independent fair flips. Use a normal approximation without continuity correction for P(X<=10 or X>=90), with Z~N(0,1).", "response_mode": "single_choice", "options": { "A": "2P(Z>=4)", "B": "2P(Z>=8)", "C": "2P(Z>=40)", "D": "P(Z>=8)" }, "correct": true, "kind": "multi_step", "topic": "bounds", "original_format": "written", "repairs": [ "Explicitly request no continuity correction; this is a tail approximation, not an exact probability." ], "source_parts": [ "16.2c" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.83, "probabilities": { "A": 0.11, "B": 0.87, "C": 0.01, "D": 0.01 }, "batch_response_ms": 379.76145799984806, "repaired": true }, { "id": "17.1", "section": 17, "context": "Pay $8 to roll two independent fair six-sided dice. Receive their final sum. A reroll, if allowed, is independent and replaces that die; each die can be rerolled at most once. Net winnings subtract entry and reroll costs.", "prompt": "Expected net winnings with no rerolls?", "response_mode": "single_choice", "options": { "A": "7", "B": "−1", "C": "−2", "D": "1" }, "correct": true, "kind": "calculation", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "17.1" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.64, "probabilities": { "D": 0.12, "A": 0.02, "C": 0.13, "B": 0.73 }, "batch_response_ms": 332.8514999993786, "repaired": false }, { "id": "17.2", "section": 17, "context": "Pay $8 to roll two independent fair six-sided dice. Receive their final sum. A reroll, if allowed, is independent and replaces that die; each die can be rerolled at most once. Net winnings subtract entry and reroll costs.", "prompt": "Reroll each initial 1 for free. Expected net winnings?", "response_mode": "single_choice", "options": { "A": "−1/6", "B": "−1", "C": "1/6", "D": "5/6" }, "correct": false, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "17.2" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.19, "probabilities": { "D": 0.29, "A": 0.21, "C": 0.4, "B": 0.1 }, "batch_response_ms": 332.8514999993786, "repaired": false }, { "id": "17.3", "section": 17, "context": "Pay $8 to roll two independent fair six-sided dice. Receive their final sum. A reroll, if allowed, is independent and replaces that die; each die can be rerolled at most once. Net winnings subtract entry and reroll costs.", "prompt": "Choose integer threshold c in advance and reroll each die <=c for free. Give optimal c and expected net winnings.", "response_mode": "single_choice", "options": { "A": "(2,1/3)", "B": "(4,2/3)", "C": "(3,3/2)", "D": "(3,1/2)" }, "correct": true, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "17.3.c", "17.3.expected" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.12, "probabilities": { "D": 0.34, "C": 0.33, "A": 0.1, "B": 0.23 }, "batch_response_ms": 332.8514999993786, "repaired": false }, { "id": "17.4", "section": 17, "context": "Pay $8 to roll two independent fair six-sided dice. Receive their final sum. A reroll, if allowed, is independent and replaces that die; each die can be rerolled at most once. Net winnings subtract entry and reroll costs.", "prompt": "Each reroll costs $1. Give optimal integer threshold c and expected net winnings.", "response_mode": "single_choice", "options": { "A": "(1,−1/6)", "B": "(3,−1/2)", "C": "(2,−1/3)", "D": "(2,1/3)" }, "correct": true, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "17.4.c", "17.4.expected" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.34, "probabilities": { "D": 0.17, "C": 0.51, "A": 0.18, "B": 0.14 }, "batch_response_ms": 332.8514999993786, "repaired": false }, { "id": "17.5", "section": 17, "context": "Pay $8 to roll two independent fair six-sided dice. Receive their final sum. A reroll, if allowed, is independent and replaces that die; each die can be rerolled at most once. Net winnings subtract entry and reroll costs.", "prompt": "Smallest integer reroll fee d such that any reroll strictly decreases conditional expected net winnings?", "response_mode": "single_choice", "options": { "A": "4", "B": "2", "C": "3", "D": "6" }, "correct": false, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "17.5" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.18, "probabilities": { "D": 0.08, "A": 0.24, "C": 0.29, "B": 0.39 }, "batch_response_ms": 332.8514999993786, "repaired": false }, { "id": "18.1a", "section": 18, "context": "Independent T1~Exp(rate lambda1),T2~Exp(rate lambda2), both rates positive. R=min(T1,T2).", "prompt": "PDF fR(x) for x>=0?", "response_mode": "single_choice", "options": { "A": "exp(−(lambda1+lambda2)x)", "B": "lambda1*lambda2 exp(−(lambda1+lambda2)x)", "C": "(lambda1+lambda2)exp(−(lambda1+lambda2)x)", "D": "lambda1 exp(−lambda1*x)+lambda2 exp(−lambda2*x)" }, "correct": true, "kind": "recognition", "topic": "distributions", "original_format": "written", "repairs": [ "State independence, required for the source formulas." ], "source_parts": [ "18.1a" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "A": 0.0, "D": 0.0, "C": 1.0 }, "batch_response_ms": 309.09583299944643, "repaired": true }, { "id": "18.1b", "section": 18, "context": "Independent T1~Exp(rate lambda1),T2~Exp(rate lambda2), both rates positive. R=min(T1,T2).", "prompt": "E[R]?", "response_mode": "single_choice", "options": { "A": "1/(lambda1+lambda2)", "B": "1/lambda1+1/lambda2", "C": "min(1/lambda1,1/lambda2)", "D": "1/(lambda1*lambda2)" }, "correct": true, "kind": "recognition", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "18.1b" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "A": 1.0, "D": 0.0, "C": 0.0 }, "batch_response_ms": 309.09583299944643, "repaired": false }, { "id": "18.1c", "section": 18, "context": "Independent T1~Exp(rate lambda1),T2~Exp(rate lambda2), both rates positive. R=min(T1,T2).", "prompt": "Choose an integral for P(T1<=T2).", "response_mode": "single_choice", "options": { "A": "integral(y=0..infinity) integral(x=0..y) lambda1*lambda2*exp(−lambda1*x−lambda2*y) dx dy", "B": "integral(x=0..infinity) exp(−(lambda1+lambda2)*x) dx", "C": "integral(x=0..infinity) lambda1*lambda2*exp(−(lambda1+lambda2)*x) dx", "D": "integral(y=0..infinity) integral(x=y..infinity) lambda1*lambda2*exp(−lambda1*x−lambda2*y) dx dy" }, "correct": true, "kind": "recognition", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "18.1c" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.87, "probabilities": { "B": 0.01, "A": 0.91, "D": 0.06, "C": 0.02 }, "batch_response_ms": 309.09583299944643, "repaired": false }, { "id": "18.1d", "section": 18, "context": "Independent T1~Exp(rate lambda1),T2~Exp(rate lambda2), both rates positive. R=min(T1,T2).", "prompt": "Evaluate P(T1<=T2).", "response_mode": "single_choice", "options": { "A": "1/2", "B": "lambda1*lambda2/(lambda1+lambda2)", "C": "lambda2/(lambda1+lambda2)", "D": "lambda1/(lambda1+lambda2)" }, "correct": true, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "18.1d" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.95, "probabilities": { "B": 0.0, "A": 0.0, "D": 0.96, "C": 0.04 }, "batch_response_ms": 309.09583299944643, "repaired": false }, { "id": "18.2a", "section": 18, "context": "Independent T1,T2~Uniform[0,10], S=min(T1,T2).", "prompt": "PDF of S for 0<=x<=10?", "response_mode": "single_choice", "options": { "A": "(10−x)/50", "B": "(10−x)²/100", "C": "x/50", "D": "1/10" }, "correct": true, "kind": "multi_step", "topic": "distributions", "original_format": "written", "repairs": [ "State independence explicitly." ], "source_parts": [ "18.2a" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.0, "A": 0.99, "D": 0.0, "C": 0.01 }, "batch_response_ms": 309.09583299944643, "repaired": true }, { "id": "18.2b", "section": 18, "context": "Independent T1,T2~Uniform[0,10], S=min(T1,T2).", "prompt": "E[S]?", "response_mode": "single_choice", "options": { "A": "20/3", "B": "5", "C": "5/2", "D": "10/3" }, "correct": true, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "18.2b" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.96, "probabilities": { "B": 0.0, "A": 0.03, "D": 0.97, "C": 0.0 }, "batch_response_ms": 309.09583299944643, "repaired": false }, { "id": "18.2c", "section": 18, "context": "Independent T1,T2~Uniform[0,10], S=min(T1,T2).", "prompt": "P(|T1−T2|<=1)?", "response_mode": "single_choice", "options": { "A": "1/10", "B": "9/100", "C": "1/5", "D": "19/100" }, "correct": true, "kind": "multi_step", "topic": "probability", "original_format": "written", "repairs": [], "source_parts": [ "18.2c" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.86, "probabilities": { "B": 0.01, "A": 0.0, "D": 0.9, "C": 0.09 }, "batch_response_ms": 309.09583299944643, "repaired": false }, { "id": "19.1", "section": 19, "context": "", "prompt": "X~Exp(rate lambda>0), Y=min(X,2/lambda). Choose a derivation of E[Y].", "response_mode": "single_choice", "options": { "A": "Integrate x*lambda*exp(−lambda*x) from 0 to 2/lambda and add (2/lambda)P(X>2/lambda); the result is (1−exp(−2))/lambda.", "B": "Discard X values above 2/lambda without assigning their clamped mass; the result is (1−3exp(−2))/lambda.", "C": "Clamping at twice the mean leaves the mean unchanged at 1/lambda.", "D": "The clamp makes Y uniform on [0,2/lambda], so E[Y]=1/lambda." }, "correct": true, "kind": "multi_step", "topic": "expectation", "original_format": "written", "repairs": [], "source_parts": [ "19.1" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.95, "probabilities": { "B": 0.04, "C": 0.0, "D": 0.0, "A": 0.96 }, "batch_response_ms": 398.3535000006668, "repaired": false }, { "id": "20.1", "section": 20, "context": "Markov states 0,1,2,3. Transition rows (destination order 0,1,2,3) are [1/2,1/4,1/4,0], [1/2,1/4,1/4,0], [1/2,0,1/4,1/4], [0,1/4,0,3/4]. State 0 is studying, 3 sleeping.", "prompt": "This chain is irreducible.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": false, "kind": "recognition", "topic": "markov", "original_format": "choice", "repairs": [], "source_parts": [ "20.1" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.01, "probabilities": { "A": 0.5, "B": 0.5 }, "batch_response_ms": 368.52491599711357, "repaired": false }, { "id": "20.2", "section": 20, "context": "Markov states 0,1,2,3. Transition rows (destination order 0,1,2,3) are [1/2,1/4,1/4,0], [1/2,1/4,1/4,0], [1/2,0,1/4,1/4], [0,1/4,0,3/4]. State 0 is studying, 3 sleeping.", "prompt": "Period of the chain?", "response_mode": "single_choice", "options": { "A": "3", "B": "2", "C": "4", "D": "1" }, "correct": true, "kind": "recognition", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "20.2" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.81, "probabilities": { "C": 0.01, "D": 0.86, "A": 0.01, "B": 0.12 }, "batch_response_ms": 368.52491599711357, "repaired": false }, { "id": "20.3", "section": 20, "context": "Markov states 0,1,2,3. Transition rows (destination order 0,1,2,3) are [1/2,1/4,1/4,0], [1/2,1/4,1/4,0], [1/2,0,1/4,1/4], [0,1/4,0,3/4]. State 0 is studying, 3 sleeping.", "prompt": "Why does the distribution converge to a unique stationary distribution from every initial state?", "response_mode": "single_choice", "options": { "A": "The rows sum to one, which alone guarantees convergence.", "B": "State 3 is absorbing and hence gives the unique stationary mass.", "C": "The finite chain is irreducible and aperiodic; self-loops give period one.", "D": "All finite chains converge uniquely, even if reducible or periodic." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "20.3" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 1.0, "D": 0.0, "A": 0.0, "B": 0.0 }, "batch_response_ms": 368.52491599711357, "repaired": false }, { "id": "20.4", "section": 20, "context": "Markov states 0,1,2,3. Transition rows (destination order 0,1,2,3) are [1/2,1/4,1/4,0], [1/2,1/4,1/4,0], [1/2,0,1/4,1/4], [0,1/4,0,3/4]. State 0 is studying, 3 sleeping.", "prompt": "Choose stationary equations, plus pi0+pi1+pi2+pi3=1.", "response_mode": "single_choice", "options": { "A": "pi0=pi1=pi2=pi3=1/4", "B": "pi0=(pi0+pi1+pi2)/2; pi1=(pi0+pi1+pi3)/4; pi2=(pi0+pi1+pi2)/4; pi3=pi2/4+3pi3/4", "C": "pi0=(pi0+pi1)/2; pi1=(pi0+pi1)/4; pi2=pi2/4; pi3=3pi3/4", "D": "pi0=pi0/2+pi1/4+pi2/4; pi1=pi0/2+pi1/4+pi2/4; pi2=pi0/2+pi2/4+pi3/4; pi3=pi1/4+3pi3/4" }, "correct": true, "kind": "recognition", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "20.4" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.96, "probabilities": { "C": 0.0, "A": 0.01, "D": 0.02, "B": 0.97 }, "batch_response_ms": 368.52491599711357, "repaired": false }, { "id": "20.5", "section": 20, "context": "Markov states 0,1,2,3. Transition rows (destination order 0,1,2,3) are [1/2,1/4,1/4,0], [1/2,1/4,1/4,0], [1/2,0,1/4,1/4], [0,1/4,0,3/4]. State 0 is studying, 3 sleeping.", "prompt": "Expected steps to first reach state 3, starting at 0?", "response_mode": "single_choice", "options": { "A": "4", "B": "8", "C": "16", "D": "12" }, "correct": false, "kind": "multi_step", "topic": "markov", "original_format": "written", "repairs": [], "source_parts": [ "20.5" ], "exam": "final_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.29, "probabilities": { "C": 0.2, "A": 0.21, "D": 0.13, "B": 0.45999999999999996 }, "batch_response_ms": 368.52491599711357, "repaired": false }, { "id": "3.1a", "section": 3, "context": "", "prompt": "P implies Q is equivalent to NOT P implies NOT Q.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.1a" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.97, "probabilities": { "B": 0.98, "A": 0.02 }, "batch_response_ms": 356.62395800318336, "repaired": false }, { "id": "3.1b", "section": 3, "context": "", "prompt": "Complete the contrapositive: (P AND Q) implies R is equivalent to NOT R implies __.", "response_mode": "single_choice", "options": { "A": "NOT P OR NOT Q", "B": "NOT P AND NOT Q", "C": "P OR Q", "D": "P AND Q" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "written", "repairs": [], "source_parts": [ "3.1b" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "D": 0.0, "A": 1.0 }, "batch_response_ms": 356.62395800318336, "repaired": false }, { "id": "3.2", "section": 3, "context": "", "prompt": "NOT exists integer x [P(x) OR Q(x)] is equivalent to forall integer x [NOT P(x) AND NOT Q(x)].", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.2" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.99, "A": 0.01 }, "batch_response_ms": 356.62395800318336, "repaired": false }, { "id": "3.3", "section": 3, "context": "", "prompt": "NOT forall natural x exists natural y [P(x) AND Q(x,y)] is equivalent to exists natural x forall natural y [NOT P(x) AND NOT Q(x,y)].", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.3" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.99, "A": 0.01 }, "batch_response_ms": 356.62395800318336, "repaired": false }, { "id": "3.4", "section": 3, "context": "", "prompt": "For all natural a and positive integer b, a/b is not sqrt(2).", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "number_theory", "original_format": "choice", "repairs": [], "source_parts": [ "3.4" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.97, "probabilities": { "B": 0.01, "A": 0.99 }, "batch_response_ms": 356.62395800318336, "repaired": false }, { "id": "3.5", "section": 3, "context": "", "prompt": "Every natural n is either a perfect square or has irrational square root.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "number_theory", "original_format": "choice", "repairs": [], "source_parts": [ "3.5" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.06, "probabilities": { "B": 0.47, "A": 0.53 }, "batch_response_ms": 356.62395800318336, "repaired": false }, { "id": "4.1", "section": 4, "context": "", "prompt": "For integers a,b,c, does a|b and a|c imply a|(b+5c)? Justify.", "response_mode": "single_choice", "options": { "A": "No: choose a=2,b=4,c=6.", "B": "No: multiplying c by 5 destroys divisibility.", "C": "Yes: b=ak,c=al imply b+5c=a(k+5l).", "D": "Yes: b+5c equals a whenever both are divisible by a." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "4.1" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.0, "C": 1.0 }, "batch_response_ms": 363.1284589973802, "repaired": false }, { "id": "4.2", "section": 4, "context": "", "prompt": "Does b²=n imply n divides b for all positive integers b,n?", "response_mode": "single_choice", "options": { "A": "Yes: any square divides its square root.", "B": "Yes: take square roots of the divisibility n|b².", "C": "No: b=1,n=1 is a counterexample.", "D": "No: b=2,n=4 is a counterexample." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "4.2" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.87, "probabilities": { "D": 0.9, "B": 0.02, "A": 0.07, "C": 0.01 }, "batch_response_ms": 363.1284589973802, "repaired": false }, { "id": "4.3", "section": 4, "context": "", "prompt": "Why must a positive nonsquare integer have an odd exponent in its prime factorization?", "response_mode": "single_choice", "options": { "A": "Every nonsquare integer is prime and has exponent one.", "B": "An even exponent would force the integer itself to be even.", "C": "All prime factors of a nonsquare must have odd exponents.", "D": "If all exponents were even, halving them gives an integer whose square is the original number, a contradiction." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "4.3" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 363.1284589973802, "repaired": false }, { "id": "5.1", "section": 5, "context": "", "prompt": "x1=1; xn=sqrt(1+x(n−1)) for n>=2. Prove xn<2.", "response_mode": "single_choice", "options": { "A": "Every term equals 1, so the result is immediate.", "B": "Base x1=1<2. If 0<=xk<2, then x(k+1)=sqrt(1+xk)=1.", "response_mode": "single_choice", "options": { "A": "A1=35; A(k+1)=2Ak+7*3^(2k+1), so the induction preserves divisibility.", "B": "A1=35 and A2=259, so all remaining cases follow without an induction step.", "C": "A1=35; A(k+1)=9Ak, proving the induction.", "D": "Both summands are separately multiples of seven." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "5.2" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.99, "C": 0.01 }, "batch_response_ms": 349.32162500263075, "repaired": false }, { "id": "6.1", "section": 6, "context": "", "prompt": "A pair present in both proposer-optimal matchings is present in every stable matching.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.1" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.55, "probabilities": { "B": 0.78, "A": 0.22 }, "batch_response_ms": 315.8796669995354, "repaired": false }, { "id": "6.2", "section": 6, "context": "", "prompt": "If some pair appears in both proposer-optimal matchings, then the entire stable matching is unique.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.2" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.44, "probabilities": { "B": 0.28, "A": 0.72 }, "batch_response_ms": 315.8796669995354, "repaired": false }, { "id": "6.3", "section": 6, "context": "", "prompt": "With two jobs and two candidates, every stable matching is optimal for one proposer side or both.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.3" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.9, "probabilities": { "B": 0.05, "A": 0.95 }, "batch_response_ms": 315.8796669995354, "repaired": false }, { "id": "6.4", "section": 6, "context": "", "prompt": "There exist instances where a candidate rejecting the less-preferred-of-two rule in job-proposing deferred acceptance can obtain a better final partner.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": false, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.4" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.06, "probabilities": { "B": 0.53, "A": 0.47 }, "batch_response_ms": 315.8796669995354, "repaired": false }, { "id": "7.1", "section": 7, "context": "", "prompt": "Smallest N such that every n-dimensional hypercube with n>=N has an even edge count?", "response_mode": "single_choice", "options": { "A": "1", "B": "3", "C": "2", "D": "4" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.1" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.12, "probabilities": { "D": 0.11, "A": 0.28, "C": 0.34, "B": 0.27 }, "batch_response_ms": 394.22216700040735, "repaired": false }, { "id": "7.2", "section": 7, "context": "", "prompt": "Kn has an even number of edges for every n>3.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": false, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "7.2" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.09, "probabilities": { "A": 0.55, "B": 0.45 }, "batch_response_ms": 394.22216700040735, "repaired": false }, { "id": "7.3a", "section": 7, "context": "", "prompt": "In Kn, how many edges cross a cut with k vertices on one side?", "response_mode": "single_choice", "options": { "A": "n−1", "B": "C(k,2)", "C": "k(n−k)", "D": "k*n" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.3a" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "D": 0.0, "A": 0.0, "C": 1.0, "B": 0.0 }, "batch_response_ms": 394.22216700040735, "repaired": false }, { "id": "7.3b", "section": 7, "context": "", "prompt": "Minimum edges crossing a nonempty proper cut of an n-vertex tree, n>=2?", "response_mode": "single_choice", "options": { "A": "n−1", "B": "0", "C": "2", "D": "1" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.3b" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 1.0, "A": 0.0, "C": 0.0, "B": 0.0 }, "batch_response_ms": 394.22216700040735, "repaired": false }, { "id": "7.3c", "section": 7, "context": "", "prompt": "Maximum edges crossing a cut of an n-vertex tree?", "response_mode": "single_choice", "options": { "A": "n", "B": "1", "C": "n−1", "D": "floor(n/2)" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.3c" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.27, "probabilities": { "D": 0.4, "A": 0.01, "C": 0.45, "B": 0.14 }, "batch_response_ms": 394.22216700040735, "repaired": false }, { "id": "7.4", "section": 7, "context": "", "prompt": "Adding a new edge either reduces the component count or creates a cycle containing that edge.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "7.4" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "A": 0.99, "B": 0.01 }, "batch_response_ms": 394.22216700040735, "repaired": false }, { "id": "7.5", "section": 7, "context": "", "prompt": "Minimum possible component count with n>=1 vertices and e edges?", "response_mode": "single_choice", "options": { "A": "n−1", "B": "e−n+1", "C": "n−e", "D": "max(n−e,1)" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.5" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.93, "probabilities": { "D": 0.95, "A": 0.0, "C": 0.05, "B": 0.0 }, "batch_response_ms": 394.22216700040735, "repaired": false }, { "id": "7.6", "section": 7, "context": "", "prompt": "Smallest edge-count threshold guaranteeing a cycle in a simple n-vertex graph?", "response_mode": "single_choice", "options": { "A": "n", "B": "2n", "C": "n+1", "D": "n−1" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.6" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.94, "probabilities": { "D": 0.01, "A": 0.95, "C": 0.04, "B": 0.0 }, "batch_response_ms": 394.22216700040735, "repaired": false }, { "id": "8.1a", "section": 8, "context": "", "prompt": "The n-dimensional hypercube is edge-colorable with n colors.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "8.1a" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.97, "probabilities": { "B": 0.02, "A": 0.98 }, "batch_response_ms": 292.3393749988463, "repaired": false }, { "id": "8.1b", "section": 8, "context": "", "prompt": "Every graph of maximum degree d is edge-colorable with d colors.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "8.1b" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.92, "probabilities": { "B": 0.96, "A": 0.04 }, "batch_response_ms": 292.3393749988463, "repaired": false }, { "id": "8.1c", "section": 8, "context": "", "prompt": "Every bipartite graph is edge-colorable with two colors.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "8.1c" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.13, "probabilities": { "B": 0.57, "A": 0.43 }, "batch_response_ms": 292.3393749988463, "repaired": false }, { "id": "8.1di", "section": 8, "context": "A bipartite graph on n>2 vertices is d-regular, and its edges partition into Hamiltonian cycles.", "prompt": "Number of edges?", "response_mode": "single_choice", "options": { "A": "nd", "B": "d/2", "C": "nd/2", "D": "n+d" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "8.1di" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.42, "probabilities": { "B": 0.0, "C": 0.57, "D": 0.0, "A": 0.43 }, "batch_response_ms": 292.3393749988463, "repaired": false }, { "id": "8.1dii", "section": 8, "context": "A bipartite graph on n>2 vertices is d-regular, and its edges partition into Hamiltonian cycles.", "prompt": "Number of cycles in this edge partition?", "response_mode": "single_choice", "options": { "A": "nd/2", "B": "d/2", "C": "d", "D": "n/2" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "8.1dii" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.92, "probabilities": { "B": 0.94, "C": 0.05, "A": 0.01, "D": 0.0 }, "batch_response_ms": 292.3393749988463, "repaired": false }, { "id": "8.1diii", "section": 8, "context": "A bipartite graph on n>2 vertices is d-regular, and its edges partition into Hamiltonian cycles.", "prompt": "Why can the graph be edge-colored with d colors?", "response_mode": "single_choice", "options": { "A": "Every d-regular graph automatically has a d-edge-coloring regardless of bipartiteness.", "B": "Each cycle gets one color, since its edges do not share endpoints.", "C": "Use the two vertex colors as edge colors for all cycles.", "D": "Each Hamiltonian cycle is even because the graph is bipartite, so alternate two colors on it; use a disjoint color pair for each of the d/2 cycles." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "8.1diii" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.95, "probabilities": { "B": 0.03, "C": 0.0, "A": 0.0, "D": 0.97 }, "batch_response_ms": 292.3393749988463, "repaired": false }, { "id": "8.2", "section": 8, "context": "", "prompt": "In a d-regular bipartite graph, d>0, prove every S on the left satisfies |N(S)|>=|S|.", "response_mode": "single_choice", "options": { "A": "Exactly d|S| edges leave S and at most d|N(S)| can meet its neighbors, so d|S|<=d|N(S)|; divide by d.", "B": "All neighbors of different left vertices are distinct, so each adds d new vertices.", "C": "Each left vertex has at least one neighbor, which alone ensures Hall’s condition.", "D": "The graph is bipartite, so |N(S)|=|S| for every subset regardless of degree." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "8.2" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "A": 1.0, "D": 0.0 }, "batch_response_ms": 292.3393749988463, "repaired": false }, { "id": "9.1", "section": 9, "context": "", "prompt": "Compute 7^50 modulo 35.", "response_mode": "single_choice", "options": { "A": "14", "B": "1", "C": "7", "D": "0" }, "correct": false, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "9.1" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.72, "probabilities": { "D": 0.79, "A": 0.11, "C": 0.1, "B": 0.0 }, "batch_response_ms": 499.06333300168626, "repaired": false }, { "id": "9.2a", "section": 9, "context": "", "prompt": "Prime q, N=q². How many residues modulo N are coprime to N?", "response_mode": "single_choice", "options": { "A": "q", "B": "q²−q", "C": "q²−1", "D": "q−1" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "9.2a" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 0.0, "A": 0.0, "D": 0.0, "B": 1.0 }, "batch_response_ms": 499.06333300168626, "repaired": false }, { "id": "9.2b", "section": 9, "context": "", "prompt": "If q is prime and gcd(a,q)=1, then a has a multiplicative inverse modulo q².", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "9.2b" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.91, "probabilities": { "A": 0.04, "B": 0.96 }, "batch_response_ms": 499.06333300168626, "repaired": false }, { "id": "9.2c", "section": 9, "context": "", "prompt": "Prime q, N=q², gcd(a,N)=1. Besides exponent 1, choose an exponent x>1 guaranteed to satisfy a^x=a mod N for every such a.", "response_mode": "single_choice", "options": { "A": "q", "B": "q−1", "C": "q(q−1)+1", "D": "q²" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [ "Use an explicit exponent >1 and the correct unit-group cardinality." ], "source_parts": [ "9.2c" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.93, "probabilities": { "C": 0.94, "A": 0.04, "D": 0.02, "B": 0.0 }, "batch_response_ms": 499.06333300168626, "repaired": true }, { "id": "9.3", "section": 9, "context": "", "prompt": "For all moduli n, ab=0 mod n implies a=0 mod n or b=0 mod n.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "9.3" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "A": 1.0, "B": 0.0 }, "batch_response_ms": 499.06333300168626, "repaired": false }, { "id": "9.4a", "section": 9, "context": "", "prompt": "If m and n are integer linear combinations of x,y, every integer linear combination of m,n is also an integer linear combination of x,y.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "number_theory", "original_format": "choice", "repairs": [], "source_parts": [ "9.4a" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "A": 0.0, "B": 1.0 }, "batch_response_ms": 499.06333300168626, "repaired": false }, { "id": "9.4b", "section": 9, "context": "", "prompt": "For positive x,y, the least positive integer linear combination is min(x,y).", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "number_theory", "original_format": "choice", "repairs": [], "source_parts": [ "9.4b" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.7, "probabilities": { "A": 0.85, "B": 0.15 }, "batch_response_ms": 499.06333300168626, "repaired": false }, { "id": "9.4c", "section": 9, "context": "", "prompt": "For positive x,y, the least positive integer linear combination is gcd(x,y).", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "number_theory", "original_format": "choice", "repairs": [], "source_parts": [ "9.4c" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.75, "probabilities": { "A": 0.13, "B": 0.87 }, "batch_response_ms": 499.06333300168626, "repaired": false }, { "id": "9.5", "section": 9, "context": "", "prompt": "If x=a mod m and x=a mod n, with coprime positive m,n, then x=a mod mn.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "9.5" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.94, "probabilities": { "A": 0.03, "B": 0.97 }, "batch_response_ms": 499.06333300168626, "repaired": false }, { "id": "9.6a", "section": 9, "context": "", "prompt": "Positive m,n have gcd q. Smallest positive k with a+kn=a mod m?", "response_mode": "single_choice", "options": { "A": "n/q", "B": "q", "C": "m/q", "D": "mn/q" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [ "Remove irrelevant x≠0 restriction; original arbitrary a could otherwise exclude the purported first k." ], "source_parts": [ "9.6a" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.93, "probabilities": { "C": 0.94, "A": 0.02, "D": 0.03, "B": 0.01 }, "batch_response_ms": 499.06333300168626, "repaired": true }, { "id": "9.6b", "section": 9, "context": "", "prompt": "Positive m,n have gcd q. How many x in 0..mn−1 satisfy x=a mod m and x=a mod n?", "response_mode": "single_choice", "options": { "A": "q", "B": "1", "C": "mn/q", "D": "m+n" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "9.6b" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.65, "probabilities": { "C": 0.01, "A": 0.74, "D": 0.0, "B": 0.25 }, "batch_response_ms": 499.06333300168626, "repaired": false }, { "id": "10.1", "section": 10, "context": "Prime p>2 has p−1=q1*...*qk, a product of distinct primes.", "prompt": "For 1<=n=1; n=0 makes the claim false." ], "source_parts": [ "10.1" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 1.0, "B": 0.0 }, "batch_response_ms": 348.27579199918546, "repaired": true }, { "id": "10.2", "section": 10, "context": "Prime p>2 has p−1=q1*...*qk, a product of distinct primes.", "prompt": "For each fixed i, prove there exists a≠0,1 modulo p with a^qi=1.", "response_mode": "single_choice", "options": { "A": "Set a=1, which satisfies both required inequalities.", "B": "Choose one a with order divisible by all qi; then each qi separately annihilates it.", "C": "Let h=(p−1)/qi. The polynomial x^h−1 has fewer than p−1 roots, so choose nonzero b with b^h≠1. Put a=b^h; then a^qi=b^(p−1)=1 by Fermat.", "D": "Choose any a≠0,1; Fermat says its qith power is one." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [ "Clarify quantifiers: a may depend on i." ], "source_parts": [ "10.2" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 1.0, "B": 0.0 }, "batch_response_ms": 348.27579199918546, "repaired": true }, { "id": "10.3", "section": 10, "context": "", "prompt": "If x has order d modulo prime p and x^q=1 mod p, prove d divides q.", "response_mode": "single_choice", "options": { "A": "The order d is always p−1 and q is always p.", "B": "Every divisor of x also divides q.", "C": "Write q=dm+r, 0<=r=dQ>=1. Euclidean division gives P=DQ+R, with R zero or deg R=dQ>=1. Euclidean division gives P=DQ+R, with R zero or deg R=dQ, required for the nonzero quotient degree formula." ], "source_parts": [ "12.2bi" ], "exam": "midterm_fa23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "D": 0.0, "B": 0.0, "A": 1.0 }, "batch_response_ms": 310.95187500250177, "repaired": true }, { "id": "12.2bii", "section": 12, "context": "Nonzero formal polynomials P,Q over a field have degrees dP>=dQ>=1. Euclidean division gives P=DQ+R, with R zero or deg R4. Hint: separate square and nonsquare factorizations.", "response_mode": "single_choice", "options": { "A": "Write n=ab with 12 and a,2a=1.", "response_mode": "single_choice", "options": { "A": "Checking the first seven n proves the rest by periodicity over the integers.", "B": "A_1=28. Both exponential terms are independently divisible by 7.", "C": "A_1=28. If 7 divides A_n, then A_(n+1)=7*3^(2n+1)+2A_n is divisible by 7.", "D": "A_1=28. The recurrence is A_(n+1)=9A_n, so the result follows." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "5.1" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.0, "C": 0.99, "A": 0.0, "D": 0.01 }, "batch_response_ms": 371.74975000016275, "repaired": false }, { "id": "6.1a", "section": 6, "context": "Jobs: A:1>2>3, B:2>3>1, C:2>3>1. Candidates: 1:B>C>A, 2:C>A>B, 3:C>A>B.", "prompt": "Job-optimal matching?", "response_mode": "single_choice", "options": { "A": "A–3, B–1, C–2", "B": "A–1, B–2, C–3", "C": "A–1, B–3, C–2", "D": "A–2, B–3, C–1" }, "correct": false, "kind": "multi_step", "topic": "matching", "original_format": "written", "repairs": [], "source_parts": [ "6.1a" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.25, "probabilities": { "C": 0.23, "A": 0.44, "D": 0.09, "B": 0.24 }, "batch_response_ms": 337.23529200142366, "repaired": false }, { "id": "6.1b", "section": 6, "context": "Jobs: A:1>2>3, B:2>3>1, C:2>3>1. Candidates: 1:B>C>A, 2:C>A>B, 3:C>A>B.", "prompt": "Candidate-optimal matching?", "response_mode": "single_choice", "options": { "A": "A–1, B–3, C–2", "B": "A–3, B–1, C–2", "C": "A–1, B–2, C–3", "D": "A–2, B–3, C–1" }, "correct": true, "kind": "multi_step", "topic": "matching", "original_format": "written", "repairs": [], "source_parts": [ "6.1b" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.4, "probabilities": { "C": 0.13, "A": 0.23, "D": 0.09, "B": 0.55 }, "batch_response_ms": 337.23529200142366, "repaired": false }, { "id": "6.1c", "section": 6, "context": "Jobs: A:1>2>3, B:2>3>1, C:2>3>1. Candidates: 1:B>C>A, 2:C>A>B, 3:C>A>B.", "prompt": "Another stable matching beyond the two proposer-optimal ones?", "response_mode": "single_choice", "options": { "A": "A–2, B–1, C–3", "B": "A–3, B–2, C–1", "C": "A–1, B–2, C–3", "D": "None" }, "correct": true, "kind": "multi_step", "topic": "matching", "original_format": "written", "repairs": [], "source_parts": [ "6.1c" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.2, "probabilities": { "C": 0.06, "A": 0.32, "D": 0.4, "B": 0.22 }, "batch_response_ms": 337.23529200142366, "repaired": false }, { "id": "6.2", "section": 6, "context": "", "prompt": "A job receiving its least-preferred candidate in the job-optimal matching must be every candidate’s least-preferred job.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.2" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.32, "probabilities": { "A": 0.66, "B": 0.34 }, "batch_response_ms": 337.23529200142366, "repaired": false }, { "id": "6.3", "section": 6, "context": "", "prompt": "If a job has the same candidate in every stable matching, that candidate must be its first choice.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.3" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.02, "probabilities": { "B": 0.51, "A": 0.49 }, "batch_response_ms": 337.23529200142366, "repaired": false }, { "id": "6.4", "section": 6, "context": "", "prompt": "If a candidate rejects all offers one day instead of retaining its favorite in job-proposing deferred acceptance, any resulting complete matching must be worse for that candidate.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": false, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.4" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.3, "probabilities": { "A": 0.65, "B": 0.35 }, "batch_response_ms": 337.23529200142366, "repaired": false }, { "id": "6.5", "section": 6, "context": "", "prompt": "A job–candidate pair present in both proposer-optimal matchings is present in every stable matching.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.5" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.55, "probabilities": { "A": 0.77, "B": 0.23 }, "batch_response_ms": 337.23529200142366, "repaired": false }, { "id": "7.1a", "section": 7, "context": "", "prompt": "A graph with n vertices and m>=n−1 edges must be connected.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "7.1a" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "A": 0.99, "B": 0.01 }, "batch_response_ms": 402.2258749973844, "repaired": false }, { "id": "7.1b", "section": 7, "context": "", "prompt": "A graph with n vertices and m<=n−1 edges must be acyclic.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "7.1b" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.88, "probabilities": { "A": 0.06, "B": 0.94 }, "batch_response_ms": 402.2258749973844, "repaired": false }, { "id": "7.1c", "section": 7, "context": "", "prompt": "Number of edges in Kn?", "response_mode": "single_choice", "options": { "A": "C(n,2)", "B": "n²", "C": "2n", "D": "n−1" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.1c" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 1.0, "B": 0.0, "C": 0.0, "D": 0.0 }, "batch_response_ms": 402.2258749973844, "repaired": false }, { "id": "7.1d", "section": 7, "context": "", "prompt": "A hypercube has n vertices. Number of edges?", "response_mode": "single_choice", "options": { "A": "n log2(n)", "B": "n log2(n)/2", "C": "n/2", "D": "2^n" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.1d" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.89, "probabilities": { "A": 0.08, "B": 0.91, "C": 0.01, "D": 0.0 }, "batch_response_ms": 402.2258749973844, "repaired": false }, { "id": "7.1e", "section": 7, "context": "", "prompt": "A graph with an Eulerian tour must have an even number of edges.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "7.1e" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.6, "probabilities": { "A": 0.2, "B": 0.8 }, "batch_response_ms": 402.2258749973844, "repaired": false }, { "id": "7.1f", "section": 7, "context": "", "prompt": "Minimum possible connected components for a simple graph with n>=1 vertices and feasible edge count m?", "response_mode": "single_choice", "options": { "A": "max(1,n−m)", "B": "n−m", "C": "n−1", "D": "max(1,m−n)" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.1f" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.96, "probabilities": { "A": 0.97, "B": 0.03, "C": 0.0, "D": 0.0 }, "batch_response_ms": 402.2258749973844, "repaired": false }, { "id": "7.1g", "section": 7, "context": "", "prompt": "A connected acyclic graph with at least two vertices has at least how many degree-one vertices? Give the sharp universal bound.", "response_mode": "single_choice", "options": { "A": "3", "B": "1", "C": "2", "D": "n−1" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [ "Require at least two vertices to exclude the one-vertex tree." ], "source_parts": [ "7.1g" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 0.0, "B": 0.0, "C": 1.0, "D": 0.0 }, "batch_response_ms": 402.2258749973844, "repaired": true }, { "id": "7.1h", "section": 7, "context": "", "prompt": "A forest has n vertices and c components. Number of edges?", "response_mode": "single_choice", "options": { "A": "c−1", "B": "n−1", "C": "n+c", "D": "n−c" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.1h" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 0.0, "B": 0.0, "C": 0.0, "D": 1.0 }, "batch_response_ms": 402.2258749973844, "repaired": false }, { "id": "7.2", "section": 7, "context": "", "prompt": "Every graph has an even number of odd-degree vertices.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "7.2" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "A": 0.99, "B": 0.01 }, "batch_response_ms": 402.2258749973844, "repaired": false }, { "id": "7.3", "section": 7, "context": "", "prompt": "Minimum edge colors for a d-dimensional hypercube?", "response_mode": "single_choice", "options": { "A": "d+1", "B": "d", "C": "2^d", "D": "2d" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.3" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.92, "probabilities": { "A": 0.05, "B": 0.94, "C": 0.0, "D": 0.01 }, "batch_response_ms": 402.2258749973844, "repaired": false }, { "id": "7.4", "section": 7, "context": "", "prompt": "Minimum edge colors for an odd cycle?", "response_mode": "single_choice", "options": { "A": "4", "B": "2", "C": "1", "D": "3" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.4" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.96, "probabilities": { "A": 0.01, "B": 0.02, "C": 0.0, "D": 0.97 }, "batch_response_ms": 402.2258749973844, "repaired": false }, { "id": "7.5", "section": 7, "context": "", "prompt": "A simple graph has n vertices and n+1 edges. Largest constant degree guaranteed for at least one vertex?", "response_mode": "single_choice", "options": { "A": "3", "B": "2", "C": "4", "D": "n−1" }, "correct": true, "kind": "multi_step", "topic": "graphs", "original_format": "written", "repairs": [ "Make n-vertex premise explicit; source omitted it in the local sentence." ], "source_parts": [ "7.5" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.6, "probabilities": { "A": 0.71, "B": 0.21, "C": 0.08, "D": 0.0 }, "batch_response_ms": 402.2258749973844, "repaired": true }, { "id": "8.1", "section": 8, "context": "", "prompt": "A connected simple planar drawing has v>2 vertices and e<3v−6. Prove some face has boundary length at least 4, counting repeated edge sides.", "response_mode": "single_choice", "options": { "A": "If every face had length 3, then 3f=2e. Euler’s v−e+f=2 would force e=3v−6, a contradiction; each face has length at least 3.", "B": "Since e<3v−6, every vertex has degree at most two, and every face is a quadrilateral.", "C": "A planar graph can have only triangular or quadrilateral faces; fewer edges forbid triangles.", "D": "Euler says f=3 for every planar graph, so one face has four sides." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "8.1" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 1.0, "C": 0.0, "B": 0.0 }, "batch_response_ms": 370.90579199866625, "repaired": false }, { "id": "8.2", "section": 8, "context": "", "prompt": "Partition the edges of a bipartite Eulerian graph into two spanning subgraphs with half of each vertex’s degree.", "response_mode": "single_choice", "options": { "A": "Color a spanning tree one color and all remaining edges the other; this halves each degree.", "B": "Alternate the edges of an Euler tour between the two sets. The tour has even length because the graph is bipartite; paired arrival/departure edges, including the wraparound, receive opposite colors at every vertex.", "C": "Assign every edge incident to the left bipartition to the first set and all others to the second.", "D": "Take any half of the edges; equal edge counts imply equal degrees at every vertex." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "8.2" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "C": 0.0, "A": 0.0, "B": 1.0 }, "batch_response_ms": 370.90579199866625, "repaired": false }, { "id": "9.1a", "section": 9, "context": "", "prompt": "For integers a,x,y, gcd(x,y)=1 and a divides xy imply a divides x or a divides y.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "9.1a" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.12, "probabilities": { "B": 0.56, "A": 0.44 }, "batch_response_ms": 338.06458300023223, "repaired": false }, { "id": "9.1b", "section": 9, "context": "", "prompt": "Choose a counterexample to: gcd(x,y)=1 and a|xy imply a|x or a|y.", "response_mode": "single_choice", "options": { "A": "a=3,x=9,y=4", "B": "a=6,x=9,y=4", "C": "a=6,x=6,y=5", "D": "a=5,x=9,y=4" }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "9.1b" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.96, "probabilities": { "B": 0.97, "D": 0.02, "C": 0.0, "A": 0.01 }, "batch_response_ms": 338.06458300023223, "repaired": false }, { "id": "9.2a", "section": 9, "context": "N=pqr for three distinct primes p,q,r.", "prompt": "Number of residues x mod N satisfying qx=0 mod N?", "response_mode": "single_choice", "options": { "A": "1", "B": "pqr", "C": "q", "D": "pr" }, "correct": false, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "9.2a" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.82, "probabilities": { "B": 0.01, "D": 0.86, "C": 0.11, "A": 0.02 }, "batch_response_ms": 338.06458300023223, "repaired": false }, { "id": "9.2b", "section": 9, "context": "N=pqr for three distinct primes p,q,r.", "prompt": "If gcd(x,N)=1, what is x^((p−1)(q−1)(r−1)) mod N?", "response_mode": "single_choice", "options": { "A": "N−1", "B": "x", "C": "0", "D": "1" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "9.2b" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.01, "D": 0.99, "C": 0.0, "A": 0.0 }, "batch_response_ms": 338.06458300023223, "repaired": false }, { "id": "9.2c", "section": 9, "context": "N=pqr for three distinct primes p,q,r.", "prompt": "Why is x^((p−1)(q−1)(r−1))=1 mod N when gcd(x,N)=1?", "response_mode": "single_choice", "options": { "A": "Being coprime to N means x=1 modulo N.", "B": "Fermat applies directly to the composite modulus pqr with exponent pqr−1.", "C": "Fermat gives value 1 modulo each prime because the exponent is a multiple of each prime minus one. CRT makes their common solution 1 modulo pqr.", "D": "The exponent is zero modulo every prime, so the power is one." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "9.2c" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "C": 1.0, "A": 0.0 }, "batch_response_ms": 338.06458300023223, "repaired": false }, { "id": "10.1", "section": 10, "context": "", "prompt": "Multiplicative order of 3 modulo 5?", "response_mode": "single_choice", "options": { "A": "3", "B": "5", "C": "2", "D": "4" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "10.1" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.91, "probabilities": { "C": 0.06, "B": 0.0, "A": 0.0, "D": 0.9400000000000001 }, "batch_response_ms": 351.1053330003051, "repaired": false }, { "id": "10.2", "section": 10, "context": "", "prompt": "Let t be the order of a modulo n, gcd(a,n)=1. Prove a^m=1 mod n iff t divides m.", "response_mode": "single_choice", "options": { "A": "Multiples of t give powers of a^t=1. Conversely write m=qt+r, 0<=r1 with n dividing 2^n−1.", "response_mode": "single_choice", "options": { "A": "Since 2^n−1 is odd, no integer n can divide it.", "B": "If n divides 2^n−1, then n divides 2 and n divides 1 separately.", "C": "Such n is odd. Let p be its smallest prime factor and t=ord_p(2)>1. Then t divides n and p−1, so a prime factor of t is a prime factor of n smaller than p, a contradiction.", "D": "Fermat says 2^n=2 mod n even for composite n, so the remainder is always one." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "10.4" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 1.0, "B": 0.0, "A": 0.0, "D": 0.0 }, "batch_response_ms": 351.1053330003051, "repaired": false }, { "id": "11.1", "section": 11, "context": "", "prompt": "7^26 modulo 10?", "response_mode": "single_choice", "options": { "A": "1", "B": "9", "C": "3", "D": "7" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "11.1" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.44, "probabilities": { "D": 0.18, "A": 0.08, "B": 0.59, "C": 0.15 }, "batch_response_ms": 466.0425000001851, "repaired": false }, { "id": "11.2", "section": 11, "context": "", "prompt": "Inverse of 7 modulo 68 in 0..67?", "response_mode": "single_choice", "options": { "A": "29", "B": "39", "C": "10", "D": "61" }, "correct": true, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "11.2" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.66, "probabilities": { "D": 0.03, "A": 0.21, "B": 0.75, "C": 0.01 }, "batch_response_ms": 466.0425000001851, "repaired": false }, { "id": "11.3", "section": 11, "context": "", "prompt": "Distinct integers x,y are congruent modulo positive m and n, gcd(m,n)=d. Sharp lower bound on |x−y|?", "response_mode": "single_choice", "options": { "A": "mn", "B": "m+n", "C": "mn/d", "D": "d" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "11.3" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.85, "probabilities": { "D": 0.1, "A": 0.01, "B": 0.0, "C": 0.89 }, "batch_response_ms": 466.0425000001851, "repaired": false }, { "id": "11.4", "section": 11, "context": "", "prompt": "Number of x in 0..504 solving 10x=5 mod 505?", "response_mode": "single_choice", "options": { "A": "1", "B": "5", "C": "10", "D": "0" }, "correct": false, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "11.4" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.51, "probabilities": { "D": 0.13, "A": 0.64, "B": 0.18, "C": 0.05 }, "batch_response_ms": 466.0425000001851, "repaired": false }, { "id": "11.5a", "section": 11, "context": "", "prompt": "For valid textbook RSA encryption E and decryption D, D(E(x))=E(D(x)) modulo N for every message x.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": false, "kind": "recognition", "topic": "rsa", "original_format": "choice", "repairs": [], "source_parts": [ "11.5a" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.3, "probabilities": { "A": 0.35, "B": 0.65 }, "batch_response_ms": 466.0425000001851, "repaired": false }, { "id": "11.5b", "section": 11, "context": "", "prompt": "Given y=E(E(x)) in valid RSA, which expression recovers x?", "response_mode": "single_choice", "options": { "A": "D(D(y))", "B": "E(E(y))", "C": "E(D(y))", "D": "D(y)" }, "correct": true, "kind": "recognition", "topic": "rsa", "original_format": "written", "repairs": [], "source_parts": [ "11.5b" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.95, "probabilities": { "D": 0.02, "A": 0.96, "B": 0.0, "C": 0.02 }, "batch_response_ms": 466.0425000001851, "repaired": false }, { "id": "11.5c", "section": 11, "context": "", "prompt": "In valid textbook RSA, a=E(x), b=E(y). What is D(ab) modulo N?", "response_mode": "single_choice", "options": { "A": "xy", "B": "x^y", "C": "x+y", "D": "x/y" }, "correct": true, "kind": "recognition", "topic": "rsa", "original_format": "written", "repairs": [], "source_parts": [ "11.5c" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "D": 0.0, "A": 0.99, "B": 0.01, "C": 0.0 }, "batch_response_ms": 466.0425000001851, "repaired": false }, { "id": "12.1", "section": 12, "context": "", "prompt": "Over GF(5), give a degree-at-most-2 polynomial through (0,1),(1,0),(3,0).", "response_mode": "single_choice", "options": { "A": "2(x−1)(x−3)", "B": "3(x−1)(x−3)", "C": "(x−1)(x−3)", "D": "2(x+1)(x+3)" }, "correct": true, "kind": "multi_step", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "12.1" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.58, "probabilities": { "D": 0.02, "B": 0.21, "A": 0.68, "C": 0.09 }, "batch_response_ms": 332.4944170017261, "repaired": false }, { "id": "12.2", "section": 12, "context": "", "prompt": "Over GF(5), ax²+bx+c passes through (0,0),(1,1),(2,4). Give (a,b,c).", "response_mode": "single_choice", "options": { "A": "(1,1,0)", "B": "(2,4,0)", "C": "(1,0,0)", "D": "(0,1,0)" }, "correct": true, "kind": "multi_step", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "12.2a", "12.2b", "12.2c" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.71, "probabilities": { "D": 0.03, "B": 0.03, "A": 0.16, "C": 0.78 }, "batch_response_ms": 332.4944170017261, "repaired": false }, { "id": "12.3", "section": 12, "context": "", "prompt": "Over GF(p), p>=5, four points have distinct x-coordinates. Number of degree-at-most-4 polynomials through them?", "response_mode": "single_choice", "options": { "A": "p²", "B": "1", "C": "p", "D": "p−1" }, "correct": false, "kind": "calculation", "topic": "polynomials", "original_format": "written", "repairs": [ "Specify distinct x-coordinates." ], "source_parts": [ "12.3" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.59, "probabilities": { "D": 0.01, "A": 0.69, "B": 0.05, "C": 0.25 }, "batch_response_ms": 332.4944170017261, "repaired": true }, { "id": "12.4a", "section": 12, "context": "Polynomial division over a field: P=QD+R, deg P=d, deg D=d′, with 1<=d′=max(s+1,6)", "B": "p>=s", "C": "p>=3", "D": "p>=max(s,5)" }, "correct": true, "kind": "recognition", "topic": "secret_sharing", "original_format": "written", "repairs": [], "source_parts": [ "13.1c" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.79, "probabilities": { "B": 0.02, "D": 0.11, "A": 0.85, "C": 0.02 }, "batch_response_ms": 394.3962499979534, "repaired": false }, { "id": "13.2a", "section": 13, "context": "P(x)=2x²+3x+3 over GF(5); evaluations at 0,1,2,3,4 are sent and only the value at x=1 is corrupted. Monic E(x)=x+b0 and Q=PE.", "prompt": "What is b0 in 0..4?", "response_mode": "single_choice", "options": { "A": "1", "B": "0", "C": "4", "D": "2" }, "correct": true, "kind": "calculation", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "13.2a" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.11, "probabilities": { "B": 0.14, "D": 0.28, "A": 0.25, "C": 0.33 }, "batch_response_ms": 394.3962499979534, "repaired": false }, { "id": "13.2bi", "section": 13, "context": "P(x)=2x²+3x+3 over GF(5); evaluations at 0,1,2,3,4 are sent and only the value at x=1 is corrupted. Monic E(x)=x+b0 and Q=PE.", "prompt": "Degree d of Q?", "response_mode": "single_choice", "options": { "A": "3", "B": "4", "C": "2", "D": "1" }, "correct": true, "kind": "recognition", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "13.2bi" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.02, "D": 0.0, "C": 0.0, "A": 0.98 }, "batch_response_ms": 394.3962499979534, "repaired": false }, { "id": "13.2bii", "section": 13, "context": "P(x)=2x²+3x+3 over GF(5); evaluations at 0,1,2,3,4 are sent and only the value at x=1 is corrupted. Monic E(x)=x+b0 and Q=PE.", "prompt": "Leading coefficient qd of Q?", "response_mode": "single_choice", "options": { "A": "1", "B": "4", "C": "3", "D": "2" }, "correct": true, "kind": "recognition", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "13.2bii" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.76, "probabilities": { "B": 0.1, "D": 0.82, "C": 0.02, "A": 0.06 }, "batch_response_ms": 394.3962499979534, "repaired": false }, { "id": "13.2biii", "section": 13, "context": "P(x)=2x²+3x+3 over GF(5); evaluations at 0,1,2,3,4 are sent and only the value at x=1 is corrupted. Monic E(x)=x+b0 and Q=PE.", "prompt": "Constant coefficient q0 in terms of b0?", "response_mode": "single_choice", "options": { "A": "2b0", "B": "3b0", "C": "b0", "D": "3+b0" }, "correct": true, "kind": "recognition", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "13.2biii" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.86, "probabilities": { "B": 0.9, "D": 0.05, "C": 0.02, "A": 0.03 }, "batch_response_ms": 394.3962499979534, "repaired": false }, { "id": "14.1", "section": 14, "context": "", "prompt": "Number of formal polynomials of degree exactly d over GF(p)?", "response_mode": "single_choice", "options": { "A": "(p−1)p^d", "B": "(p−1)^(d+1)", "C": "p^d", "D": "p^(d+1)" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "14.1" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "C": 0.01, "A": 0.99, "D": 0.0 }, "batch_response_ms": 403.18283399756183, "repaired": false }, { "id": "14.2", "section": 14, "context": "", "prompt": "Every degree-exactly-one polynomial over a field has exactly one root.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "choice", "repairs": [], "source_parts": [ "14.2" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.73, "probabilities": { "B": 0.86, "A": 0.14 }, "batch_response_ms": 403.18283399756183, "repaired": false }, { "id": "14.3", "section": 14, "context": "", "prompt": "Choose a quadratic over GF(p) having x=1 as its only distinct root, for every prime p.", "response_mode": "single_choice", "options": { "A": "x²−1", "B": "(x−1)²", "C": "x²+1", "D": "x(x−1)" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "14.3" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.99, "C": 0.0, "A": 0.01, "D": 0.0 }, "batch_response_ms": 403.18283399756183, "repaired": false }, { "id": "14.4", "section": 14, "context": "", "prompt": "Number of distinct formal polynomials a(x−r)² over GF(p), a nonzero?", "response_mode": "single_choice", "options": { "A": "(p−1)C(p,2)", "B": "(p−1)p", "C": "p−1", "D": "p²" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "14.4" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.7, "probabilities": { "B": 0.78, "C": 0.04, "A": 0.17, "D": 0.01 }, "batch_response_ms": 403.18283399756183, "repaired": false }, { "id": "14.5", "section": 14, "context": "", "prompt": "Number of a(x−r)(x−s), a nonzero and r≠s, over GF(p)?", "response_mode": "single_choice", "options": { "A": "C(p,2)", "B": "p²", "C": "(p−1)p(p−1)", "D": "(p−1)C(p,2)" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "14.5" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.79, "probabilities": { "B": 0.01, "C": 0.14, "A": 0.01, "D": 0.84 }, "batch_response_ms": 403.18283399756183, "repaired": false }, { "id": "14.6", "section": 14, "context": "", "prompt": "Number of degree-exactly-two polynomials over GF(p) with no field roots?", "response_mode": "single_choice", "options": { "A": "(p−1)(p²−C(p,2)−p)", "B": "(p−1)C(p,2)−p", "C": "(p−1)(p²−C(p,2))", "D": "(p−1)p²" }, "correct": true, "kind": "multi_step", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "14.6" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.25, "probabilities": { "B": 0.2, "C": 0.31, "A": 0.45, "D": 0.04 }, "batch_response_ms": 403.18283399756183, "repaired": false }, { "id": "14.7", "section": 14, "context": "", "prompt": "For fixed distinct field elements r1,...,rk, count polynomials product (x−ri)^bi with each bi>=1 and sum bi=d>=k.", "response_mode": "single_choice", "options": { "A": "C(d,k)", "B": "k^d", "C": "C(d+k−1,k−1)", "D": "C(d−1,k−1)" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [ "Require distinct roots so different exponent tuples give distinct formal polynomials." ], "source_parts": [ "14.7" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.95, "probabilities": { "B": 0.0, "C": 0.01, "A": 0.03, "D": 0.96 }, "batch_response_ms": 403.18283399756183, "repaired": true }, { "id": "15.1a", "section": 15, "context": "", "prompt": "Number of unordered 13-card hands from 52 cards?", "response_mode": "single_choice", "options": { "A": "C(64,13)", "B": "C(52,13)", "C": "52!/39!", "D": "52^13" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "15.1a" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 0.0, "B": 1.0, "C": 0.0, "D": 0.0 }, "batch_response_ms": 391.0184999986086, "repaired": false }, { "id": "15.1b", "section": 15, "context": "", "prompt": "Deal all 52 distinct cards to four named players, 13 each, unordered within each hand. Number of deals?", "response_mode": "single_choice", "options": { "A": "C(52,13)", "B": "52!/(4!*(13!)^4)", "C": "52!/(13!)^4", "D": "4^52" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "15.1b" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.85, "probabilities": { "A": 0.0, "B": 0.11, "C": 0.89, "D": 0.0 }, "batch_response_ms": 391.0184999986086, "repaired": false }, { "id": "15.2", "section": 15, "context": "", "prompt": "Count integer k-tuples with xi>=−1 and sum xi=n, n>=0, k>=1.", "response_mode": "single_choice", "options": { "A": "C(n+2k−1,k−1)", "B": "C(n+k−1,k−1)", "C": "k^n", "D": "C(n−1,k−1)" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "15.2" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.86, "probabilities": { "A": 0.9, "B": 0.1, "C": 0.0, "D": 0.0 }, "batch_response_ms": 391.0184999986086, "repaired": false }, { "id": "15.3a", "section": 15, "context": "", "prompt": "Number of anagrams of CANTSTANFURD?", "response_mode": "single_choice", "options": { "A": "12!/2!", "B": "11!/(2!2!2!)", "C": "12!/(3!3!)", "D": "12!/(2!2!2!)" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "15.3a" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.69, "probabilities": { "A": 0.16, "B": 0.06, "C": 0.01, "D": 0.77 }, "batch_response_ms": 391.0184999986086, "repaired": false }, { "id": "15.3b", "section": 15, "context": "", "prompt": "Let x be the number of anagrams of CANTSTANFURD. How many put C before S before D, without an adjacency requirement?", "response_mode": "single_choice", "options": { "A": "x/2", "B": "x/12", "C": "x/6", "D": "x/3" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "15.3b" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.92, "probabilities": { "A": 0.0, "B": 0.04, "C": 0.94, "D": 0.02 }, "batch_response_ms": 391.0184999986086, "repaired": false }, { "id": "15.3c", "section": 15, "context": "", "prompt": "Let x be the number of anagrams of CANTSTANFURD. How many put both As before S, without an adjacency requirement?", "response_mode": "single_choice", "options": { "A": "x/2", "B": "2x/3", "C": "x/6", "D": "x/3" }, "correct": false, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "15.3c" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.31, "probabilities": { "A": 0.48, "B": 0.06, "C": 0.14, "D": 0.32 }, "batch_response_ms": 391.0184999986086, "repaired": false }, { "id": "15.4", "section": 15, "context": "", "prompt": "Number of length-n bitstrings with k ones?", "response_mode": "single_choice", "options": { "A": "C(n,k)", "B": "2^k", "C": "n!/k!", "D": "n^k" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "15.4" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 1.0, "B": 0.0, "C": 0.0, "D": 0.0 }, "batch_response_ms": 391.0184999986086, "repaired": false }, { "id": "15.5", "section": 15, "context": "", "prompt": "Distribute five identical dollar bills among four named people, allowing zero. Number of allocations?", "response_mode": "single_choice", "options": { "A": "C(5,4)", "B": "C(8,3)", "C": "5^4", "D": "4^5" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "15.5" ], "exam": "midterm_fa24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 0.0, "B": 1.0, "C": 0.0, "D": 0.0 }, "batch_response_ms": 391.0184999986086, "repaired": false }, { "id": "1.1", "section": 1, "context": "", "prompt": "If 6 is prime, then trees can walk.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "1.1" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.51, "probabilities": { "A": 0.76, "B": 0.24 }, "batch_response_ms": 370.6732499995269, "repaired": false }, { "id": "1.2", "section": 1, "context": "", "prompt": "Is NOT(P implies Q) equivalent to (NOT P) AND (NOT Q)?", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "1.2" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "A": 0.99, "B": 0.01 }, "batch_response_ms": 370.6732499995269, "repaired": false }, { "id": "1.3", "section": 1, "context": "", "prompt": "For integers a,m,n, prove: if a does not divide mn, then a divides neither m nor n. Which argument is valid?", "response_mode": "single_choice", "options": { "A": "Contrapositive: if m=ak or n=ak for an integer k, then respectively mn=a(kn) or mn=a(km), so a divides mn.", "B": "If a does not divide m or n, their product cannot be divisible by a.", "C": "If a divides mn, it must divide both m and n; take the converse.", "D": "Since a does not divide mn, neither m nor n can have any divisor in common with a." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "1.3" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 1.0, "B": 0.0, "C": 0.0 }, "batch_response_ms": 370.6732499995269, "repaired": false }, { "id": "2.1", "section": 2, "context": "J means jumping, G on ground, M in motion, H healing, W touching a wall.", "prompt": "Express: Bloof is not healing unless it is not in motion.", "response_mode": "single_choice", "options": { "A": "NOT M implies NOT H", "B": "M implies NOT H", "C": "NOT H implies M", "D": "H implies M" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "written", "repairs": [], "source_parts": [ "2.1" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.98, "D": 0.01, "A": 0.0, "C": 0.01 }, "batch_response_ms": 385.4925830019056, "repaired": false }, { "id": "2.2", "section": 2, "context": "J means jumping, G on ground, M in motion, H healing, W touching a wall.", "prompt": "Give the contrapositive of J implies (NOT G AND M).", "response_mode": "single_choice", "options": { "A": "(G OR NOT M) implies NOT J", "B": "(NOT G AND M) implies J", "C": "(G AND NOT M) implies NOT J", "D": "NOT J implies (G OR NOT M)" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "written", "repairs": [], "source_parts": [ "2.2" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 1.0, "C": 0.0 }, "batch_response_ms": 385.4925830019056, "repaired": false }, { "id": "2.3", "section": 2, "context": "J means jumping, G on ground, M in motion, H healing, W touching a wall. Rules: H implies NOT M; J implies (NOT G AND M); NOT(G AND W); NOT G implies (M OR J).", "prompt": "H is true. What can be concluded about W?", "response_mode": "single_choice", "options": { "A": "True", "B": "Could be either", "C": "False" }, "correct": false, "kind": "multi_step", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.3" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.88, "probabilities": { "B": 0.92, "A": 0.02, "C": 0.06 }, "batch_response_ms": 385.4925830019056, "repaired": false }, { "id": "3.1", "section": 3, "context": "", "prompt": "Which proves that an integer n is odd iff 5n+3 is even?", "response_mode": "single_choice", "options": { "A": "If n=2k+1, then 5n+3 is even; this alone proves both implications.", "B": "For n=2k+1, 5n+3=2(5k+4). Conversely, if n=2k then 5n+3=2(5k+1)+1 is odd, proving the reverse implication by contrapositive.", "C": "An odd integer plus 3 is always even, regardless of the factor 5; therefore every integer n is odd.", "D": "If 5n+3=2k, then n=2(k-3)/5 is even, so n is odd." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "3.1" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 1.0, "C": 0.0, "D": 0.0, "A": 0.0 }, "batch_response_ms": 341.8095839988382, "repaired": false }, { "id": "3.2", "section": 3, "context": "", "prompt": "Prove 7 divides A_n=2^(n+2)+3^(2n+1), for integers n>=0. Which induction works?", "response_mode": "single_choice", "options": { "A": "A_0=7. If 7 divides A_k, then A_(k+1)=2A_k+7*3^(2k+1), so 7 also divides A_(k+1).", "B": "A_0=7 and A_(k+1)=9A_k+7, so divisibility follows.", "C": "Check n=0 and n=1; two successful cases establish every n.", "D": "A_0=7 and A_(k+1)=2A_k, so divisibility follows." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "3.2" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.93, "probabilities": { "B": 0.04, "C": 0.0, "D": 0.0, "A": 0.96 }, "batch_response_ms": 341.8095839988382, "repaired": false }, { "id": "4.1a", "section": 4, "context": "Jobs prefer A:2>3>1, B:2>3>1, C:1>3>2. Candidates prefer 1:A>B>C, 2:A>C>B, 3:C>A>B.", "prompt": "Is matching A–1, B–2, C–3 stable? Include a valid reason.", "response_mode": "single_choice", "options": { "A": "No; C and 2 form a blocking pair.", "B": "No; A and 2 form a blocking pair.", "C": "Yes; every candidate has a different job.", "D": "Yes; candidate 3 has its top choice." }, "correct": true, "kind": "multi_step", "topic": "matching", "original_format": "written", "repairs": [], "source_parts": [ "4.1a" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.53, "probabilities": { "C": 0.02, "A": 0.3, "B": 0.64, "D": 0.04 }, "batch_response_ms": 326.9495829990774, "repaired": false }, { "id": "4.1b", "section": 4, "context": "Jobs prefer A:2>3>1, B:2>3>1, C:1>3>2. Candidates prefer 1:A>B>C, 2:A>C>B, 3:C>A>B.", "prompt": "Which matching is stable?", "response_mode": "single_choice", "options": { "A": "A–1, B–3, C–2", "B": "A–3, B–2, C–1", "C": "A–1, B–2, C–3", "D": "A–2, B–1, C–3" }, "correct": false, "kind": "multi_step", "topic": "matching", "original_format": "written", "repairs": [], "source_parts": [ "4.1b" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.3, "probabilities": { "C": 0.48, "A": 0.23, "B": 0.25, "D": 0.04 }, "batch_response_ms": 326.9495829990774, "repaired": false }, { "id": "4.2", "section": 4, "context": "", "prompt": "In a stable matching, it is impossible for two partners to be each other’s least preferred option.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": false, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "4.2" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.11, "probabilities": { "A": 0.55, "B": 0.45 }, "batch_response_ms": 326.9495829990774, "repaired": false }, { "id": "4.3", "section": 4, "context": "", "prompt": "If a job has the same partner in every stable matching, that partner must be its first choice.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "4.3" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.13, "probabilities": { "A": 0.43, "B": 0.57 }, "batch_response_ms": 326.9495829990774, "repaired": false }, { "id": "4.4", "section": 4, "context": "", "prompt": "Job-proposing and candidate-proposing deferred acceptance produce the same matching. How many stable matchings exist?", "response_mode": "single_choice", "options": { "A": "Exactly one", "B": "None", "C": "Cannot be determined", "D": "More than one" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "4.4" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.84, "probabilities": { "C": 0.12, "A": 0.88, "B": 0.0, "D": 0.0 }, "batch_response_ms": 326.9495829990774, "repaired": false }, { "id": "5.1", "section": 5, "context": "", "prompt": "K4 has an Eulerian tour.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "5.1" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.67, "probabilities": { "B": 0.16, "A": 0.84 }, "batch_response_ms": 296.81454099772964, "repaired": false }, { "id": "5.2", "section": 5, "context": "", "prompt": "K6 is planar.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "5.2" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "A": 1.0 }, "batch_response_ms": 296.81454099772964, "repaired": false }, { "id": "5.3", "section": 5, "context": "", "prompt": "A graph has vertex degrees 4,3,3,2,1,1. How many edges?", "response_mode": "single_choice", "options": { "A": "8", "B": "14", "C": "6", "D": "7" }, "correct": false, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "5.3" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.44, "probabilities": { "B": 0.0, "C": 0.59, "D": 0.34, "A": 0.07 }, "batch_response_ms": 296.81454099772964, "repaired": false }, { "id": "5.4", "section": 5, "context": "", "prompt": "In the 5-dimensional hypercube, what is the distance between 10010 and 01000?", "response_mode": "single_choice", "options": { "A": "4", "B": "3", "C": "5", "D": "2" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "5.4" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.62, "probabilities": { "B": 0.71, "C": 0.0, "D": 0.09, "A": 0.2 }, "batch_response_ms": 296.81454099772964, "repaired": false }, { "id": "6.1", "section": 6, "context": "The undirected train graph has exactly these edges: Oakland–Berkeley, Berkeley–Richmond, Oakland–Orinda, Orinda–Walnut Creek, Oakland–Market Street, Market Street–Mission Street, Mission Street–Daly City, Oakland–Coliseum, Coliseum–Fremont.", "prompt": "Is this graph a tree?", "response_mode": "single_choice", "options": { "A": "No", "B": "Yes" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "6.1" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.51, "probabilities": { "A": 0.24, "B": 0.76 }, "batch_response_ms": 561.8327080010204, "repaired": false }, { "id": "6.2", "section": 6, "context": "The undirected train graph has exactly these edges: Oakland–Berkeley, Berkeley–Richmond, Oakland–Orinda, Orinda–Walnut Creek, Oakland–Market Street, Market Street–Mission Street, Mission Street–Daly City, Oakland–Coliseum, Coliseum–Fremont.", "prompt": "Is this graph bipartite?", "response_mode": "single_choice", "options": { "A": "No", "B": "Yes" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "6.2" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.23, "probabilities": { "A": 0.39, "B": 0.61 }, "batch_response_ms": 561.8327080010204, "repaired": false }, { "id": "6.3", "section": 6, "context": "The undirected train graph has exactly these edges: Oakland–Berkeley, Berkeley–Richmond, Oakland–Orinda, Orinda–Walnut Creek, Oakland–Market Street, Market Street–Mission Street, Mission Street–Daly City, Oakland–Coliseum, Coliseum–Fremont.", "prompt": "What is the minimum number of colors for a proper edge coloring?", "response_mode": "single_choice", "options": { "A": "3", "B": "5", "C": "4", "D": "2" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "6.3" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.2, "probabilities": { "D": 0.01, "A": 0.25, "B": 0.34, "C": 0.4 }, "batch_response_ms": 561.8327080010204, "repaired": false }, { "id": "6.4", "section": 6, "context": "The undirected train graph has exactly these edges: Oakland–Berkeley, Berkeley–Richmond, Oakland–Orinda, Orinda–Walnut Creek, Oakland–Market Street, Market Street–Mission Street, Mission Street–Daly City, Oakland–Coliseum, Coliseum–Fremont.", "prompt": "Does the graph have an Eulerian walk? If not, choose an edge to add that creates one.", "response_mode": "single_choice", "options": { "A": "No; add Berkeley–Coliseum.", "B": "Yes; no added edge needed.", "C": "No; add Oakland–Daly City.", "D": "No; add Walnut Creek–Fremont." }, "correct": false, "kind": "multi_step", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "6.Euler.1" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.07, "probabilities": { "D": 0.21, "A": 0.24, "B": 0.25, "C": 0.3 }, "batch_response_ms": 561.8327080010204, "repaired": false }, { "id": "6.5", "section": 6, "context": "The undirected train graph has exactly these edges: Oakland–Berkeley, Berkeley–Richmond, Oakland–Orinda, Orinda–Walnut Creek, Oakland–Market Street, Market Street–Mission Street, Mission Street–Daly City, Oakland–Coliseum, Coliseum–Fremont.", "prompt": "Does the graph have an Eulerian tour? If not, choose two edges that create one.", "response_mode": "single_choice", "options": { "A": "No; add Walnut Creek–Fremont and Daly City–Richmond.", "B": "No; add Oakland–Fremont and Oakland–Richmond.", "C": "Yes; no added edges needed.", "D": "No; add Berkeley–Coliseum and Orinda–Mission Street." }, "correct": true, "kind": "multi_step", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "6.Euler.2" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.46, "probabilities": { "D": 0.12, "A": 0.59, "B": 0.23, "C": 0.06 }, "batch_response_ms": 561.8327080010204, "repaired": false }, { "id": "6.6", "section": 6, "context": "The undirected train graph has exactly these edges: Oakland–Berkeley, Berkeley–Richmond, Oakland–Orinda, Orinda–Walnut Creek, Oakland–Market Street, Market Street–Mission Street, Mission Street–Daly City, Oakland–Coliseum, Coliseum–Fremont.", "prompt": "A new OAK station and edge OAK–Coliseum are added. Further added edges cannot touch OAK. If an Eulerian walk is then possible, what must hold?", "response_mode": "single_choice", "options": { "A": "OAK must be an interior vertex, since it is newly added.", "B": "OAK can be anywhere on a closed Eulerian tour.", "C": "OAK must be visited twice, since it has degree one.", "D": "OAK is an endpoint, because its only incident edge cannot be used to both enter and leave without repetition." }, "correct": true, "kind": "multi_step", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "6.Euler.3" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.93, "probabilities": { "D": 0.95, "A": 0.01, "B": 0.02, "C": 0.02 }, "batch_response_ms": 561.8327080010204, "repaired": false }, { "id": "7.1", "section": 7, "context": "", "prompt": "Which induction proves the sum of vertex degrees is twice the edge count?", "response_mode": "single_choice", "options": { "A": "Induct on vertices. A graph on k vertices has degree sum 2k. Adding any vertex adds exactly two to that sum.", "B": "Removing a vertex always removes one edge and decreases the degree sum by two.", "C": "Every edge touches two vertices, so each vertex has degree two and the sum is twice the vertex count.", "D": "Induct on edges. With zero edges the sum is zero. Remove one edge from a graph with k+1 edges; the remaining degree sum is 2k. Restoring the edge adds one degree at each endpoint, yielding 2k+2." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "7.1" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 1.0, "A": 0.0, "C": 0.0, "B": 0.0 }, "batch_response_ms": 433.6140000013984, "repaired": false }, { "id": "8.1", "section": 8, "context": "", "prompt": "For every positive integer n, 5 divides 6^n−1.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": false, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "8.1" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.14, "probabilities": { "A": 0.57, "B": 0.43 }, "batch_response_ms": 392.3504590020457, "repaired": false }, { "id": "8.2", "section": 8, "context": "", "prompt": "If xy=0 modulo 6, then x=0 or y=0 modulo 6.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "8.2" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "A": 0.99, "B": 0.01 }, "batch_response_ms": 392.3504590020457, "repaired": false }, { "id": "8.3", "section": 8, "context": "", "prompt": "For every positive odd m, 9 divides 4^m+5^m.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "8.3" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.64, "probabilities": { "A": 0.82, "B": 0.18 }, "batch_response_ms": 392.3504590020457, "repaired": false }, { "id": "8.4", "section": 8, "context": "", "prompt": "If p>3 and both p and p+2 are prime, what is p modulo 3?", "response_mode": "single_choice", "options": { "A": "0", "B": "2", "C": "1", "D": "Not determined" }, "correct": true, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "8.4" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.39, "probabilities": { "C": 0.41, "A": 0.04, "D": 0.01, "B": 0.54 }, "batch_response_ms": 392.3504590020457, "repaired": false }, { "id": "8.5", "section": 8, "context": "", "prompt": "What is the last decimal digit of 9^68?", "response_mode": "single_choice", "options": { "A": "1", "B": "8", "C": "6", "D": "9" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "8.5" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.86, "probabilities": { "C": 0.05, "A": 0.89, "D": 0.06, "B": 0.0 }, "batch_response_ms": 392.3504590020457, "repaired": false }, { "id": "8.6", "section": 8, "context": "", "prompt": "RSA has p=5, q=7 and public exponent e=5. Which is a valid decryption exponent d?", "response_mode": "single_choice", "options": { "A": "24", "B": "7", "C": "12", "D": "5" }, "correct": true, "kind": "multi_step", "topic": "rsa", "original_format": "written", "repairs": [], "source_parts": [ "8.6" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.66, "probabilities": { "C": 0.07, "A": 0.09, "D": 0.75, "B": 0.09 }, "batch_response_ms": 392.3504590020457, "repaired": false }, { "id": "8.7", "section": 8, "context": "", "prompt": "In valid textbook RSA, a=x^e mod N and b=y^e mod N. Does decrypting ab recover xy modulo N? Choose the justification.", "response_mode": "single_choice", "options": { "A": "No; RSA only preserves multiplication when x=y.", "B": "Yes; (ab)^d=a^d b^d=xy mod N.", "C": "Yes; because ed=1 as an equality of integers.", "D": "No; decryption recovers x+y because ciphertext multiplication becomes addition." }, "correct": true, "kind": "multi_step", "topic": "rsa", "original_format": "written", "repairs": [], "source_parts": [ "8.7" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 0.01, "A": 0.0, "D": 0.0, "B": 0.99 }, "batch_response_ms": 392.3504590020457, "repaired": false }, { "id": "9.1", "section": 9, "context": "", "prompt": "Positive integers p,q are coprime; c,d are positive. How many residues x modulo cdpq solve cx=ca mod cp and dx=db mod dq?", "response_mode": "single_choice", "options": { "A": "cd: cancel the integer factors to get x=a mod p and x=b mod q; CRT gives one class mod pq, which has cd lifts mod cdpq.", "B": "1: CRT always gives one class regardless of the requested modulus.", "C": "gcd(c,d): the two scaling factors determine all compatibility constraints.", "D": "pq: each residue mod cd gives one solution." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "9.1" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "D": 0.0, "B": 0.0, "A": 1.0, "C": 0.0 }, "batch_response_ms": 383.8550000000396, "repaired": false }, { "id": "10.1a", "section": 10, "context": "", "prompt": "Polynomials f,g over a field have degrees k>j. What is deg(f+g)?", "response_mode": "single_choice", "options": { "A": "Could be any value below k", "B": "k+j", "C": "j", "D": "k" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "10.1a" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.71, "probabilities": { "D": 0.79, "B": 0.0, "A": 0.21, "C": 0.0 }, "batch_response_ms": 311.4189580010134, "repaired": false }, { "id": "10.1b", "section": 10, "context": "", "prompt": "Nonzero polynomials f,g over a field have degrees k,j. What is deg(fg)?", "response_mode": "single_choice", "options": { "A": "k−j", "B": "k*j", "C": "k+j", "D": "max(k,j)" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "10.1b" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.0, "C": 1.0 }, "batch_response_ms": 311.4189580010134, "repaired": false }, { "id": "10.2", "section": 10, "context": "", "prompt": "Over GF(5), which degree-at-most-4 polynomial is equal as a function to 4x^84+x^23+2x^5+3?", "response_mode": "single_choice", "options": { "A": "4x^4+x^3+2x+3", "B": "4x^4+x^2+2x+3", "C": "4x+x^3+2x+3", "D": "4+x^3+2x+3" }, "correct": true, "kind": "multi_step", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "10.2" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.79, "probabilities": { "D": 0.03, "A": 0.84, "B": 0.05, "C": 0.08 }, "batch_response_ms": 311.4189580010134, "repaired": false }, { "id": "10.3", "section": 10, "context": "", "prompt": "A degree-2 polynomial over GF(7) passes through (0,1),(3,0),(4,0). Which is it?", "response_mode": "single_choice", "options": { "A": "4(x−3)(x−4)", "B": "3(x−3)(x−4)", "C": "3(x−2)(x−4)", "D": "(x−3)(x−4)" }, "correct": true, "kind": "multi_step", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "10.3" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.45, "probabilities": { "D": 0.17, "B": 0.58, "A": 0.25, "C": 0.0 }, "batch_response_ms": 311.4189580010134, "repaired": false }, { "id": "10.4", "section": 10, "context": "", "prompt": "How many formal polynomials of degree at most 4 over GF(5) pass through two given points with distinct x-coordinates?", "response_mode": "single_choice", "options": { "A": "5", "B": "625", "C": "125", "D": "25" }, "correct": false, "kind": "calculation", "topic": "polynomials", "original_format": "written", "repairs": [ "Specify distinct x-coordinates, needed for independent constraints." ], "source_parts": [ "10.4" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.58, "probabilities": { "D": 0.05, "B": 0.6799999999999999, "A": 0.02, "C": 0.25 }, "batch_response_ms": 311.4189580010134, "repaired": true }, { "id": "11.1", "section": 11, "context": "There are 2 professors, 4 head TAs, 7 TAs. Allowed coalitions contain both professors, or one professor and at least three heads, or one professor, two heads and at least four TAs. No other coalition should reconstruct. Each share uses a distinct nonzero field coordinate.", "prompt": "Choose a valid Shamir configuration (degree; shares per professor, head TA, TA; sufficient prime bound). Secret s is a nonnegative integer.", "response_mode": "single_choice", "options": { "A": "(23; 12,3,1; prime p>=max(s+1,44))", "B": "(23; 11,4,1; prime p>=max(s+1,46))", "C": "(22; 12,4,1; prime p>=max(s+1,48))", "D": "(23; 12,4,1; prime p>=max(s+1,48))" }, "correct": true, "kind": "multi_step", "topic": "secret_sharing", "original_format": "written", "repairs": [], "source_parts": [ "11.1a", "11.1b", "11.1c", "11.1d", "11.1e" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.82, "probabilities": { "B": 0.03, "A": 0.07, "C": 0.04, "D": 0.86 }, "batch_response_ms": 349.7750829992583, "repaired": false }, { "id": "11.2", "section": 11, "context": "", "prompt": "A polynomial of degree 9 is transmitted by evaluations. At most two values are corrupted. How many evaluations suffice for unique recovery?", "response_mode": "single_choice", "options": { "A": "12", "B": "13", "C": "18", "D": "14" }, "correct": false, "kind": "calculation", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "11.2" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.7, "probabilities": { "B": 0.77, "A": 0.08, "C": 0.01, "D": 0.14 }, "batch_response_ms": 349.7750829992583, "repaired": false }, { "id": "12.1", "section": 12, "context": "", "prompt": "How many length-30 bit strings contain exactly 12 ones?", "response_mode": "single_choice", "options": { "A": "C(30,12)", "B": "30^12", "C": "2^12", "D": "30!/12!" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "12.1" ], "exam": "midterm_fa25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "D": 0.0, "A": 1.0 }, "batch_response_ms": 373.97812500057626, "repaired": false }, { "id": "12.2", "section": 12, "context": "", "prompt": "How many formal polynomials of degree exactly d over GF(p), where 0<=d0 integers, prove some nonempty subset has sum divisible by N. Hint: prefix sums.", "response_mode": "single_choice", "options": { "A": "Each integer has a distinct residue, so one must be zero.", "B": "The sum of N integers is always divisible by N.", "C": "If a prefix sum is 0 mod N, use that prefix. Otherwise N prefix sums occupy N−1 nonzero residues, so two coincide; their difference is a nonempty block with sum 0 mod N.", "D": "Take the empty subset, whose sum is zero." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [ "Require nonempty subset explicitly, as intended by the supplied proof." ], "source_parts": [ "4.2" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 }, "batch_response_ms": 424.6948750005686, "repaired": true }, { "id": "5.1", "section": 5, "context": "F1=F2=1, Fj=F(j−1)+F(j−2) for j>=3. To prove every natural m is a sum of distinct nonconsecutive-index Fibonacci numbers, assume the claim for all smaller naturals and choose largest k with Fk<=m, m>=2.", "prompt": "Why is m−Fk=Fk+F(k−1)=F(k+1), contradicting maximality of k.", "C": "Because every Fibonacci number is greater than m.", "D": "Because m=Fk by the definition of largest k." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "5.1" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 1.0, "C": 0.0, "A": 0.0, "D": 0.0 }, "batch_response_ms": 308.4194170005503, "repaired": false }, { "id": "5.2", "section": 5, "context": "F1=F2=1, Fj=F(j−1)+F(j−2) for j>=3. To prove every natural m is a sum of distinct nonconsecutive-index Fibonacci numbers, assume the claim for all smaller naturals and choose largest k with Fk<=m, m>=2. It has been established that m−Fk=2 define Rn=sqrt(n) and Rk=sqrt(k*R(k+1)) for k=2 vertices?", "response_mode": "single_choice", "options": { "A": "1", "B": "n−1", "C": "2", "D": "0" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "8.4" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "D": 0.0, "A": 1.0, "C": 0.0, "B": 0.0 }, "batch_response_ms": 312.59441699876334, "repaired": false }, { "id": "8.5a", "section": 8, "context": "", "prompt": "Maximum edges in a bipartite graph with fixed parts A,B?", "response_mode": "single_choice", "options": { "A": "C(|A|+|B|,2)", "B": "|A|*|B|", "C": "min(|A|,|B|)", "D": "|A|+|B|−1" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "8.5a" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.0, "B": 1.0 }, "batch_response_ms": 312.59441699876334, "repaired": false }, { "id": "8.5b", "section": 8, "context": "", "prompt": "Every graph of maximum degree at most two is bipartite.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "8.5b" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.9, "probabilities": { "A": 0.05, "B": 0.95 }, "batch_response_ms": 312.59441699876334, "repaired": false }, { "id": "8.5c", "section": 8, "context": "", "prompt": "Smallest constant D such that every nonempty bipartite planar graph has a vertex of degree at most D?", "response_mode": "single_choice", "options": { "A": "4", "B": "3", "C": "5", "D": "2" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "8.5c" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.46, "probabilities": { "D": 0.02, "A": 0.21, "C": 0.17, "B": 0.6 }, "batch_response_ms": 312.59441699876334, "repaired": false }, { "id": "8.5d", "section": 8, "context": "", "prompt": "For the three-dimensional hypercube, choose one side A of a bipartition.", "response_mode": "single_choice", "options": { "A": "{001,011,101,111}", "B": "{000,011,110,101}", "C": "{000,001,010,011}", "D": "{000,111}" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "8.5d" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.47, "probabilities": { "D": 0.01, "A": 0.38, "C": 0.01, "B": 0.6 }, "batch_response_ms": 312.59441699876334, "repaired": false }, { "id": "8.6", "section": 8, "context": "", "prompt": "Largest possible vertex degree in a simple planar graph on n vertices?", "response_mode": "single_choice", "options": { "A": "6", "B": "5", "C": "3n−6", "D": "n−1" }, "correct": false, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "8.6" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.28, "probabilities": { "D": 0.45, "A": 0.04, "C": 0.05, "B": 0.46 }, "batch_response_ms": 312.59441699876334, "repaired": false }, { "id": "8.7", "section": 8, "context": "", "prompt": "In a simple n-vertex graph, n>=3, what sharp threshold on du+dv guarantees distinct vertices u,v share a neighbor?", "response_mode": "single_choice", "options": { "A": "n−1", "B": "n", "C": "2n−2", "D": "n+1" }, "correct": false, "kind": "multi_step", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "8.7" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.49, "probabilities": { "D": 0.05, "A": 0.61, "C": 0.07, "B": 0.27 }, "batch_response_ms": 312.59441699876334, "repaired": false }, { "id": "8.8", "section": 8, "context": "", "prompt": "Must a connected planar graph with more than 100 vertices have at least three vertices of degree below six?", "response_mode": "single_choice", "options": { "A": "Yes: every vertex of a planar graph has degree below six.", "B": "Yes: if at most two have degree below six, the degree sum is at least 6(n−2)+2=6n−10, exceeding the planar bound 6n−12.", "C": "No: a planar complete graph on 101 vertices is a counterexample.", "D": "No: any long path has fewer than three such vertices." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "8.8" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.0, "B": 1.0 }, "batch_response_ms": 312.59441699876334, "repaired": false }, { "id": "8.9", "section": 8, "context": "", "prompt": "Remove a degree-d vertex v from G. The rest has a given k-coloring. What bound follows when restoring v?", "response_mode": "single_choice", "options": { "A": "k+d", "B": "max(k,1+min(d,k))", "C": "min(k,d)", "D": "k−1" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "8.9" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.83, "probabilities": { "D": 0.01, "A": 0.11, "C": 0.01, "B": 0.87 }, "batch_response_ms": 312.59441699876334, "repaired": false }, { "id": "9.1a", "section": 9, "context": "", "prompt": "Every 3-regular graph has an even number of vertices.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "9.1a" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.93, "probabilities": { "B": 0.97, "A": 0.03 }, "batch_response_ms": 287.6444579997042, "repaired": false }, { "id": "9.1b", "section": 9, "context": "", "prompt": "Justify the parity constraint for a 3-regular graph on n vertices.", "response_mode": "single_choice", "options": { "A": "Every vertex can be paired with a unique neighbor, without needing a matching theorem.", "B": "Every 3-regular graph is bipartite with equal parts.", "C": "The handshake lemma gives 3n=2e, so n is even.", "D": "All graphs with odd maximum degree have even order." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "9.1b" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 0.0, "C": 1.0 }, "batch_response_ms": 287.6444579997042, "repaired": false }, { "id": "9.2a", "section": 9, "context": "", "prompt": "Every 4-regular graph has an odd number of vertices.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "9.2a" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.87, "probabilities": { "B": 0.06, "A": 0.94 }, "batch_response_ms": 287.6444579997042, "repaired": false }, { "id": "9.2b", "section": 9, "context": "", "prompt": "Give a counterexample to the claim that every 4-regular graph has an odd number of vertices.", "response_mode": "single_choice", "options": { "A": "K5: five vertices, each degree four.", "B": "K6 with the three edges of a perfect matching removed: six vertices, each degree four.", "C": "K6: six vertices, each degree five.", "D": "A six-cycle: six vertices, each degree two." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "9.2b" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 1.0, "C": 0.0 }, "batch_response_ms": 287.6444579997042, "repaired": false }, { "id": "9.3", "section": 9, "context": "", "prompt": "Prove any simple n-vertex graph, n>=2, has two equal degrees.", "response_mode": "single_choice", "options": { "A": "The degree sum is even, so every individual degree is even.", "B": "Every simple graph has two leaves.", "C": "Degrees lie in 0..n−1, but 0 and n−1 cannot both occur. At most n−1 possible values remain for n vertices, so pigeonhole applies.", "D": "There are n possible degrees and n vertices, which alone forces a repetition." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "9.3" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "B": 0.0, "C": 1.0 }, "batch_response_ms": 287.6444579997042, "repaired": false }, { "id": "10.1", "section": 10, "context": "A function is k-uniform when each output has either zero or exactly k distinct preimages.", "prompt": "A={0,1,2,3}, B={0,1,2}; f(0)=f(1)=0 and f(2)=f(3)=1. Find k.", "response_mode": "single_choice", "options": { "A": "Unknown", "B": "1", "C": "2", "D": "3" }, "correct": true, "kind": "recognition", "topic": "functions", "original_format": "written", "repairs": [], "source_parts": [ "10.1" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.89, "probabilities": { "D": 0.0, "A": 0.07, "C": 0.92, "B": 0.01 }, "batch_response_ms": 459.4066669997119, "repaired": false }, { "id": "10.2", "section": 10, "context": "A function is k-uniform when each output has either zero or exactly k distinct preimages.", "prompt": "g is k1-uniform and h is k2-uniform. What k is guaranteed for g composed with h?", "response_mode": "single_choice", "options": { "A": "k1+k2", "B": "k1*k2", "C": "Unknown; the composition need not be uniform.", "D": "max(k1,k2)" }, "correct": false, "kind": "recognition", "topic": "functions", "original_format": "written", "repairs": [], "source_parts": [ "10.2" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.61, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.29, "B": 0.71 }, "batch_response_ms": 459.4066669997119, "repaired": false }, { "id": "10.3a", "section": 10, "context": "A function is k-uniform when each output has either zero or exactly k distinct preimages.", "prompt": "f(x)=ax mod prime m, a nonzero, domain and codomain all residues. Find k.", "response_mode": "single_choice", "options": { "A": "1", "B": "a", "C": "m", "D": "Unknown" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "10.3a" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "D": 0.0, "A": 1.0, "C": 0.0, "B": 0.0 }, "batch_response_ms": 459.4066669997119, "repaired": false }, { "id": "10.3b", "section": 10, "context": "A function is k-uniform when each output has either zero or exactly k distinct preimages.", "prompt": "f(x)=ax mod m, gcd(a,m)=d, domain and codomain all residues. Find k.", "response_mode": "single_choice", "options": { "A": "Unknown", "B": "1", "C": "d", "D": "m/d" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "10.3b" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.84, "probabilities": { "D": 0.05, "C": 0.88, "A": 0.06, "B": 0.01 }, "batch_response_ms": 459.4066669997119, "repaired": false }, { "id": "10.4", "section": 10, "context": "A function is k-uniform when each output has either zero or exactly k distinct preimages.", "prompt": "m=pq for distinct primes; f(x)=ax mod m on the unit residues, gcd(a,m)=1. Find k.", "response_mode": "single_choice", "options": { "A": "Unknown", "B": "(p−1)(q−1)", "C": "1", "D": "p−1" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "10.4" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.92, "probabilities": { "D": 0.0, "C": 0.94, "A": 0.04, "B": 0.02 }, "batch_response_ms": 459.4066669997119, "repaired": false }, { "id": "10.5", "section": 10, "context": "A function is k-uniform when each output has either zero or exactly k distinct preimages.", "prompt": "m=pq for distinct primes; f(x)=ax mod m on unit residues, gcd(a,m)=p. Codomain all residues. Find k.", "response_mode": "single_choice", "options": { "A": "q−1", "B": "p", "C": "p−1", "D": "1" }, "correct": false, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [ "Specify finite residue domain and distinct primes." ], "source_parts": [ "10.5" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.09, "probabilities": { "D": 0.3, "A": 0.32, "C": 0.3, "B": 0.08 }, "batch_response_ms": 459.4066669997119, "repaired": true }, { "id": "11.1", "section": 11, "context": "", "prompt": "x=3 mod 5 and x=2 mod 11. Find x mod 55 in 0..54.", "response_mode": "single_choice", "options": { "A": "13", "B": "23", "C": "2", "D": "3" }, "correct": false, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "11.1" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.65, "probabilities": { "C": 0.01, "D": 0.02, "A": 0.23, "B": 0.74 }, "batch_response_ms": 341.4315000009083, "repaired": false }, { "id": "11.2a", "section": 11, "context": "", "prompt": "x=a mod 10 and x=b mod 15, where b−a is divisible by 5. Number of solutions modulo 150?", "response_mode": "single_choice", "options": { "A": "0", "B": "5", "C": "1", "D": "10" }, "correct": false, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "11.2a" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.16, "probabilities": { "C": 0.37, "D": 0.33, "A": 0.02, "B": 0.28 }, "batch_response_ms": 341.4315000009083, "repaired": false }, { "id": "11.2b", "section": 11, "context": "", "prompt": "x=a mod 10 and x=b mod 15, where b−a is not divisible by 5. Number of solutions modulo 150?", "response_mode": "single_choice", "options": { "A": "0", "B": "5", "C": "1", "D": "15" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "11.2b" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.89, "probabilities": { "C": 0.05, "D": 0.01, "A": 0.91, "B": 0.03 }, "batch_response_ms": 341.4315000009083, "repaired": false }, { "id": "11.3", "section": 11, "context": "", "prompt": "For every integer a and prime p, p divides a^p−a.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "11.3" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.91, "probabilities": { "A": 0.04, "B": 0.96 }, "batch_response_ms": 341.4315000009083, "repaired": false }, { "id": "11.4", "section": 11, "context": "", "prompt": "For distinct primes p,q and every integer a, choose a guaranteed divisor of a*(a^((p−1)(q−1))−1).", "response_mode": "single_choice", "options": { "A": "pq", "B": "p²q", "C": "p+q", "D": "p²q²" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [ "Ask guaranteed divisor among alternatives. Original largest-divisor wording can fail: e.g. p=3,q=5 gives an extra universal factor 2." ], "source_parts": [ "11.4" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "C": 0.0, "D": 0.0, "A": 0.99, "B": 0.01 }, "batch_response_ms": 341.4315000009083, "repaired": true }, { "id": "11.5a", "section": 11, "context": "", "prompt": "RSA ciphertexts are y1=x1^e and y2=x2^e modulo N. Ciphertext of x1*x2?", "response_mode": "single_choice", "options": { "A": "(y1*y2)^d mod N", "B": "y1*y2 mod N", "C": "y1+y2 mod N", "D": "(y1+y2)^e mod N" }, "correct": true, "kind": "recognition", "topic": "rsa", "original_format": "written", "repairs": [], "source_parts": [ "11.5a" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 0.0, "D": 0.0, "A": 0.01, "B": 0.99 }, "batch_response_ms": 341.4315000009083, "repaired": false }, { "id": "11.5b", "section": 11, "context": "", "prompt": "In valid RSA, y1=x1^e and y2=x2^e modulo N, with decryption exponent d. Express x1*x2 modulo N.", "response_mode": "single_choice", "options": { "A": "(y1*y2)^e mod N", "B": "(y1*y2)^d mod N", "C": "y1*y2 mod N", "D": "(y1+y2)^d mod N" }, "correct": true, "kind": "recognition", "topic": "rsa", "original_format": "written", "repairs": [], "source_parts": [ "11.5b" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "A": 0.0, "D": 0.0, "B": 1.0 }, "batch_response_ms": 341.4315000009083, "repaired": false }, { "id": "12.1", "section": 12, "context": "p=2^n−1 is a Mersenne prime, n>=2.", "prompt": "Least nonnegative residue of 2^n modulo p?", "response_mode": "single_choice", "options": { "A": "2", "B": "0", "C": "1", "D": "p−1" }, "correct": false, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "12.1" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.69, "probabilities": { "B": 0.0, "D": 0.01, "C": 0.22, "A": 0.77 }, "batch_response_ms": 302.1441249984491, "repaired": false }, { "id": "12.2", "section": 12, "context": "p=2^n−1 is a Mersenne prime, n>=2.", "prompt": "For positive integer a, least nonnegative residue of (2^a)^n modulo p?", "response_mode": "single_choice", "options": { "A": "0", "B": "1", "C": "a", "D": "2^a" }, "correct": false, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "12.2" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.48, "probabilities": { "B": 0.34, "D": 0.61, "C": 0.02, "A": 0.03 }, "batch_response_ms": 302.1441249984491, "repaired": false }, { "id": "12.3", "section": 12, "context": "p=2^n−1 is a Mersenne prime, n>=2.", "prompt": "Show there are at least n distinct residues x with x^n=1 mod p.", "response_mode": "single_choice", "options": { "A": "Use x=2^j for j=0,...,n−1. They are distinct positive integers below p and each nth power is (2^n)^j=1 mod p.", "B": "Use x=2^(jn), j=1,...,n; these are all distinct modulo p.", "C": "A degree-n polynomial must have at least n roots over every field.", "D": "Use all residues 0,...,n−1; every small residue has nth power one." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "12.3" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "C": 0.0, "A": 1.0 }, "batch_response_ms": 302.1441249984491, "repaired": false }, { "id": "13.1", "section": 13, "context": "", "prompt": "Choose a polynomial over GF(7) through (1,0),(2,5).", "response_mode": "single_choice", "options": { "A": "5x+1", "B": "5x+2", "C": "x−1", "D": "2x+5" }, "correct": true, "kind": "calculation", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "13.1" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.79, "probabilities": { "A": 0.1, "C": 0.01, "B": 0.84, "D": 0.05 }, "batch_response_ms": 414.7680419991957, "repaired": false }, { "id": "13.2", "section": 13, "context": "", "prompt": "Over GF(p), p>d, how many formal polynomials of degree at most d pass through d−1 points with distinct x-coordinates?", "response_mode": "single_choice", "options": { "A": "p²", "B": "p^(d−1)", "C": "p", "D": "1" }, "correct": true, "kind": "calculation", "topic": "polynomials", "original_format": "written", "repairs": [ "Distinct x-coordinates explicitly required." ], "source_parts": [ "13.2" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.81, "probabilities": { "A": 0.86, "C": 0.09, "B": 0.05, "D": 0.0 }, "batch_response_ms": 414.7680419991957, "repaired": true }, { "id": "13.3a", "section": 13, "context": "The normalized Lagrange basis polynomials Delta1,...,Delta(d+1) correspond to d+1 points with distinct x-coordinates over a sufficiently large field, d>=1.", "prompt": "How many distinct roots does Delta1 have?", "response_mode": "single_choice", "options": { "A": "d", "B": "d+1", "C": "d−1", "D": "Unknown" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "13.3a" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.97, "probabilities": { "A": 0.98, "C": 0.0, "B": 0.0, "D": 0.02 }, "batch_response_ms": 414.7680419991957, "repaired": false }, { "id": "13.3bi", "section": 13, "context": "The normalized Lagrange basis polynomials Delta1,...,Delta(d+1) correspond to d+1 points with distinct x-coordinates over a sufficiently large field, d>=1.", "prompt": "Formal degree of Delta1*Delta2?", "response_mode": "single_choice", "options": { "A": "d+1", "B": "2d", "C": "d", "D": "Unknown" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "13.3bi" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.78, "probabilities": { "A": 0.02, "C": 0.13, "B": 0.83, "D": 0.02 }, "batch_response_ms": 414.7680419991957, "repaired": false }, { "id": "13.3bii", "section": 13, "context": "The normalized Lagrange basis polynomials Delta1,...,Delta(d+1) correspond to d+1 points with distinct x-coordinates over a sufficiently large field, d>=1.", "prompt": "Number of distinct roots of Delta1*Delta2?", "response_mode": "single_choice", "options": { "A": "d+1", "B": "2d", "C": "d", "D": "Unknown" }, "correct": false, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "13.3bii" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.42, "probabilities": { "A": 0.09, "C": 0.27, "B": 0.56, "D": 0.07 }, "batch_response_ms": 414.7680419991957, "repaired": false }, { "id": "13.3c", "section": 13, "context": "", "prompt": "Every real polynomial of degree d has exactly d distinct real roots.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "choice", "repairs": [], "source_parts": [ "13.3c" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "A": 0.0, "B": 1.0 }, "batch_response_ms": 414.7680419991957, "repaired": false }, { "id": "14.1", "section": 14, "context": "", "prompt": "Packets sufficient to recover n message symbols with at most e erased packets and k corruptions?", "response_mode": "single_choice", "options": { "A": "n+2e+k", "B": "n+e+2k", "C": "n+e+k", "D": "n+2(e+k)" }, "correct": true, "kind": "recognition", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "14.1" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.44, "probabilities": { "D": 0.2, "A": 0.21, "C": 0.01, "B": 0.58 }, "batch_response_ms": 311.50424999941606, "repaired": false }, { "id": "14.2", "section": 14, "context": "", "prompt": "Minimal error locator is x²−1 over GF(13). Give the error coordinates in 0..12.", "response_mode": "single_choice", "options": { "A": "{1,2}", "B": "{0,1}", "C": "{1,12}", "D": "{1,6}" }, "correct": true, "kind": "calculation", "topic": "coding", "original_format": "written", "repairs": [ "Specify minimal error locator, excluding arbitrary extra roots from padding." ], "source_parts": [ "14.2" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "D": 0.0, "A": 0.0, "C": 1.0, "B": 0.0 }, "batch_response_ms": 311.50424999941606, "repaired": true }, { "id": "14.3", "section": 14, "context": "", "prompt": "Shamir shares secret 5 with ten participants, any three reconstruct. Smallest prime modulus, with nonzero distinct share coordinates?", "response_mode": "single_choice", "options": { "A": "7", "B": "5", "C": "11", "D": "13" }, "correct": true, "kind": "calculation", "topic": "secret_sharing", "original_format": "written", "repairs": [], "source_parts": [ "14.3" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.24, "probabilities": { "D": 0.42, "A": 0.14, "C": 0.43, "B": 0.01 }, "batch_response_ms": 311.50424999941606, "repaired": false }, { "id": "14.4", "section": 14, "context": "", "prompt": "Design a scheme requiring a strict majority of both a five-person group and a seven-person group.", "response_mode": "single_choice", "options": { "A": "Use a random degree-one master polynomial with secret at zero. Share one of its nonzero-coordinate values using degree two among the five people, and another using degree three among the seven; use one sufficiently large field.", "B": "Use one degree-two polynomial and give all twelve people one share each.", "C": "Use a degree-one master polynomial and give every participant both required master points.", "D": "Use two independent polynomials of degrees two and three, each with the full secret at zero; either group then suffices." }, "correct": true, "kind": "multi_step", "topic": "secret_sharing", "original_format": "written", "repairs": [], "source_parts": [ "14.4" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.57, "probabilities": { "C": 0.05, "A": 0.68, "D": 0.24, "B": 0.03 }, "batch_response_ms": 311.50424999941606, "repaired": false }, { "id": "15.1", "section": 15, "context": "", "prompt": "The set of finite subsets of a countably infinite set is uncountable.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "15.1" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.01, "A": 0.99 }, "batch_response_ms": 337.7193750020524, "repaired": false }, { "id": "15.2", "section": 15, "context": "", "prompt": "The set of all subsets of a countably infinite set is uncountable.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "countability", "original_format": "choice", "repairs": [], "source_parts": [ "15.2" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "A": 1.0 }, "batch_response_ms": 337.7193750020524, "repaired": false }, { "id": "15.3a", "section": 15, "context": "", "prompt": "Every real number has a finite program that halts on input n and returns its nth digit.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "computability", "original_format": "choice", "repairs": [], "source_parts": [ "15.3a" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.95, "probabilities": { "B": 0.03, "A": 0.97 }, "batch_response_ms": 337.7193750020524, "repaired": false }, { "id": "15.3b", "section": 15, "context": "", "prompt": "Why are some real numbers not computable digit-by-digit?", "response_mode": "single_choice", "options": { "A": "Every real has infinitely many digits, so none is computable.", "B": "There are countably many finite programs but uncountably many reals, so programs cannot cover all reals.", "C": "A program can output only finitely many digits over all inputs.", "D": "Uncomputable reals must all be rational." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "15.3b" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 1.0, "D": 0.0, "C": 0.0, "A": 0.0 }, "batch_response_ms": 337.7193750020524, "repaired": false }, { "id": "16.1", "section": 16, "context": "", "prompt": "Distinct anagrams of ABRACADABRA?", "response_mode": "single_choice", "options": { "A": "11!/5!", "B": "11!/(5!2!2!)", "C": "11!/(4!2!2!)", "D": "10!/(5!2!2!)" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "16.1" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.97, "probabilities": { "D": 0.0, "A": 0.01, "C": 0.02, "B": 0.97 }, "batch_response_ms": 352.52466699967044, "repaired": false }, { "id": "16.2", "section": 16, "context": "", "prompt": "Strings with exactly four As and three Bs?", "response_mode": "single_choice", "options": { "A": "4!*3!", "B": "C(7,3)", "C": "2^7", "D": "7!" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "16.2" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.99, "A": 0.01, "D": 0.0, "C": 0.0 }, "batch_response_ms": 352.52466699967044, "repaired": false }, { "id": "16.3", "section": 16, "context": "", "prompt": "Distribute n1 identical apples and n2 identical bananas among k named people, allowing zero. Count?", "response_mode": "single_choice", "options": { "A": "C(n1+n2+k−1,k−1)", "B": "C(n1+k−1,k−1)*C(n2+k−1,k−1)", "C": "k^(n1+n2)", "D": "C(n1,k)*C(n2,k)" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "16.3" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 1.0, "A": 0.0, "D": 0.0, "C": 0.0 }, "batch_response_ms": 352.52466699967044, "repaired": false }, { "id": "16.4", "section": 16, "context": "", "prompt": "Walk starts at 0 and ends at n>=0 after n+2k unit steps, each ±1, unrestricted positions. Number of step sequences?", "response_mode": "single_choice", "options": { "A": "2^(n+2k)", "B": "C(n+2k,n)", "C": "C(n+k,k)", "D": "C(n+2k,k)" }, "correct": true, "kind": "multi_step", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "16.4" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.29, "probabilities": { "B": 0.29, "A": 0.21, "D": 0.45999999999999996, "C": 0.04 }, "batch_response_ms": 352.52466699967044, "repaired": false }, { "id": "16.5a", "section": 16, "context": "", "prompt": "Sn counts ordered sums of ones and twos equaling n. Give a summation.", "response_mode": "single_choice", "options": { "A": "sum(i=0..n) C(n,i)", "B": "sum(i=0..floor(n/2)) C(n−i,i)", "C": "sum(i=0..floor(n/2)) 2^i", "D": "sum(i=0..floor(n/2)) C(n,i)" }, "correct": true, "kind": "multi_step", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "16.5a" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "A": 0.0, "B": 1.0, "D": 0.0, "C": 0.0 }, "batch_response_ms": 352.52466699967044, "repaired": false }, { "id": "16.5b", "section": 16, "context": "", "prompt": "With S0=S1=1, recurrence for ordered sums of ones and twos equaling n, n>=2?", "response_mode": "single_choice", "options": { "A": "Sn=2S(n−1)", "B": "Sn=S(n−1)+S(n−2)", "C": "Sn=S(n−1)+1", "D": "Sn=S(n−1)*S(n−2)" }, "correct": true, "kind": "recognition", "topic": "recurrences", "original_format": "written", "repairs": [], "source_parts": [ "16.5b" ], "exam": "midterm_sp23", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 1.0, "A": 0.0, "D": 0.0, "C": 0.0 }, "batch_response_ms": 352.52466699967044, "repaired": false }, { "id": "1.1", "section": 1, "context": "", "prompt": "A judge says: if Bob stole the diamonds, he did not act alone. A truthful attorney denies that statement. What must be true?", "response_mode": "single_choice", "options": { "A": "Bob stole the diamonds, and did so alone.", "B": "Bob may have stolen the diamonds, but we do not know for sure.", "C": "Bob stole the diamonds with someone else.", "D": "Bob did not steal the diamonds.", "E": "None of the above." }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "1.1" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.95, "probabilities": { "C": 0.0, "E": 0.01, "A": 0.96, "B": 0.0, "D": 0.03 }, "batch_response_ms": 397.95404100004816, "repaired": false }, { "id": "2.1", "section": 2, "context": "", "prompt": "If the halting problem is computable, then pigs can fly.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "computability", "original_format": "choice", "repairs": [], "source_parts": [ "2.1" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.66, "probabilities": { "A": 0.17, "B": 0.83 }, "batch_response_ms": 318.3907499987981, "repaired": false }, { "id": "2.2", "section": 2, "context": "", "prompt": "forall x [P(x) implies Q(x)] is equivalent to NOT exists x [NOT P(x) AND Q(x)].", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.2" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "A": 0.99, "B": 0.01 }, "batch_response_ms": 318.3907499987981, "repaired": false }, { "id": "2.3", "section": 2, "context": "", "prompt": "[Q(0) AND forall n>=0 ((P(n) implies Q(n+1)) AND (Q(n) implies P(n)))] implies forall n>=0 P(n).", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.3" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.43, "probabilities": { "A": 0.29, "B": 0.71 }, "batch_response_ms": 318.3907499987981, "repaired": false }, { "id": "2.4", "section": 2, "context": "", "prompt": "[Q(0) AND forall n>=0 ((P(n) implies Q(n+1)) AND (Q(n) implies P(n+1)))] implies forall n>=0 P(n).", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.4" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.65, "probabilities": { "A": 0.18, "B": 0.82 }, "batch_response_ms": 318.3907499987981, "repaired": false }, { "id": "2.5", "section": 2, "context": "", "prompt": "There is an algorithm deciding whether an input finite decimal string is the corresponding prefix of pi’s decimal expansion.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "computability", "original_format": "choice", "repairs": [], "source_parts": [ "2.5" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.63, "probabilities": { "A": 0.19, "B": 0.81 }, "batch_response_ms": 318.3907499987981, "repaired": false }, { "id": "2.6", "section": 2, "context": "", "prompt": "There is an algorithm deciding, for arbitrary program P and strings x,y, whether P on x outputs y.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "computability", "original_format": "choice", "repairs": [], "source_parts": [ "2.6" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.98, "probabilities": { "A": 0.01, "B": 0.99 }, "batch_response_ms": 318.3907499987981, "repaired": false }, { "id": "3.1", "section": 3, "context": "", "prompt": "A pair present in both the job-proposing and candidate-proposing stable matchings is present in every stable matching.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "3.1" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.82, "probabilities": { "B": 0.09, "A": 0.91 }, "batch_response_ms": 349.6895000025688, "repaired": false }, { "id": "3.2", "section": 3, "context": "", "prompt": "For every n, there can be a job-optimal stable matching containing a pair who rank each other last.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "3.2" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.62, "probabilities": { "B": 0.19, "A": 0.81 }, "batch_response_ms": 349.6895000025688, "repaired": false }, { "id": "3.3", "section": 3, "context": "", "prompt": "Two participants who rank each other first must be paired in every stable matching.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "3.3" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.95, "probabilities": { "B": 0.03, "A": 0.97 }, "batch_response_ms": 349.6895000025688, "repaired": false }, { "id": "3.4", "section": 3, "context": "", "prompt": "Two disjoint pairs, each pair ranking one another last, cannot both occur in a stable matching.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "3.4" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.15, "probabilities": { "B": 0.57, "A": 0.43 }, "batch_response_ms": 349.6895000025688, "repaired": false }, { "id": "3.5", "section": 3, "context": "", "prompt": "If job J is ranked last by every candidate other than C, and C is ranked last by every job other than J, then J and C must be matched in every stable matching.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "3.5" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.59, "probabilities": { "B": 0.79, "A": 0.21 }, "batch_response_ms": 349.6895000025688, "repaired": false }, { "id": "4.a", "section": 4, "context": "P: powered on. M: in motion. A: alarm activated. R: rerouting. S: senses an obstacle.", "prompt": "Express: the robot is not in motion unless powered on.", "response_mode": "single_choice", "options": { "A": "P AND M", "B": "NOT M implies NOT P", "C": "P implies M", "D": "M implies P" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "written", "repairs": [], "source_parts": [ "4.a" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "C": 0.0, "A": 0.0 }, "batch_response_ms": 323.58054199721664, "repaired": false }, { "id": "4.b", "section": 4, "context": "P: powered on. M: in motion. A: alarm activated. R: rerouting. S: senses an obstacle.", "prompt": "Contrapositive of: if powered on, it is in motion or its alarm is activated.", "response_mode": "single_choice", "options": { "A": "NOT P implies (NOT M AND NOT A)", "B": "(NOT M AND NOT A) implies NOT P", "C": "(NOT M OR NOT A) implies NOT P", "D": "(M OR A) implies P" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "written", "repairs": [], "source_parts": [ "4.b" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.0, "B": 1.0 }, "batch_response_ms": 323.58054199721664, "repaired": false }, { "id": "4.c", "section": 4, "context": "P: powered on. M: in motion. A: alarm activated. R: rerouting. S: senses an obstacle. Rules: M=>P; (S AND A)=>(NOT R OR NOT P); (P AND M AND R)=>S; (P AND S)=>(A OR NOT M).", "prompt": "Given M true, what follows about R,S?", "response_mode": "single_choice", "options": { "A": "R true; S could be either", "B": "R false; S true", "C": "R false; S could be either", "D": "R true; S true", "E": "R false; S false", "F": "R true; S false" }, "correct": true, "kind": "multi_step", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "4.c" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.38, "probabilities": { "D": 0.1, "E": 0.18, "B": 0.03, "F": 0.08, "A": 0.14, "C": 0.47 }, "batch_response_ms": 323.58054199721664, "repaired": false }, { "id": "4.d", "section": 4, "context": "", "prompt": "L(t) means low battery at t, C(t) means charging. Express: whenever battery is low, it charges at some strictly later time.", "response_mode": "single_choice", "options": { "A": "forall t>=0, C(t) implies exists u>t L(u)", "B": "forall t>=0, L(t) implies exists u>t C(u)", "C": "forall t>=0, L(t) implies C(t)", "D": "exists u>=0, forall t>=0, L(t) implies C(u)" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "written", "repairs": [], "source_parts": [ "4.d" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.0, "B": 1.0 }, "batch_response_ms": 323.58054199721664, "repaired": false }, { "id": "5.a", "section": 5, "context": "", "prompt": "Find the fatal error and explanation in this attempted induction that 2^n=1 for all n>=0: base 2^0=1; write n+1=i+j with positive i,j<=n; apply strong induction to i,j; multiply 2^i 2^j=1.", "response_mode": "single_choice", "options": { "A": "The base case is wrong because 2^0=0.", "B": "The multiplication identity 2^(i+j)=2^i*2^j is false.", "C": "Strong induction cannot assume two earlier cases.", "D": "The decomposition fails at n=0: 1 cannot be the sum of two positive integers." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "5.a" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 0.0, "D": 1.0 }, "batch_response_ms": 294.62887499903445, "repaired": false }, { "id": "5.b", "section": 5, "context": "", "prompt": "Find the error: base triangle angle sum is 180; assume all n-gons have angle sum 180(n−2); add a triangle to an n-gon to make an (n+1)-gon; the angle sum increases by 180; conclude all (n+1)-gons satisfy the formula.", "response_mode": "single_choice", "options": { "A": "The construction has not shown that every (n+1)-gon is obtainable this way; start with an arbitrary larger polygon and justify removing a triangle.", "B": "The base case fails because triangles have angle sum 360.", "C": "Adding the three angles of a triangle gives 360, not 180.", "D": "Induction can never prove a geometric statement." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "5.b" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "C": 0.0, "A": 1.0, "B": 0.0 }, "batch_response_ms": 294.62887499903445, "repaired": false }, { "id": "6.a", "section": 6, "context": "", "prompt": "F0=0,F1=1,Fn=F(n−1)+F(n−2). Which induction proves sum(Fi²,i=0..n)=Fn*F(n+1)?", "response_mode": "single_choice", "options": { "A": "Every Fibonacci number equals its index, so the claim reduces to the sum of squares formula.", "B": "Square the recurrence and discard its cross term to make the sum telescope.", "C": "Base n=0 gives zero on both sides. Add F(k+1)² to Fk*F(k+1), factor to F(k+1)[Fk+F(k+1)]=F(k+1)F(k+2).", "D": "Base n=0 holds. Add F(k+1) instead of its square, obtaining the next product." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "6.a" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 }, "batch_response_ms": 581.02966700244, "repaired": false }, { "id": "6.b", "section": 6, "context": "", "prompt": "Prove sum(i³,i=1..n)=[sum(i,i=1..n)]². You may use sum(i)=n(n+1)/2.", "response_mode": "single_choice", "options": { "A": "Base n=1 holds. By induction, the next sum is [k(k+1)/2]²+(k+1)³=(k+1)²(k+2)²/4, the required next square.", "B": "Cube the formula for sum(i), because the cube of a sum is the sum of cubes.", "C": "The next difference between the squared sums is (k+1)², equal to the next cube.", "D": "Checking n=1 and n=2 suffices, since both sides are polynomials of arbitrary degree." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "6.b" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 1.0, "C": 0.0 }, "batch_response_ms": 581.02966700244, "repaired": false }, { "id": "7.a", "section": 7, "context": "Vertices a,b,c,d,e,f. Edges ab,af,bf,bc,fc,fe,ce,cd,de. There are no other edges.", "prompt": "Which one new edge creates an Eulerian tour?", "response_mode": "single_choice", "options": { "A": "b–d", "B": "a–e", "C": "a–d", "D": "b–e" }, "correct": false, "kind": "multi_step", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.a" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.03, "probabilities": { "C": 0.28, "D": 0.2, "A": 0.25, "B": 0.27 }, "batch_response_ms": 420.4808750000666, "repaired": false }, { "id": "7.b", "section": 7, "context": "Vertices a,b,c,d,e,f,g. Edges ab,ad,bc,bf,ce,de,ef,fg,eg. There are no other edges.", "prompt": "Which one vertex must be removed so the remaining graph has a Hamiltonian cycle?", "response_mode": "single_choice", "options": { "A": "g", "B": "b", "C": "c", "D": "a" }, "correct": false, "kind": "multi_step", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.b" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.22, "probabilities": { "C": 0.16, "D": 0.41, "A": 0.1, "B": 0.33 }, "batch_response_ms": 420.4808750000666, "repaired": false }, { "id": "7.c", "section": 7, "context": "", "prompt": "Prove a tree with a degree-k vertex has at least k leaves.", "response_mode": "single_choice", "options": { "A": "The handshake lemma says every tree has exactly k leaves regardless of its other degrees.", "B": "All k neighbors must be leaves because trees have no cycles.", "C": "For k>=2, remove that vertex; its k components each contain a leaf of the original tree, by following a longest path away from the removed vertex. The k=0,1 cases hold directly.", "D": "Remove the vertex; all remaining vertices become isolated and therefore leaves." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "7.c" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 1.0, "D": 0.0, "A": 0.0, "B": 0.0 }, "batch_response_ms": 420.4808750000666, "repaired": false }, { "id": "7.d", "section": 7, "context": "", "prompt": "A connected graph has 80 vertices, 135 edges and girth 5. Is it planar? Justify.", "response_mode": "single_choice", "options": { "A": "No. Euler and 5f<=2e imply e<=5(v−2)/3=130, violated by 135.", "B": "Yes. The inequality e<=3v−6 holds and is sufficient for planarity.", "C": "No. Any graph with a five-cycle is nonplanar.", "D": "Yes. Odd girth guarantees a planar embedding." }, "correct": true, "kind": "multi_step", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.d" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "D": 0.0, "A": 1.0, "B": 0.0 }, "batch_response_ms": 420.4808750000666, "repaired": false }, { "id": "8.ai", "section": 8, "context": "", "prompt": "Compute 1!+3!+5!+...+99! modulo 24.", "response_mode": "single_choice", "options": { "A": "1", "B": "6", "C": "7", "D": "0" }, "correct": true, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "8.ai" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.49, "probabilities": { "B": 0.18, "D": 0.05, "C": 0.61, "A": 0.16 }, "batch_response_ms": 364.05420799928834, "repaired": false }, { "id": "8.aii", "section": 8, "context": "", "prompt": "Solve 5x=7 mod 11, x in 0..10.", "response_mode": "single_choice", "options": { "A": "2", "B": "5", "C": "9", "D": "8" }, "correct": false, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "8.aii" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.26, "probabilities": { "B": 0.1, "D": 0.37, "C": 0.45, "A": 0.08 }, "batch_response_ms": 364.05420799928834, "repaired": false }, { "id": "8.aiii", "section": 8, "context": "", "prompt": "Units digit of 7^96?", "response_mode": "single_choice", "options": { "A": "1", "B": "3", "C": "9", "D": "7" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "8.aiii" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.86, "probabilities": { "B": 0.0, "D": 0.0, "C": 0.1, "A": 0.9 }, "batch_response_ms": 364.05420799928834, "repaired": false }, { "id": "8.aiv", "section": 8, "context": "RSA uses primes p=7,q=11 and public exponent e=11.", "prompt": "Give the public key (N,e).", "response_mode": "single_choice", "options": { "A": "(77,7)", "B": "(77,11)", "C": "(60,11)", "D": "(18,11)" }, "correct": true, "kind": "calculation", "topic": "rsa", "original_format": "written", "repairs": [], "source_parts": [ "8.aiv" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 1.0, "D": 0.0, "C": 0.0, "A": 0.0 }, "batch_response_ms": 364.05420799928834, "repaired": false }, { "id": "8.av", "section": 8, "context": "RSA uses primes p=7,q=11 and public exponent e=11.", "prompt": "Encrypt message 3.", "response_mode": "single_choice", "options": { "A": "27", "B": "11", "C": "33", "D": "47" }, "correct": true, "kind": "multi_step", "topic": "rsa", "original_format": "written", "repairs": [], "source_parts": [ "8.av" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.31, "probabilities": { "B": 0.06, "D": 0.48, "C": 0.13, "A": 0.33 }, "batch_response_ms": 364.05420799928834, "repaired": false }, { "id": "8.avi", "section": 8, "context": "RSA uses primes p=7,q=11 and public exponent e=11.", "prompt": "Decrypt ciphertext 47.", "response_mode": "single_choice", "options": { "A": "3", "B": "7", "C": "47", "D": "11" }, "correct": false, "kind": "multi_step", "topic": "rsa", "original_format": "written", "repairs": [], "source_parts": [ "8.avi" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.23, "probabilities": { "B": 0.21, "D": 0.04, "C": 0.43, "A": 0.32 }, "batch_response_ms": 364.05420799928834, "repaired": false }, { "id": "8.b", "section": 8, "context": "", "prompt": "Three products appear every 5,3,11 days and were last available 3,1,2 days ago respectively. Earliest number of days from today when all are available? Hint: CRT.", "response_mode": "single_choice", "options": { "A": "97", "B": "42", "C": "165", "D": "152" }, "correct": false, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "8.b" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.32, "probabilities": { "B": 0.16, "D": 0.23, "C": 0.12, "A": 0.49 }, "batch_response_ms": 364.05420799928834, "repaired": false }, { "id": "9.a", "section": 9, "context": "Over GF(p), prime p>20, a polynomial P has degree at most 2. Treasure coordinates are (P(10),P(20)). Hints give values at distinct field coordinates other than 10 and 20.", "prompt": "Hints needed to determine the treasure location?", "response_mode": "single_choice", "options": { "A": "3", "B": "4", "C": "1", "D": "2" }, "correct": false, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [ "Use degree at most 2 to match the source’s interpolation/counting convention." ], "source_parts": [ "9.a" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.31, "probabilities": { "B": 0.01, "C": 0.16, "A": 0.35, "D": 0.48 }, "batch_response_ms": 369.33124999995925, "repaired": true }, { "id": "9.b", "section": 9, "context": "Over GF(p), prime p>20, a polynomial P has degree at most 2. Treasure coordinates are (P(10),P(20)). Hints give values at distinct field coordinates other than 10 and 20.", "prompt": "After one hint, number of still-possible treasure locations?", "response_mode": "single_choice", "options": { "A": "p−1", "B": "p²", "C": "p", "D": "1" }, "correct": false, "kind": "calculation", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "9.b" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.57, "probabilities": { "B": 0.12, "C": 0.68, "A": 0.18, "D": 0.02 }, "batch_response_ms": 369.33124999995925, "repaired": false }, { "id": "9.c", "section": 9, "context": "Over GF(p), prime p>20, a polynomial P has degree at most 2. Treasure coordinates are (P(10),P(20)). Hints give values at distinct field coordinates other than 10 and 20.", "prompt": "After two hints, number of still-possible treasure locations?", "response_mode": "single_choice", "options": { "A": "p−1", "B": "1", "C": "p²", "D": "p" }, "correct": false, "kind": "calculation", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "9.c" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "B" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.79, "probabilities": { "B": 0.85, "C": 0.02, "A": 0.03, "D": 0.1 }, "batch_response_ms": 369.33124999995925, "repaired": false }, { "id": "9.d", "section": 9, "context": "Over GF(p), prime p>20, a polynomial P has degree at most 2. Treasure coordinates are (P(10),P(20)). Hints give values at distinct field coordinates other than 10 and 20.", "prompt": "Hints P(0)=5,P(1)=5,P(3)=11. Find P. Hint: solve for coefficients rather than use Lagrange.", "response_mode": "single_choice", "options": { "A": "x²−x+5", "B": "x²+x+5", "C": "x+5", "D": "2x²−2x+5" }, "correct": true, "kind": "multi_step", "topic": "polynomials", "original_format": "written", "repairs": [ "Allow coordinates 0 and 1, used by source despite its earlier k>=2 restriction." ], "source_parts": [ "9.d" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.65, "probabilities": { "B": 0.05, "C": 0.07, "A": 0.73, "D": 0.15 }, "batch_response_ms": 369.33124999995925, "repaired": true }, { "id": "9.e", "section": 9, "context": "Over GF(p), prime p>20, a polynomial P has degree at most 2. Treasure coordinates are (P(10),P(20)). Hints give values at distinct field coordinates other than 10 and 20.", "prompt": "For p=89 and hints P(0)=5,P(1)=5,P(3)=11, find the treasure coordinates.", "response_mode": "single_choice", "options": { "A": "(16,69)", "B": "(6,29)", "C": "(5,11)", "D": "(95,385)" }, "correct": false, "kind": "multi_step", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "9.e" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.32, "probabilities": { "B": 0.23, "C": 0.21, "A": 0.48, "D": 0.08 }, "batch_response_ms": 369.33124999995925, "repaired": false }, { "id": "9.f", "section": 9, "context": "Over GF(p), prime p>20, a polynomial P has degree at most 2. Treasure coordinates are (P(10),P(20)). Hints give values at distinct field coordinates other than 10 and 20.", "prompt": "At most one hint is corrupted. Minimum hints to detect whether a corruption occurred, using only the hints and no visits to treasure locations?", "response_mode": "single_choice", "options": { "A": "5", "B": "2", "C": "3", "D": "4" }, "correct": false, "kind": "recognition", "topic": "coding", "original_format": "written", "repairs": [ "Disallow physical location checks; source accepted an alternative three-hint strategy using extra observations." ], "source_parts": [ "9.f" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.44, "probabilities": { "B": 0.12, "C": 0.58, "A": 0.12, "D": 0.18 }, "batch_response_ms": 369.33124999995925, "repaired": true }, { "id": "9.g", "section": 9, "context": "Over GF(p), prime p>20, a polynomial P has degree at most 2. Treasure coordinates are (P(10),P(20)). Hints give values at distinct field coordinates other than 10 and 20.", "prompt": "One hint is corrupted at an unknown location. Minimum hints guaranteeing recovery of the correct polynomial?", "response_mode": "single_choice", "options": { "A": "6", "B": "4", "C": "3", "D": "5" }, "correct": true, "kind": "recognition", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "9.g" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.26, "probabilities": { "B": 0.33, "C": 0.11, "A": 0.12, "D": 0.44 }, "batch_response_ms": 369.33124999995925, "repaired": false }, { "id": "10.a", "section": 10, "context": "", "prompt": "Is the set of integer grid points (x,y) with gcd(x,y)=1 countable? Justify.", "response_mode": "single_choice", "options": { "A": "No; the plane is uncountable.", "B": "No; coprimality prevents enumeration.", "C": "Yes; it is a subset of the countable set Z×Z.", "D": "Yes; it is finite." }, "correct": true, "kind": "multi_step", "topic": "countability", "original_format": "written", "repairs": [], "source_parts": [ "10.a" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 }, "batch_response_ms": 299.58137499852455, "repaired": false }, { "id": "10.b", "section": 10, "context": "", "prompt": "A vehicle starts at the origin facing up and chooses left, straight or right at each grid intersection. Are its finite routes countable?", "response_mode": "single_choice", "options": { "A": "No; three choices at every step make even finite strings uncountable.", "B": "No; routes may have arbitrary finite length.", "C": "Yes; finite routes are finite strings over a three-symbol alphabet.", "D": "Yes; there are only three possible routes." }, "correct": true, "kind": "multi_step", "topic": "countability", "original_format": "written", "repairs": [], "source_parts": [ "10.b" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 }, "batch_response_ms": 299.58137499852455, "repaired": false }, { "id": "10.c", "section": 10, "context": "", "prompt": "A vehicle starts at the origin facing up and chooses left, straight or right at each grid intersection. Are all infinite routes countable?", "response_mode": "single_choice", "options": { "A": "No; there are only finitely many infinite routes.", "B": "Yes; the union of all finite routes contains every infinite route.", "C": "No; restricting to two turn choices embeds all infinite binary sequences.", "D": "Yes; each route visits only countably many intersections." }, "correct": true, "kind": "multi_step", "topic": "countability", "original_format": "written", "repairs": [], "source_parts": [ "10.c" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.0, "D": 0.01, "A": 0.0, "C": 0.99 }, "batch_response_ms": 299.58137499852455, "repaired": false }, { "id": "10.d", "section": 10, "context": "", "prompt": "At each (x,y) in Z² there are |x|+|y| parking spaces. Is the set of all spaces countable?", "response_mode": "single_choice", "options": { "A": "Yes; there is a finite total number of spaces.", "B": "No; the number at an intersection is unbounded across the grid.", "C": "Yes; a countable union of finite sets is countable.", "D": "No; a union of infinitely many finite sets is always uncountable." }, "correct": true, "kind": "multi_step", "topic": "countability", "original_format": "written", "repairs": [], "source_parts": [ "10.d" ], "exam": "midterm_sp24", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.01, "D": 0.0, "A": 0.0, "C": 0.99 }, "batch_response_ms": 299.58137499852455, "repaired": false }, { "id": "2.1", "section": 2, "context": "Another exam question says a valid RSA signature with public key (55,3) is attached to one of the possible TA counts, 12 or 13, with signature 23.", "prompt": "How many TAs are there this semester?", "response_mode": "single_choice", "options": { "A": "13", "B": "14", "C": "12", "D": "11" }, "correct": false, "kind": "multi_step", "topic": "course_context", "original_format": "written", "repairs": [ "Supply original Q9.3 cross-question clue to avoid testing unavailable course attendance context." ], "source_parts": [ "2.1" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.38, "probabilities": { "C": 0.43, "D": 0.01, "A": 0.55, "B": 0.01 }, "batch_response_ms": 347.33437500108266, "repaired": true }, { "id": "3.1a", "section": 3, "context": "", "prompt": "(NOT P AND R) OR (NOT P AND Q) is equivalent to NOT P AND (R OR Q).", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.1a" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 1.0, "A": 0.0 }, "batch_response_ms": 393.1597920018248, "repaired": false }, { "id": "3.1b", "section": 3, "context": "", "prompt": "(P AND Q) AND (NOT P OR NOT Q) is equivalent to P AND NOT P.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": false, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.1b" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.12, "probabilities": { "B": 0.44, "A": 0.56 }, "batch_response_ms": 393.1597920018248, "repaired": false }, { "id": "3.1c", "section": 3, "context": "", "prompt": "(P implies Q) AND (P implies NOT Q) is equivalent to NOT P.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.1c" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.01, "A": 0.99 }, "batch_response_ms": 393.1597920018248, "repaired": false }, { "id": "3.1d", "section": 3, "context": "", "prompt": "(P implies R) AND (P implies NOT R) is equivalent to NOT R.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.1d" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.97, "probabilities": { "B": 0.02, "A": 0.98 }, "batch_response_ms": 393.1597920018248, "repaired": false }, { "id": "3.2a", "section": 3, "context": "", "prompt": "Negate forall n in N, P(n), without using forall.", "response_mode": "single_choice", "options": { "A": "exists n in N, P(n)", "B": "NOT exists n in N, P(n)", "C": "exists n in N, NOT P(n)", "D": "exists n in N, P(n) AND NOT P(n)" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "written", "repairs": [], "source_parts": [ "3.2a" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 }, "batch_response_ms": 393.1597920018248, "repaired": false }, { "id": "3.2b", "section": 3, "context": "", "prompt": "N includes zero. For every predicate P, [NOT(forall n P(n)) AND P(0)] implies exists n [P(n) AND NOT P(n+1)].", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.2b" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.42, "probabilities": { "B": 0.29, "A": 0.71 }, "batch_response_ms": 393.1597920018248, "repaired": false }, { "id": "3.2c", "section": 3, "context": "", "prompt": "forall n [Q(n) AND P(n)] is equivalent to (forall n Q(n)) AND (forall n P(n)).", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.2c" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "A": 1.0 }, "batch_response_ms": 393.1597920018248, "repaired": false }, { "id": "3.2d", "section": 3, "context": "", "prompt": "forall n [Q(n) implies P(n)] is equivalent to (forall n NOT Q(n)) OR (forall n P(n)).", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "3.2d" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.97, "probabilities": { "B": 0.01, "A": 0.99 }, "batch_response_ms": 393.1597920018248, "repaired": false }, { "id": "4.1", "section": 4, "context": "", "prompt": "Three people each like exactly one other person. Must someone be liked by nobody? Give a proof or counterexample.", "response_mode": "single_choice", "options": { "A": "Yes: three preferences among three people force two preferences to share a target.", "B": "No: A likes B, B likes C, and C likes A; everyone is liked once.", "C": "Yes: whoever is least popular receives no preference.", "D": "No: let everyone like themselves." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "4.1" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "D": 0.0, "A": 0.0, "B": 1.0 }, "batch_response_ms": 342.6203749986598, "repaired": false }, { "id": "4.2", "section": 4, "context": "", "prompt": "Positive integers a,b,c satisfy a²+b²=c² and gcd(a,b,c)=1. Why must c be odd?", "response_mode": "single_choice", "options": { "A": "Every square is odd, so the sum of two squares is odd.", "B": "If c is even, then a+b=c and both legs are automatically even.", "C": "Squares are 0 or 1 mod 4. If c were even, both a,b would have to be even (two odd squares sum to 2 mod 4), contradicting the common gcd 1.", "D": "Since the gcd is 1, a,b,c must all be odd, proving the claim." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [ "Use the intended common-gcd definition explicitly, consistent with the official grading note." ], "source_parts": [ "4.2" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 1.0, "A": 0.0, "D": 0.0, "B": 0.0 }, "batch_response_ms": 342.6203749986598, "repaired": true }, { "id": "5.1", "section": 5, "context": "", "prompt": "Prove sum(i³,i=1..n)=[n(n+1)/2]² by induction. Which step is valid? You may use sum(i)=n(n+1)/2.", "response_mode": "single_choice", "options": { "A": "Cube the identity sum(i)=n(n+1)/2; sums commute with cubing.", "B": "Base n=1 holds. The next cube is (n+1)², so adding it gives the next square of a triangular number.", "C": "Base n=1 holds. Adding (n+1)³ to [n(n+1)/2]² gives (n+1)²[n²+4(n+1)]/4=[(n+1)(n+2)/2]².", "D": "Because the formula holds for n=1 and n=2, it holds for all n without another step." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "5.1" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 1.0, "B": 0.0 }, "batch_response_ms": 479.07479100103956, "repaired": false }, { "id": "6.1", "section": 6, "context": "", "prompt": "Among all preference instances with n jobs and n candidates, what is the maximum possible number of blocking pairs in a complete pairing?", "response_mode": "single_choice", "options": { "A": "n(n−1)", "B": "n−1", "C": "n²", "D": "n(n−1)/2" }, "correct": true, "kind": "calculation", "topic": "matching", "original_format": "written", "repairs": [], "source_parts": [ "6.1" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.71, "probabilities": { "B": 0.0, "C": 0.06, "D": 0.16, "A": 0.78 }, "batch_response_ms": 421.0574999997334, "repaired": false }, { "id": "6.2", "section": 6, "context": "", "prompt": "If a stable matching gives one particular job its best attainable stable partner, must that candidate have its worst attainable stable partner?", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.2" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.16, "probabilities": { "B": 0.58, "A": 0.42 }, "batch_response_ms": 421.0574999997334, "repaired": false }, { "id": "6.3", "section": 6, "context": "", "prompt": "In job-proposing deferred acceptance, if a candidate rejects n−1 distinct jobs, that candidate ends with its favorite job.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": false, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.3" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.16, "probabilities": { "B": 0.58, "A": 0.42 }, "batch_response_ms": 421.0574999997334, "repaired": false }, { "id": "6.4", "section": 6, "context": "", "prompt": "If job J is rejected by C, and J′ ends with C, then J’s eventual partner has a worse rank in J’s list than C has in J′’s list.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": false, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.4" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.2, "probabilities": { "B": 0.4, "A": 0.6 }, "batch_response_ms": 421.0574999997334, "repaired": false }, { "id": "6.5a", "section": 6, "context": "", "prompt": "If a job and candidate rank each other first, they are paired in every stable matching.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "6.5a" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.96, "probabilities": { "B": 0.02, "A": 0.98 }, "batch_response_ms": 421.0574999997334, "repaired": false }, { "id": "6.5b", "section": 6, "context": "", "prompt": "Construct, for each n>=2, an instance with exactly one stable matching.", "response_mode": "single_choice", "options": { "A": "Let rankings be arbitrary; the deferred acceptance output proves uniqueness.", "B": "Make every job rank C1 first; this alone uniquely determines every candidate’s partner.", "C": "Make Ji and Ci rank each other first for all i. Every such pair must be matched, otherwise it blocks; their combined matching is stable.", "D": "Let each Ji rank Ci last and each Ci rank Ji last; these pairs must be stable." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "6.5b" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.0, "C": 0.99, "A": 0.01, "D": 0.0 }, "batch_response_ms": 421.0574999997334, "repaired": false }, { "id": "6.5c", "section": 6, "context": "", "prompt": "Construct an instance with exactly two stable matchings for every n>=2.", "response_mode": "single_choice", "options": { "A": "Take two arbitrary complete pairings and declare both stable; preferences do not affect stability.", "B": "Give all jobs the same ranking and all candidates the same ranking; this yields exactly two stable matchings.", "C": "On the first two pairs use J1:C2>C1, J2:C1>C2, C1:J1>J2, C2:J2>J1; rank outsiders below these. For i>2, Ji and Ci rank each other first. The forced outer pairs leave exactly the two stable pairings of the first block.", "D": "Make every Ji and Ci rank each other first; then both the identity and any swap are stable." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "6.5c" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "C": 1.0, "A": 0.0, "D": 0.0 }, "batch_response_ms": 421.0574999997334, "repaired": false }, { "id": "7.1a", "section": 7, "context": "", "prompt": "A connected simple planar graph has v>=3 vertices. Maximum number of faces?", "response_mode": "single_choice", "options": { "A": "3v−6", "B": "2v−4", "C": "2v−2", "D": "v−2" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.1a" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.89, "probabilities": { "B": 0.91, "C": 0.06, "A": 0.02, "D": 0.01 }, "batch_response_ms": 311.75304200223763, "repaired": false }, { "id": "7.1b", "section": 7, "context": "", "prompt": "A connected simple planar graph contains a cycle and has girth at least g>=3. Complete the bound e <= [g/(g−2)]v − __.", "response_mode": "single_choice", "options": { "A": "g/(g−2)", "B": "g−2", "C": "2", "D": "2g/(g−2)" }, "correct": true, "kind": "multi_step", "topic": "graphs", "original_format": "written", "repairs": [ "Require a cycle; without it a tiny tree can violate the stated girth bound for arbitrarily large g." ], "source_parts": [ "7.1b" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.4, "probabilities": { "B": 0.03, "C": 0.06, "A": 0.36, "D": 0.55 }, "batch_response_ms": 311.75304200223763, "repaired": true }, { "id": "7.1c", "section": 7, "context": "", "prompt": "Why is every planar graph with no cycle shorter than six edges 3-colorable?", "response_mode": "single_choice", "options": { "A": "The average degree is below three, so every vertex has degree at most two and the graph is a disjoint union of paths.", "B": "Every nonempty subgraph has a vertex of degree at most two: forests have leaves, and the planar girth bound gives average degree below three for cyclic components. Remove such a vertex, inductively 3-color the rest, then restore it using a color absent from its at most two neighbors.", "C": "Euler’s formula gives exactly three faces, which can be used as vertex colors.", "D": "Every planar graph is bipartite, so two colors suffice." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "7.1c" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "B": 1.0, "A": 0.0, "D": 0.0 }, "batch_response_ms": 311.75304200223763, "repaired": false }, { "id": "7.2a", "section": 7, "context": "", "prompt": "How many edges are in the complement of a tree on n vertices?", "response_mode": "single_choice", "options": { "A": "C(n,2)−n+1", "B": "C(n−1,2)−1", "C": "C(n,2)−n", "D": "n−1" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.2a" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 1.0, "D": 0.0 }, "batch_response_ms": 311.75304200223763, "repaired": false }, { "id": "7.2b", "section": 7, "context": "", "prompt": "How many edges are in the complement of a d-dimensional hypercube?", "response_mode": "single_choice", "options": { "A": "C(2^d,2)−d*2^(d−1)", "B": "2^d−d", "C": "C(2^d,2)−d*2^d", "D": "C(d,2)−2^d" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.2b" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.99, "probabilities": { "C": 0.0, "B": 0.0, "A": 1.0, "D": 0.0 }, "batch_response_ms": 311.75304200223763, "repaired": false }, { "id": "7.3a", "section": 7, "context": "Doubling a graph means taking two disjoint copies and adding an edge between each original vertex and its corresponding copy.", "prompt": "If G has n vertices and m edges, how many edges does its double have?", "response_mode": "single_choice", "options": { "A": "m+2n", "B": "2m+n", "C": "2m+2n", "D": "2m" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.3a" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 1.0, "C": 0.0, "A": 0.0, "D": 0.0 }, "batch_response_ms": 311.75304200223763, "repaired": false }, { "id": "7.3b", "section": 7, "context": "Doubling a graph means taking two disjoint copies and adding an edge between each original vertex and its corresponding copy.", "prompt": "If G is bipartite, its double is bipartite.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": false, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "7.3b" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.25, "probabilities": { "B": 0.62, "A": 0.38 }, "batch_response_ms": 311.75304200223763, "repaired": false }, { "id": "7.3c", "section": 7, "context": "Doubling a graph means taking two disjoint copies and adding an edge between each original vertex and its corresponding copy.", "prompt": "How many doublings turn a k-dimensional hypercube into a d-dimensional hypercube, k=2, what is the smallest universal upper bound on colors needed for its double?", "response_mode": "single_choice", "options": { "A": "2c", "B": "c", "C": "c+1", "D": "c−1" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "7.3e" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.27, "probabilities": { "B": 0.46, "C": 0.31, "A": 0.23, "D": 0.0 }, "batch_response_ms": 311.75304200223763, "repaired": false }, { "id": "7.3f", "section": 7, "context": "Doubling a graph means taking two disjoint copies and adding an edge between each original vertex and its corresponding copy.", "prompt": "If G contains a Hamiltonian cycle, its double also contains a Hamiltonian cycle.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "7.3f" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.71, "probabilities": { "B": 0.15, "A": 0.85 }, "batch_response_ms": 311.75304200223763, "repaired": false }, { "id": "7.3g", "section": 7, "context": "Doubling a graph means taking two disjoint copies and adding an edge between each original vertex and its corresponding copy.", "prompt": "If a graph with at least two vertices contains an Eulerian tour, its double contains an Eulerian tour.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": false, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "7.3g" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.51, "probabilities": { "B": 0.76, "A": 0.24 }, "batch_response_ms": 311.75304200223763, "repaired": false }, { "id": "8.1", "section": 8, "context": "", "prompt": "Prove the complement of a disconnected graph is connected.", "response_mode": "single_choice", "options": { "A": "For vertices u,v in different original components, uv is a complement edge. For u,v in the same component, choose w in another component; uw and wv are complement edges, giving a path.", "B": "Every original component becomes a complete graph, so all components join automatically.", "C": "A disconnected graph always has an isolated vertex, which becomes adjacent to all vertices in the complement.", "D": "The complement has more edges than the original, and any denser graph is connected." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "8.1" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "C": 0.0, "A": 1.0, "D": 0.0 }, "batch_response_ms": 384.1220419999445, "repaired": false }, { "id": "9.1", "section": 9, "context": "", "prompt": "Inverse of 5 modulo 24, in 0..23?", "response_mode": "single_choice", "options": { "A": "19", "B": "1", "C": "7", "D": "5" }, "correct": false, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "9.1" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.8, "probabilities": { "C": 0.01, "B": 0.01, "D": 0.13, "A": 0.85 }, "batch_response_ms": 379.47658299890463, "repaired": false }, { "id": "9.2", "section": 9, "context": "", "prompt": "Compute 2^75 modulo 35. Hint: consider RSA with p=7,q=5,e=5.", "response_mode": "single_choice", "options": { "A": "2", "B": "1", "C": "8", "D": "32" }, "correct": false, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "9.2" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.18, "probabilities": { "A": 0.17, "B": 0.1, "C": 0.35, "D": 0.38 }, "batch_response_ms": 379.47658299890463, "repaired": false }, { "id": "9.3", "section": 9, "context": "", "prompt": "Public key is (N,e)=(55,3). Received signed messages are (12,23) and (13,23), where the second value is the signature m^d mod N. Which verifies, and why? Hint: CRT may help.", "response_mode": "single_choice", "options": { "A": "Both, because they carry the same signature.", "B": "Neither, because a signature must equal the message.", "C": "12, because 23³=12 mod 55.", "D": "13, because 23³=13 mod 55." }, "correct": false, "kind": "multi_step", "topic": "rsa", "original_format": "written", "repairs": [], "source_parts": [ "9.3" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.29, "probabilities": { "C": 0.42, "B": 0.04, "D": 0.47, "A": 0.07 }, "batch_response_ms": 379.47658299890463, "repaired": false }, { "id": "9.4", "section": 9, "context": "", "prompt": "For positive integers y=0, prove a^k=a^(k mod (p−1)) mod p.", "response_mode": "single_choice", "options": { "A": "Write k=q(p−1)+r. Fermat gives a^(p−1)=1, so a^k=(a^(p−1))^q*a^r=a^r.", "B": "Reduce the base modulo p−1 and retain the exponent; these are equivalent operations.", "C": "Fermat gives a^p=1, so exponents repeat modulo p.", "D": "All powers of nonzero residues are equal, so the exponent is irrelevant." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [ "State a nonzero explicitly, needed for Fermat exponent reduction." ], "source_parts": [ "9.6b" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 1.0, "B": 0.0, "C": 0.0, "D": 0.0 }, "batch_response_ms": 379.47658299890463, "repaired": true }, { "id": "9.6c", "section": 9, "context": "", "prompt": "If gcd(k,p−1)=1, multiplication by k permutes the residue classes modulo p−1.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "9.6c" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.01, "A": 0.99 }, "batch_response_ms": 379.47658299890463, "repaired": false }, { "id": "9.6d", "section": 9, "context": "", "prompt": "Prime p, positive k, and a^k=1 mod p. Why must gcd(k,p−1)>1 or a=1 mod p?", "response_mode": "single_choice", "options": { "A": "If gcd(k,p−1)=1, choose t with kt=1 mod p−1. Since a is nonzero, Fermat gives a=a^(kt)=(a^k)^t=1 mod p.", "B": "If a is not one, it is a primitive root by definition.", "C": "Fermat implies every k divides p−1, so their gcd exceeds one.", "D": "The equation a^k=1 implies a=k=1 as integers." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "9.6d" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "D": 0.0, "A": 1.0 }, "batch_response_ms": 379.47658299890463, "repaired": false }, { "id": "9.6e", "section": 9, "context": "", "prompt": "If a^k=1 mod prime p, k>0, and a is not 1 mod p, then gcd(k,p−1)>1.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "9.6e" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.17, "probabilities": { "A": 0.42, "B": 0.58 }, "batch_response_ms": 379.47658299890463, "repaired": false }, { "id": "10.1", "section": 10, "context": "", "prompt": "Over GF(5), ax²+bx+c passes through (0,1),(1,2),(2,0). Give (a,b,c).", "response_mode": "single_choice", "options": { "A": "(1,0,1)", "B": "(1,1,0)", "C": "(2,4,1)", "D": "(0,1,1)" }, "correct": false, "kind": "multi_step", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "10.1a", "10.1b", "10.1c" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.67, "probabilities": { "B": 0.01, "D": 0.01, "A": 0.23, "C": 0.75 }, "batch_response_ms": 343.57004200137453, "repaired": false }, { "id": "10.2a", "section": 10, "context": "A degree-one polynomial over GF(5) was evaluated; received (0,1),(1,3),(2,1),(3,2), with at most one corrupted value.", "prompt": "Recover the polynomial.", "response_mode": "single_choice", "options": { "A": "3x+1", "B": "2x+1", "C": "x+1", "D": "2x+3" }, "correct": true, "kind": "multi_step", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "10.2a" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.35, "probabilities": { "B": 0.51, "D": 0.09, "A": 0.36, "C": 0.04 }, "batch_response_ms": 343.57004200137453, "repaired": false }, { "id": "10.2b", "section": 10, "context": "A degree-one polynomial over GF(5) was evaluated; received (0,1),(1,3),(2,1),(3,2), with at most one corrupted value.", "prompt": "Find the monic minimal error locator.", "response_mode": "single_choice", "options": { "A": "x", "B": "x−3", "C": "x−2", "D": "x−1" }, "correct": true, "kind": "multi_step", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "10.2b" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.3, "probabilities": { "D": 0.26, "B": 0.18, "A": 0.09, "C": 0.47 }, "batch_response_ms": 343.57004200137453, "repaired": false }, { "id": "10.3", "section": 10, "context": "", "prompt": "Over GF(p), P has degree exactly d with 02d and d>=1, P and Q have degree exactly d, with r and s distinct roots respectively.", "prompt": "Maximum distinct roots of PQ?", "response_mode": "single_choice", "options": { "A": "max(r,s)", "B": "r*s", "C": "min(r,s)", "D": "r+s" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [ "Quantify sufficiently large prime as p>2d to permit disjoint root sets." ], "source_parts": [ "10.4a" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.01, "D": 0.98, "A": 0.01, "C": 0.0 }, "batch_response_ms": 343.57004200137453, "repaired": true }, { "id": "10.4b", "section": 10, "context": "Over GF(p) with p>2d and d>=1, P and Q have degree exactly d, with r and s distinct roots respectively.", "prompt": "Minimum distinct roots of PQ?", "response_mode": "single_choice", "options": { "A": "r+s", "B": "max(r,s)", "C": "min(r,s)", "D": "0" }, "correct": false, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "10.4b" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.33, "probabilities": { "B": 0.44, "D": 0.03, "A": 0.49, "C": 0.04 }, "batch_response_ms": 343.57004200137453, "repaired": false }, { "id": "10.4c", "section": 10, "context": "Over GF(p) with p>2d and d>=1, P and Q have degree exactly d, with r and s distinct roots respectively.", "prompt": "Degree of PQ?", "response_mode": "single_choice", "options": { "A": "d²", "B": "d", "C": "At most d−1", "D": "2d" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "10.4c" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.99, "probabilities": { "B": 0.0, "D": 1.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 343.57004200137453, "repaired": false }, { "id": "10.4d", "section": 10, "context": "Over GF(p) with p>2d and d>=1, P and Q have degree exactly d, with r and s distinct roots respectively.", "prompt": "Maximum degree of P+Q?", "response_mode": "single_choice", "options": { "A": "0", "B": "2d", "C": "d−1", "D": "d" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [], "source_parts": [ "10.4d" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.77, "probabilities": { "B": 0.07, "D": 0.84, "A": 0.0, "C": 0.09 }, "batch_response_ms": 343.57004200137453, "repaired": false }, { "id": "10.5", "section": 10, "context": "", "prompt": "Why does a degree-one polynomial over a field have exactly one root?", "response_mode": "single_choice", "options": { "A": "All field elements have inverses, including zero, so every polynomial has a unique root.", "B": "The root is always the constant coefficient b.", "C": "Write P(x)=mx+b with m nonzero. The equation has the unique solution x=−m^(-1)b.", "D": "A degree-one polynomial has at most one root, which alone proves existence." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "10.5" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "D": 0.0, "A": 0.0, "C": 1.0 }, "batch_response_ms": 343.57004200137453, "repaired": false }, { "id": "10.6a", "section": 10, "context": "", "prompt": "Over GF(p), count degree-exactly-two polynomials that split into linear factors, including repeated roots.", "response_mode": "single_choice", "options": { "A": "p²", "B": "(p−1)C(p,2)", "C": "(p−1)(C(p,2)+p)", "D": "(p−1)p²" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "10.6a" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.8, "probabilities": { "B": 0.02, "D": 0.06, "A": 0.07, "C": 0.85 }, "batch_response_ms": 343.57004200137453, "repaired": false }, { "id": "10.6b", "section": 10, "context": "", "prompt": "Over GF(p), count all formal polynomials of degree exactly two.", "response_mode": "single_choice", "options": { "A": "(p−1)(C(p,2)+p)", "B": "(p−1)³", "C": "p³", "D": "(p−1)p²" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "10.6b" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.85, "probabilities": { "B": 0.06, "D": 0.89, "A": 0.05, "C": 0.0 }, "batch_response_ms": 343.57004200137453, "repaired": false }, { "id": "11.1", "section": 11, "context": "70 groups each hold one point of a degree-49 shared polynomial. Within each group, 10 delegates hold evaluations of a degree-5 polynomial encoding that group’s point. Adversaries know all polynomials and choose their groups. Berlekamp–Welch is applied at both levels.", "prompt": "What is the largest number of adversarial delegates guaranteed tolerable? Give the threshold argument.", "response_mode": "single_choice", "options": { "A": "32: each group corrects two bad shares, so corrupting a group needs three delegates; the outer code corrects ten bad groups, and defeating that guarantee needs eleven groups, costing 33 delegates.", "B": "30: three delegates per group times ten groups is the first failure.", "C": "22: the outer code tolerates eleven groups and each needs two delegates.", "D": "20: each of ten bad groups needs two delegates, so the 21st defeats the scheme." }, "correct": true, "kind": "multi_step", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "11.1" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.79, "probabilities": { "C": 0.01, "D": 0.11, "A": 0.85, "B": 0.03 }, "batch_response_ms": 283.5174169995298, "repaired": false }, { "id": "12.1", "section": 12, "context": "", "prompt": "Distinct arrangements of BAA?", "response_mode": "single_choice", "options": { "A": "3", "B": "1", "C": "6", "D": "2" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "12.1" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.82, "probabilities": { "D": 0.12, "C": 0.01, "A": 0.86, "B": 0.01 }, "batch_response_ms": 400.84479200231726, "repaired": false }, { "id": "12.2", "section": 12, "context": "", "prompt": "Distinct arrangements of 126?", "response_mode": "single_choice", "options": { "A": "3", "B": "6", "C": "1", "D": "9" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "12.2" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.34, "probabilities": { "D": 0.05, "C": 0.3, "A": 0.13, "B": 0.52 }, "batch_response_ms": 400.84479200231726, "repaired": false }, { "id": "12.3", "section": 12, "context": "", "prompt": "n distinct bears, k distinct children, n>k. Assign exactly one distinct bear to each child. How many assignments?", "response_mode": "single_choice", "options": { "A": "k^n", "B": "C(n,k)", "C": "n^k", "D": "n!/(n−k)!" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "12.3" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.99, "probabilities": { "D": 1.0, "C": 0.0, "A": 0.0, "B": 0.0 }, "batch_response_ms": 400.84479200231726, "repaired": false }, { "id": "12.4", "section": 12, "context": "", "prompt": "Number of positive integer n-tuples summing to k, k>=n>=1?", "response_mode": "single_choice", "options": { "A": "C(k+n−1,n−1)", "B": "C(k−1,n−1)", "C": "n^k", "D": "C(k,n)" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "12.4" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.99, "probabilities": { "D": 0.0, "C": 0.0, "A": 0.0, "B": 1.0 }, "batch_response_ms": 400.84479200231726, "repaired": false }, { "id": "12.5", "section": 12, "context": "", "prompt": "Number of n-bit strings with exactly k ones, 0<=k<=n?", "response_mode": "single_choice", "options": { "A": "n!/k!", "B": "C(n,k)", "C": "2^k", "D": "n^k" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [], "source_parts": [ "12.5" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 0.0, "C": 0.0, "B": 1.0 }, "batch_response_ms": 400.84479200231726, "repaired": false }, { "id": "12.6", "section": 12, "context": "", "prompt": "Circular arrangements of n distinct beads, identifying rotations but not reflections?", "response_mode": "single_choice", "options": { "A": "(n−1)!", "B": "(n−1)!/2", "C": "2^n", "D": "n!" }, "correct": true, "kind": "calculation", "topic": "counting", "original_format": "written", "repairs": [ "Explicitly identify rotations only, consistent with the source answer." ], "source_parts": [ "12.6" ], "exam": "midterm_sp25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "C": 0.0, "A": 1.0, "B": 0.0 }, "batch_response_ms": 400.84479200231726, "repaired": true }, { "id": "2.1a", "section": 2, "context": "", "prompt": "What is the truth status of (P AND NOT P) implies Q?", "response_mode": "single_choice", "options": { "A": "Could be either", "B": "True", "C": "False" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.1a" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.99, "C": 0.0, "A": 0.01 }, "batch_response_ms": 357.25633300171467, "repaired": false }, { "id": "2.1b", "section": 2, "context": "", "prompt": "What is the truth status of P implies (Q OR NOT Q)?", "response_mode": "single_choice", "options": { "A": "Could be either", "B": "False", "C": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.1b" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.0, "C": 0.99, "A": 0.01 }, "batch_response_ms": 357.25633300171467, "repaired": false }, { "id": "2.1c", "section": 2, "context": "", "prompt": "What is the truth status of (P OR NOT P) implies (Q AND NOT Q)?", "response_mode": "single_choice", "options": { "A": "Could be either", "B": "False", "C": "True" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.1c" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.82, "probabilities": { "B": 0.88, "C": 0.01, "A": 0.11 }, "batch_response_ms": 357.25633300171467, "repaired": false }, { "id": "2.1d", "section": 2, "context": "", "prompt": "What is the truth status of (P OR Q) implies (P AND Q)?", "response_mode": "single_choice", "options": { "A": "False", "B": "Could be either", "C": "True" }, "correct": false, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.1d" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.35, "probabilities": { "B": 0.43, "C": 0.0, "A": 0.5700000000000001 }, "batch_response_ms": 357.25633300171467, "repaired": false }, { "id": "2.2a", "section": 2, "context": "", "prompt": "Over nonempty domains, are forall x [P(x) implies exists y Q(x,y)] and forall x exists y [P(x) implies Q(x,y)] equivalent?", "response_mode": "single_choice", "options": { "A": "Equivalent", "B": "Not equivalent" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.2a" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.81, "probabilities": { "B": 0.1, "A": 0.9 }, "batch_response_ms": 357.25633300171467, "repaired": false }, { "id": "2.2b", "section": 2, "context": "", "prompt": "Are exists x [P(x) implies forall y R(x,y)] and forall x [P(x) OR exists y R(x,y)] equivalent?", "response_mode": "single_choice", "options": { "A": "Not equivalent", "B": "Equivalent" }, "correct": true, "kind": "recognition", "topic": "logic", "original_format": "choice", "repairs": [], "source_parts": [ "2.2b" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.83, "probabilities": { "B": 0.09, "A": 0.91 }, "batch_response_ms": 357.25633300171467, "repaired": false }, { "id": "2.3", "section": 2, "context": "", "prompt": "R(x)=NOT P(x), S(x,y)=NOT Q(x,y). Rewrite NOT[(forall x P(x)) AND (exists x forall y Q(x,y))] without NOT.", "response_mode": "single_choice", "options": { "A": "(exists x R(x)) AND (forall x exists y S(x,y))", "B": "(exists x R(x)) OR (forall x exists y S(x,y))", "C": "(forall x R(x)) OR (exists x forall y S(x,y))", "D": "(forall x R(x)) AND (exists x forall y S(x,y))" }, "correct": true, "kind": "multi_step", "topic": "logic", "original_format": "written", "repairs": [], "source_parts": [ "2.3" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.93, "probabilities": { "B": 0.95, "C": 0.05, "D": 0.0, "A": 0.0 }, "batch_response_ms": 357.25633300171467, "repaired": false }, { "id": "3.1", "section": 3, "context": "", "prompt": "For integers a,b,c, prove a does not divide 2b+3c implies a does not divide b or a does not divide c.", "response_mode": "single_choice", "options": { "A": "Contrapositive: b=ak and c=al imply 2b+3c=a(2k+3l), so a divides the sum.", "B": "Since a does not divide the sum, it cannot divide either 2b or 3c individually.", "C": "If a fails to divide b and c, it also fails to divide every linear combination.", "D": "If a divides 2b+3c, then it divides b and c separately; use the converse." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "3.1" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "D": 0.0, "B": 0.0, "A": 1.0 }, "batch_response_ms": 447.66695799989975, "repaired": false }, { "id": "3.2a", "section": 3, "context": "", "prompt": "State AM–GM for two nonnegative numbers x,y.", "response_mode": "single_choice", "options": { "A": "sqrt(x+y) >= x*y", "B": "x+y >= xy", "C": "(x+y)/2 >= sqrt(xy)", "D": "(x+y)/2 <= sqrt(xy)" }, "correct": true, "kind": "recognition", "topic": "inequalities", "original_format": "written", "repairs": [], "source_parts": [ "3.2a" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 1.0, "D": 0.0, "B": 0.0, "A": 0.0 }, "batch_response_ms": 447.66695799989975, "repaired": false }, { "id": "3.2b", "section": 3, "context": "", "prompt": "Which proves AM–GM for nonnegative x,y?", "response_mode": "single_choice", "options": { "A": "(x−y)^2>=0 gives (x+y)^2>=4xy; both sides of x+y>=2sqrt(xy) are nonnegative, so taking square roots is valid.", "B": "Since x>=y and y>=x, adding yields equality in AM–GM.", "C": "Squaring any inequality preserves its direction, even for negative sides; apply this to sqrt(xy).", "D": "x^2+y^2>=0 gives x+y>=2sqrt(xy) directly by distributing the square root over a sum." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "3.2b" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "D": 0.0, "B": 0.0, "A": 1.0 }, "batch_response_ms": 447.66695799989975, "repaired": false }, { "id": "4.1a", "section": 4, "context": "S0=3, S1=2, S2=1; for n>=3, Sn=S(n−1)+S(n−2)+2S(n−3).", "prompt": "Smallest integer c with Sn<=c*2^n for all n>=0?", "response_mode": "single_choice", "options": { "A": "3", "B": "2", "C": "4", "D": "1" }, "correct": false, "kind": "multi_step", "topic": "recurrences", "original_format": "written", "repairs": [], "source_parts": [ "4.1a" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.1, "probabilities": { "C": 0.07, "B": 0.33, "A": 0.31, "D": 0.29 }, "batch_response_ms": 336.0027919989079, "repaired": false }, { "id": "4.1b", "section": 4, "context": "S0=3, S1=2, S2=1; for n>=3, Sn=S(n−1)+S(n−2)+2S(n−3).", "prompt": "Which is a valid induction proving Sn<=3*2^n?", "response_mode": "single_choice", "options": { "A": "Check n=0,1,2. Strong induction gives S(k+1)<=3*2^k+3*2^(k−1)+6*2^(k−2)=3*2^(k+1).", "B": "Check the first three terms and use S(k+1)=2Sk.", "C": "Every term is at most twice its predecessor, including S3, so ordinary induction is immediate.", "D": "Check n=0 only. Assume Sk<=3*2^k; this also establishes bounds on earlier terms without any additional hypothesis." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "4.1b" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "A": 1.0, "B": 0.0, "D": 0.0 }, "batch_response_ms": 336.0027919989079, "repaired": false }, { "id": "4.1c", "section": 4, "context": "S0=3, S1=2, S2=1; for n>=3, Sn=S(n−1)+S(n−2)+2S(n−3).", "prompt": "Smallest integer c with Sn<=c*2^n for every n>=4?", "response_mode": "single_choice", "options": { "A": "1", "B": "3", "C": "2", "D": "0" }, "correct": true, "kind": "multi_step", "topic": "recurrences", "original_format": "written", "repairs": [ "Official intermediate terms miscomputed; reference bound remains 1." ], "source_parts": [ "4.1c" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.49, "probabilities": { "C": 0.19, "B": 0.16, "A": 0.62, "D": 0.03 }, "batch_response_ms": 336.0027919989079, "repaired": true }, { "id": "4.2", "section": 4, "context": "", "prompt": "Which induction proves 7 divides 11^n−4^n for n>=0?", "response_mode": "single_choice", "options": { "A": "Base n=1 is 7. Since both bases are odd, the difference must be divisible by 7.", "B": "The first seven values suffice because every exponential sequence has period seven.", "C": "Base n=0 is zero. Write 11^(k+1)−4^(k+1)=7*11^k+4(11^k−4^k), so the inductive divisibility persists.", "D": "Base n=0 is zero. Multiply the difference by 7 to obtain the next difference." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "4.2" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 1.0, "B": 0.0, "A": 0.0, "D": 0.0 }, "batch_response_ms": 336.0027919989079, "repaired": false }, { "id": "5.1a", "section": 5, "context": "Jobs: A:3>1>2, B:2>3>1, C:3>2>1. Candidates: 1:B>C>A, 2:C>A>B, 3:A>C>B.", "prompt": "Find the job-optimal stable matching.", "response_mode": "single_choice", "options": { "A": "A–3, B–1, C–2", "B": "A–2, B–1, C–3", "C": "A–3, B–2, C–1", "D": "A–1, B–2, C–3" }, "correct": true, "kind": "multi_step", "topic": "matching", "original_format": "written", "repairs": [], "source_parts": [ "5.1a" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.24, "probabilities": { "B": 0.06, "D": 0.11, "A": 0.42, "C": 0.41 }, "batch_response_ms": 329.3755000013334, "repaired": false }, { "id": "5.1b", "section": 5, "context": "Jobs: A:3>1>2, B:2>3>1, C:3>2>1. Candidates: 1:B>C>A, 2:C>A>B, 3:A>C>B.", "prompt": "Find the candidate-optimal stable matching.", "response_mode": "single_choice", "options": { "A": "A–3, B–2, C–1", "B": "A–2, B–1, C–3", "C": "A–1, B–2, C–3", "D": "A–3, B–1, C–2" }, "correct": false, "kind": "multi_step", "topic": "matching", "original_format": "written", "repairs": [], "source_parts": [ "5.1b" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.52, "probabilities": { "D": 0.08, "B": 0.06, "A": 0.63, "C": 0.23 }, "batch_response_ms": 329.3755000013334, "repaired": false }, { "id": "5.1c", "section": 5, "context": "Jobs: A:3>1>2, B:2>3>1, C:3>2>1. Candidates: 1:B>C>A, 2:C>A>B, 3:A>C>B.", "prompt": "Is there any other stable matching beyond the outputs of the two proposer directions?", "response_mode": "single_choice", "options": { "A": "No; the two extremes coincide.", "B": "Yes: A–2, B–1, C–3.", "C": "Yes: A–1, B–2, C–3.", "D": "Yes: A–3, B–2, C–1." }, "correct": true, "kind": "multi_step", "topic": "matching", "original_format": "written", "repairs": [], "source_parts": [ "5.1c" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.26, "probabilities": { "B": 0.14, "D": 0.14, "A": 0.45, "C": 0.27 }, "batch_response_ms": 329.3755000013334, "repaired": false }, { "id": "5.2", "section": 5, "context": "", "prompt": "If every job has a different first-choice candidate, there must be exactly one stable matching.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "5.2" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.09, "probabilities": { "B": 0.45, "A": 0.55 }, "batch_response_ms": 329.3755000013334, "repaired": false }, { "id": "5.3", "section": 5, "context": "", "prompt": "In the job-optimal stable matching, at least one job must get its first-choice candidate.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "5.3" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.22, "probabilities": { "B": 0.61, "A": 0.39 }, "batch_response_ms": 329.3755000013334, "repaired": false }, { "id": "5.4", "section": 5, "context": "", "prompt": "A job can receive its least-preferred candidate in the job-optimal stable matching.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "5.4" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.29, "probabilities": { "B": 0.65, "A": 0.35 }, "batch_response_ms": 329.3755000013334, "repaired": false }, { "id": "5.5", "section": 5, "context": "", "prompt": "If a job is matched to its favorite candidate, that candidate cannot participate in a blocking pair.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": false, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "5.5" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.02, "probabilities": { "B": 0.49, "A": 0.51 }, "batch_response_ms": 329.3755000013334, "repaired": false }, { "id": "5.6", "section": 5, "context": "", "prompt": "For every preference instance, modifying only candidate preferences can change the job-proposing algorithm’s output.", "response_mode": "single_choice", "options": { "A": "False", "B": "True" }, "correct": true, "kind": "recognition", "topic": "matching", "original_format": "choice", "repairs": [], "source_parts": [ "5.6" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.48, "probabilities": { "B": 0.26, "A": 0.74 }, "batch_response_ms": 329.3755000013334, "repaired": false }, { "id": "5.7", "section": 5, "context": "", "prompt": "For every n>=2, construct preferences where job-proposing matches every job to its favorite and every candidate to its least favorite.", "response_mode": "single_choice", "options": { "A": "All jobs and candidates have identical ranking lists; everyone therefore receives their first choice on one side and last on the other.", "B": "Each Ji and Ci rank one another first; jobs get their favorite while candidates get their least favorite.", "C": "Each job ranks a different candidate last and proposes to that candidate first, as required by deferred acceptance.", "D": "Each Ji ranks Ci first, and each Ci ranks Ji last; complete all remaining rankings strictly in any order. All jobs propose to distinct first choices, so those proposals are accepted." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "5.7" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.95, "probabilities": { "B": 0.01, "D": 0.97, "A": 0.02, "C": 0.0 }, "batch_response_ms": 329.3755000013334, "repaired": false }, { "id": "6.1a", "section": 6, "context": "", "prompt": "A six-vertex graph with four edges is connected.", "response_mode": "single_choice", "options": { "A": "False", "B": "Could be either", "C": "True" }, "correct": false, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "6.1a" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.24, "probabilities": { "C": 0.03, "B": 0.49, "A": 0.48 }, "batch_response_ms": 367.22316699888324, "repaired": false }, { "id": "6.1b", "section": 6, "context": "", "prompt": "A six-vertex graph with six edges is connected.", "response_mode": "single_choice", "options": { "A": "Could be either", "B": "True", "C": "False" }, "correct": false, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "6.1b" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.26, "probabilities": { "C": 0.51, "B": 0.09, "A": 0.4 }, "batch_response_ms": 367.22316699888324, "repaired": false }, { "id": "6.1c", "section": 6, "context": "", "prompt": "A six-vertex graph with six edges is acyclic.", "response_mode": "single_choice", "options": { "A": "Could be either", "B": "True", "C": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "6.1c" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.68, "probabilities": { "C": 0.79, "B": 0.01, "A": 0.2 }, "batch_response_ms": 367.22316699888324, "repaired": false }, { "id": "6.1d", "section": 6, "context": "", "prompt": "A six-vertex graph with thirteen edges is connected.", "response_mode": "single_choice", "options": { "A": "True", "B": "False", "C": "Could be either" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "6.1d" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.55, "probabilities": { "C": 0.13, "B": 0.18, "A": 0.69 }, "batch_response_ms": 367.22316699888324, "repaired": false }, { "id": "6.1e", "section": 6, "context": "", "prompt": "A six-vertex graph with thirteen edges is planar.", "response_mode": "single_choice", "options": { "A": "False", "B": "Could be either", "C": "True" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "6.1e" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.85, "probabilities": { "C": 0.08, "B": 0.02, "A": 0.9 }, "batch_response_ms": 367.22316699888324, "repaired": false }, { "id": "6.1f", "section": 6, "context": "", "prompt": "Maximum edges of a simple graph on six vertices?", "response_mode": "single_choice", "options": { "A": "15", "B": "12", "C": "18", "D": "30" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "6.1f" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 1.0, "D": 0.0 }, "batch_response_ms": 367.22316699888324, "repaired": false }, { "id": "6.2", "section": 6, "context": "", "prompt": "Minimum edges of an undirected graph with V vertices and C connected components?", "response_mode": "single_choice", "options": { "A": "C−1", "B": "V+C", "C": "V−1", "D": "V−C" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "6.2" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 0.0, "D": 1.0 }, "batch_response_ms": 367.22316699888324, "repaired": false }, { "id": "6.3a", "section": 6, "context": "", "prompt": "K7 has an Eulerian tour.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "6.3a" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.62, "probabilities": { "B": 0.19, "A": 0.81 }, "batch_response_ms": 367.22316699888324, "repaired": false }, { "id": "6.3b", "section": 6, "context": "", "prompt": "Why does K7 have an Eulerian tour?", "response_mode": "single_choice", "options": { "A": "Every complete graph has all degrees odd.", "B": "It is planar and connected.", "C": "It has an odd number of vertices, which alone guarantees a tour.", "D": "It is connected and each vertex has even degree 6." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "6.3b" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 0.0, "D": 1.0 }, "batch_response_ms": 367.22316699888324, "repaired": false }, { "id": "6.4a", "section": 6, "context": "", "prompt": "K8 has an Eulerian tour.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "graphs", "original_format": "choice", "repairs": [], "source_parts": [ "6.4a" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.59, "probabilities": { "B": 0.79, "A": 0.21 }, "batch_response_ms": 367.22316699888324, "repaired": false }, { "id": "6.4b", "section": 6, "context": "", "prompt": "Why does K8 lack an Eulerian tour?", "response_mode": "single_choice", "options": { "A": "It is disconnected.", "B": "It has too many edges to be bipartite, which forbids Eulerian tours.", "C": "It has an even number of vertices, and no even-order graph has a tour.", "D": "Each vertex has odd degree 7, whereas a connected Eulerian graph requires all degrees even." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "6.4b" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "B": 0.0, "A": 0.0, "D": 1.0 }, "batch_response_ms": 367.22316699888324, "repaired": false }, { "id": "6.5", "section": 6, "context": "", "prompt": "A connected planar embedding has 15 vertices and 4 faces including the outer face. How many edges?", "response_mode": "single_choice", "options": { "A": "13", "B": "17", "C": "19", "D": "11" }, "correct": true, "kind": "calculation", "topic": "graphs", "original_format": "written", "repairs": [], "source_parts": [ "6.5" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.42, "probabilities": { "C": 0.37, "B": 0.57, "A": 0.02, "D": 0.04 }, "batch_response_ms": 367.22316699888324, "repaired": false }, { "id": "7.1", "section": 7, "context": "", "prompt": "Prove that a graph all of whose vertices have odd degree has an even number of vertices.", "response_mode": "single_choice", "options": { "A": "Every vertex must have an odd number of neighbors, so all vertices can be paired along disjoint edges.", "B": "Each odd degree is at least one, so the vertex count equals twice the edge count.", "C": "The degree sum is twice the number of edges and therefore even. A sum of an odd number of odd integers is odd, so the number of vertices must be even.", "D": "All odd-degree graphs are complete, and complete graphs have even order." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "7.1" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "A": 0.0, "C": 1.0, "D": 0.0 }, "batch_response_ms": 281.9140419996984, "repaired": false }, { "id": "7.2", "section": 7, "context": "", "prompt": "In a finite tree with at least two vertices, start walking along unused edges until stuck. Why does this reach a degree-one vertex?", "response_mode": "single_choice", "options": { "A": "A tree has no edges, so the starting vertex already has degree one.", "B": "Every vertex of a tree has degree at most two; hence any arbitrary walk ends at a leaf.", "C": "A repeated vertex would create a cycle. The walk is finite and its terminal vertex was not previously visited; only its arrival edge has been used there, so being stuck means it has no other incident edge.", "D": "Each step decreases the total degree by two, so eventually every vertex has degree one." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "7.2" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 1.0, "probabilities": { "B": 0.0, "A": 0.0, "C": 1.0, "D": 0.0 }, "batch_response_ms": 281.9140419996984, "repaired": false }, { "id": "8.1", "section": 8, "context": "", "prompt": "Compute 5^42 modulo 41.", "response_mode": "single_choice", "options": { "A": "1", "B": "0", "C": "5", "D": "25" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "8.1" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.27, "probabilities": { "D": 0.46, "B": 0.01, "A": 0.29, "C": 0.24 }, "batch_response_ms": 757.2612500007381, "repaired": false }, { "id": "8.2", "section": 8, "context": "", "prompt": "Last digit of 519^487?", "response_mode": "single_choice", "options": { "A": "5", "B": "9", "C": "7", "D": "1" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "8.2" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.63, "probabilities": { "B": 0.72, "D": 0.05, "A": 0.07, "C": 0.16 }, "batch_response_ms": 757.2612500007381, "repaired": false }, { "id": "8.3", "section": 8, "context": "", "prompt": "Compute 43^21 modulo 55.", "response_mode": "single_choice", "options": { "A": "43", "B": "1", "C": "23", "D": "12" }, "correct": true, "kind": "multi_step", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "8.3" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.15, "probabilities": { "B": 0.08, "D": 0.25, "A": 0.36, "C": 0.31 }, "batch_response_ms": 757.2612500007381, "repaired": false }, { "id": "8.4", "section": 8, "context": "", "prompt": "Choose nonzero a modulo m for which ax=0 modulo m has a nonzero solution.", "response_mode": "single_choice", "options": { "A": "a=4,m=9", "B": "a=2,m=5", "C": "a=3,m=7", "D": "a=2,m=4" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "8.4" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 1.0, "B": 0.0, "A": 0.0, "C": 0.0 }, "batch_response_ms": 757.2612500007381, "repaired": false }, { "id": "8.5", "section": 8, "context": "", "prompt": "For positive integers k1?", "response_mode": "single_choice", "options": { "A": "3", "B": "2", "C": "7", "D": "5" }, "correct": false, "kind": "multi_step", "topic": "rsa", "original_format": "written", "repairs": [ "Require e>1 explicitly; without it, exponent 1 is mathematically invertible." ], "source_parts": [ "8.8a" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.64, "probabilities": { "B": 0.2, "D": 0.06, "A": 0.73, "C": 0.01 }, "batch_response_ms": 757.2612500007381, "repaired": true }, { "id": "8.8b", "section": 8, "context": "", "prompt": "RSA has p=11,q=7,e=29. What is the least positive decryption exponent?", "response_mode": "single_choice", "options": { "A": "31", "B": "29", "C": "59", "D": "7" }, "correct": false, "kind": "multi_step", "topic": "rsa", "original_format": "written", "repairs": [], "source_parts": [ "8.8b" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.48, "probabilities": { "B": 0.2, "D": 0.09, "A": 0.1, "C": 0.61 }, "batch_response_ms": 757.2612500007381, "repaired": false }, { "id": "9.1", "section": 9, "context": "", "prompt": "Without induction, why is 11^n−4^n divisible by 7 for all nonnegative n?", "response_mode": "single_choice", "options": { "A": "11−4=7, so 11^n−4^n=7^n as integers.", "B": "11=4 mod 7, so exponentiating preserves congruence: 11^n=4^n mod 7.", "C": "All differences of powers with odd and even bases are multiples of 7.", "D": "Both 11 and 4 are zero modulo 7." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "9.1" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 1.0, "probabilities": { "C": 0.0, "A": 0.0, "D": 0.0, "B": 1.0 }, "batch_response_ms": 288.97745900030714, "repaired": false }, { "id": "9.2", "section": 9, "context": "", "prompt": "If a has multiplicative order p−1 modulo prime p, why do a^0,...,a^(p−2) cover all nonzero residues?", "response_mode": "single_choice", "options": { "A": "They are nonzero. A repeated pair i1.", "response_mode": "single_choice", "options": { "A": "2", "B": "0", "C": "m−1", "D": "1" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "9.3a" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.85, "probabilities": { "D": 0.89, "A": 0.01, "B": 0.0, "C": 0.1 }, "batch_response_ms": 288.97745900030714, "repaired": false }, { "id": "9.3b", "section": 9, "context": "", "prompt": "The residue m−1 is a nontrivial square root of 1 modulo m, where trivial roots mean ±1.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" }, "correct": true, "kind": "recognition", "topic": "modular_arithmetic", "original_format": "choice", "repairs": [], "source_parts": [ "9.3b" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.55, "probabilities": { "A": 0.22, "B": 0.78 }, "batch_response_ms": 288.97745900030714, "repaired": false }, { "id": "9.3c", "section": 9, "context": "", "prompt": "Why can a prime modulus p have no square root of 1 other than ±1?", "response_mode": "single_choice", "options": { "A": "x^2−1=(x−1)(x+1); a prime dividing this product divides one factor, so x=1 or −1 mod p.", "B": "A polynomial of degree two always has exactly one root.", "C": "One can cancel x−1 from the equation even when x=1.", "D": "Every composite modulus has no roots of 1, so primes have only the trivial ones." }, "correct": true, "kind": "multi_step", "topic": "proof", "original_format": "written", "repairs": [], "source_parts": [ "9.3c" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "D": 0.0, "A": 1.0, "B": 0.0, "C": 0.0 }, "batch_response_ms": 288.97745900030714, "repaired": false }, { "id": "9.3d", "section": 9, "context": "", "prompt": "Which is a nontrivial square root of 1 modulo 15?", "response_mode": "single_choice", "options": { "A": "4", "B": "14", "C": "0", "D": "1" }, "correct": true, "kind": "calculation", "topic": "modular_arithmetic", "original_format": "written", "repairs": [], "source_parts": [ "9.3d" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 0.73, "probabilities": { "D": 0.0, "A": 0.8, "B": 0.2, "C": 0.0 }, "batch_response_ms": 288.97745900030714, "repaired": false }, { "id": "10.1a", "section": 10, "context": "P,Q are distinct formal polynomials over GF(p), with degrees d1,d2=2. Smallest possible degree of P−Q?", "response_mode": "single_choice", "options": { "A": "d1", "B": "Undefined because it must be zero", "C": "0", "D": "d1−1" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [ "Equal degrees added: with unequal degrees the printed reference 0 is false." ], "source_parts": [ "10.1b" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.73, "probabilities": { "B": 0.0, "D": 0.15, "A": 0.05, "C": 0.8 }, "batch_response_ms": 364.7012920009729, "repaired": true }, { "id": "10.1c", "section": 10, "context": "P,Q are distinct formal polynomials over GF(p), with degrees d1,d2=2. Smallest possible number of x values where P(x)=Q(x)?", "response_mode": "single_choice", "options": { "A": "1", "B": "d1", "C": "0", "D": "p" }, "correct": true, "kind": "recognition", "topic": "polynomials", "original_format": "written", "repairs": [ "Equal degrees added: degree-one versus constant polynomials necessarily agree once." ], "source_parts": [ "10.1d" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.87, "probabilities": { "B": 0.05, "D": 0.0, "A": 0.05, "C": 0.9 }, "batch_response_ms": 364.7012920009729, "repaired": true }, { "id": "10.2a", "section": 10, "context": "Shamir sharing over GF(7), where any two of three participants can reconstruct the secret P(0).", "prompt": "What polynomial degree is used?", "response_mode": "single_choice", "options": { "A": "0", "B": "2", "C": "1", "D": "3" }, "correct": true, "kind": "recognition", "topic": "secret_sharing", "original_format": "written", "repairs": [], "source_parts": [ "10.2a" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "C" ], "expected": [ "C" ], "valid_response": true, "confidence": 0.95, "probabilities": { "B": 0.03, "D": 0.0, "A": 0.0, "C": 0.97 }, "batch_response_ms": 364.7012920009729, "repaired": false }, { "id": "10.2b", "section": 10, "context": "Shamir sharing over GF(7), where any two of three participants can reconstruct the secret P(0).", "prompt": "Shares are (1,6) and (2,2). What is the secret?", "response_mode": "single_choice", "options": { "A": "6", "B": "5", "C": "4", "D": "3" }, "correct": true, "kind": "multi_step", "topic": "secret_sharing", "original_format": "written", "repairs": [], "source_parts": [ "10.2b" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "D" ], "expected": [ "D" ], "valid_response": true, "confidence": 0.11, "probabilities": { "B": 0.29, "D": 0.33, "A": 0.07, "C": 0.31 }, "batch_response_ms": 364.7012920009729, "repaired": false }, { "id": "10.2c", "section": 10, "context": "Shamir sharing over GF(7), where any two of three participants can reconstruct the secret P(0).", "prompt": "Shares are (1,6) and (2,2). What should a third share at x=3 contain?", "response_mode": "single_choice", "options": { "A": "4", "B": "5", "C": "3", "D": "1" }, "correct": false, "kind": "multi_step", "topic": "secret_sharing", "original_format": "written", "repairs": [], "source_parts": [ "10.2c" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.18, "probabilities": { "B": 0.23, "D": 0.24, "A": 0.38, "C": 0.15 }, "batch_response_ms": 364.7012920009729, "repaired": false }, { "id": "10.3", "section": 10, "context": "", "prompt": "A k-threshold scheme is reconstructed from k shares. An attacker controls coordinate x1 and knows the other k−1 shares. How can it force the reconstructed secret to v?", "response_mode": "single_choice", "options": { "A": "Interpolate the original k shares first, then multiply only its own share by v.", "B": "Interpolate the unique degree-at-most-(k−1) polynomial through the other shares and (0,v), then announce its value at x1.", "C": "Announce the pair (0,v) as its share even though its assigned coordinate is x1.", "D": "Set its share value to v; interpolation always copies any participant’s value to the constant term." }, "correct": true, "kind": "multi_step", "topic": "secret_sharing", "original_format": "written", "repairs": [], "source_parts": [ "10.3" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.98, "probabilities": { "B": 0.99, "D": 0.0, "A": 0.0, "C": 0.01 }, "batch_response_ms": 364.7012920009729, "repaired": false }, { "id": "11.1a", "section": 11, "context": "", "prompt": "To transmit n field symbols by Reed–Solomon and correct up to k corrupt evaluations, how many packets suffice?", "response_mode": "single_choice", "options": { "A": "n+2k−1", "B": "n+2k", "C": "2n+k", "D": "n+k" }, "correct": true, "kind": "recognition", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "11.1a" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.91, "probabilities": { "A": 0.01, "B": 0.93, "C": 0.0, "D": 0.05 }, "batch_response_ms": 379.65533299939125, "repaired": false }, { "id": "11.1b", "section": 11, "context": "", "prompt": "For n message symbols used as coefficients, what is the degree bound of the message polynomial?", "response_mode": "single_choice", "options": { "A": "At most n−1", "B": "At most n+2k", "C": "Exactly n", "D": "Exactly k" }, "correct": true, "kind": "recognition", "topic": "coding", "original_format": "written", "repairs": [], "source_parts": [ "11.1b" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "A" ], "expected": [ "A" ], "valid_response": true, "confidence": 1.0, "probabilities": { "A": 1.0, "B": 0.0, "C": 0.0, "D": 0.0 }, "batch_response_ms": 379.65533299939125, "repaired": false }, { "id": "11.1c", "section": 11, "context": "", "prompt": "With at most k errors, what is the degree bound of the minimal error locator E(x), the product over actual error locations?", "response_mode": "single_choice", "options": { "A": "At most n−1", "B": "At most k", "C": "Exactly 2k", "D": "Exactly n+k" }, "correct": true, "kind": "recognition", "topic": "coding", "original_format": "written", "repairs": [ "Specify minimal locator; padded Berlekamp–Welch locators need not have exactly the actual error count as degree." ], "source_parts": [ "11.1c" ], "exam": "midterm_su25", "sample": "follow_up", "selected": [ "B" ], "expected": [ "B" ], "valid_response": true, "confidence": 0.97, "probabilities": { "A": 0.01, "B": 0.98, "C": 0.01, "D": 0.0 }, "batch_response_ms": 379.65533299939125, "repaired": true }, { "id": "11.1d", "section": 11, "context": "", "prompt": "There are exactly m=0 and zero otherwise. Give an integral for P(A_t=1), clipped at 1?", "response_mode": "single_choice", "options": { "A": "1-(1-p^5)^12", "B": "min(1,12*p^5)", "C": "min(1,12*p^10)", "D": "min(1,p^5)" }, "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 2, "repairs": [], "reference_text": "B. min(1,12*p^5)", "robot_text": "B. min(1,12*p^5)", "output": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "C": 0.0, "A": 0.0, "B": 1.0, "D": 0.0 } }, "request_question": { "type": "choice", "instructions": "Answer the problem in `problems[4]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "1-(1-p^5)^12", "B": "min(1,12*p^5)", "C": "min(1,12*p^10)", "D": "min(1,p^5)" } }, "batch_ms": 439.7859159998916, "batch_items": 6, "topic": "Bingo & variance", "source_page": 26, "exam": "final_fa25", "sample": "discovery" }, { "id": "Q13.6", "section": 13, "context": "A 5 by 5 bingo card represents 25 booths. Melody visits each booth independently with probability p; its square is marked exactly when visited. There is no free square. The 12 possible bingo lines are 5 rows, 5 columns and 2 diagonals. X is the number of completely marked lines.", "prompt": "Assume E[X]<2. Which is the standard two-sided Chebyshev upper-bound expression applied to P(X>=2), in terms of E[X] and Var(X), clipped at 1?", "response_mode": "single_choice", "options": { "A": "min(1,(2-E[X])^2/Var(X))", "B": "min(1,Var(X)/(2-E[X]))", "C": "min(1,Var(X)/(2-E[X])^2)", "D": "min(1,Var(X)/(2+E[X])^2)" }, "correct": true, "valid_response": true, "selected": [ "C" ], "expected": [ "C" ], "adapted_weight": 3, "repairs": [ "chebyshev_condition" ], "reference_text": "C. min(1,Var(X)/(2-E[X])^2)", "robot_text": "C. min(1,Var(X)/(2-E[X])^2)", "output": { "type": "choice", "choice": "C", "confidence": 0.97, "probabilities": { "C": 0.98, "A": 0.01, "B": 0.01, "D": 0.0 } }, "request_question": { "type": "choice", "instructions": "Answer the problem in `problems[5]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "min(1,(2-E[X])^2/Var(X))", "B": "min(1,Var(X)/(2-E[X]))", "C": "min(1,Var(X)/(2-E[X])^2)", "D": "min(1,Var(X)/(2+E[X])^2)" } }, "batch_ms": 439.7859159998916, "batch_items": 6, "topic": "Bingo & variance", "source_page": 26, "exam": "final_fa25", "sample": "discovery" }, { "id": "Q14", "section": 14, "context": "", "prompt": "Events A,B have nonzero probability and indicators I_A,I_B. Which argument proves that Cov(I_A,I_B)>0 implies P(A|B)>P(A)?", "response_mode": "single_choice", "options": { "A": "Cov(I_A,I_B)=P(A union B)-P(A)P(B)=P(B)[P(A|B)-P(A)]. Because P(B)>0, positive covariance forces the bracket to be positive.", "B": "Cov(I_A,I_B)=P(A intersection B)-P(A)P(B)=P(A)[P(A|B)-P(A)]. Because P(A)>0, positive covariance forces the bracket to be positive.", "C": "Cov(I_A,I_B)=P(A intersection B)-P(A)P(B)=P(B)[P(A|B)-P(A)]. Because P(B)>0, positive covariance forces the bracket to be positive.", "D": "Cov(I_A,I_B)=P(A)P(B)-P(A intersection B)=P(B)[P(A|B)-P(A)]. Because P(B)>0, positive covariance forces the bracket to be positive." }, "correct": true, "valid_response": true, "selected": [ "C" ], "expected": [ "C" ], "adapted_weight": 5, "repairs": [], "reference_text": "C. Cov(I_A,I_B)=P(A intersection B)-P(A)P(B)=P(B)[P(A|B)-P(A)]. Because P(B)>0, positive covariance forces the bracket to be positive.", "robot_text": "C. Cov(I_A,I_B)=P(A intersection B)-P(A)P(B)=P(B)[P(A|B)-P(A)]. Because P(B)>0, positive covariance forces the bracket to be positive.", "output": { "type": "choice", "choice": "C", "confidence": 0.97, "probabilities": { "A": 0.02, "B": 0.0, "C": 0.98, "D": 0.0 } }, "request_question": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Cov(I_A,I_B)=P(A union B)-P(A)P(B)=P(B)[P(A|B)-P(A)]. Because P(B)>0, positive covariance forces the bracket to be positive.", "B": "Cov(I_A,I_B)=P(A intersection B)-P(A)P(B)=P(A)[P(A|B)-P(A)]. Because P(A)>0, positive covariance forces the bracket to be positive.", "C": "Cov(I_A,I_B)=P(A intersection B)-P(A)P(B)=P(B)[P(A|B)-P(A)]. Because P(B)>0, positive covariance forces the bracket to be positive.", "D": "Cov(I_A,I_B)=P(A)P(B)-P(A intersection B)=P(B)[P(A|B)-P(A)]. Because P(B)>0, positive covariance forces the bracket to be positive." } }, "batch_ms": 384.9657089995162, "batch_items": 1, "topic": "Covariance", "source_page": 27, "exam": "final_fa25", "sample": "discovery" }, { "id": "Q15.1", "section": 15, "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.", "prompt": "Let X~Gaussian(3,6) and Y~Exponential(3) be independent. What is P(X=Y)?", "response_mode": "single_choice", "options": { "A": "0", "B": "Cannot be determined from the stated information", "C": "1", "D": "1/2" }, "correct": true, "valid_response": true, "selected": [ "A" ], "expected": [ "A" ], "adapted_weight": 2, "repairs": [ "independence_added" ], "reference_text": "A. 0", "robot_text": "A. 0", "output": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "B": 0.0, "D": 0.0, "A": 1.0, "C": 0.0 } }, "request_question": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "0", "B": "Cannot be determined from the stated information", "C": "1", "D": "1/2" } }, "batch_ms": 377.990958000737, "batch_items": 5, "topic": "Continuous variables", "source_page": 28, "exam": "final_fa25", "sample": "discovery" }, { "id": "Q15.2", "section": 15, "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.", "prompt": "For X~Gaussian(3,6), which statement is true about P(X<4)?", "response_mode": "single_choice", "options": { "A": "<0.5", "B": "=0.5", "C": ">0.5", "D": "Not enough information" }, "correct": true, "valid_response": true, "selected": [ "C" ], "expected": [ "C" ], "adapted_weight": 2, "repairs": [], "reference_text": "C. >0.5", "robot_text": "C. >0.5", "output": { "type": "choice", "choice": "C", "confidence": 0.38, "probabilities": { "B": 0.02, "D": 0.02, "C": 0.54, "A": 0.42 } }, "request_question": { "type": "choice", "instructions": "Answer the problem in `problems[1]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "<0.5", "B": "=0.5", "C": ">0.5", "D": "Not enough information" } }, "batch_ms": 377.990958000737, "batch_items": 5, "topic": "Continuous variables", "source_page": 28, "exam": "final_fa25", "sample": "discovery" }, { "id": "Q15.3", "section": 15, "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.\n\nA, B and C are independent Uniform(0,12) random variables. X indicates that C lies between A and B.", "prompt": "What is P(X=1 | A,B), in terms of A and B?", "response_mode": "single_choice", "options": { "A": "1-abs(A-B)/12", "B": "abs(A-B)/12", "C": "(A-B)/12", "D": "abs(A-B)/144" }, "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 2, "repairs": [], "reference_text": "B. abs(A-B)/12", "robot_text": "B. abs(A-B)/12", "output": { "type": "choice", "choice": "B", "confidence": 0.96, "probabilities": { "B": 0.97, "D": 0.0, "C": 0.0, "A": 0.03 } }, "request_question": { "type": "choice", "instructions": "Answer the problem in `problems[2]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "1-abs(A-B)/12", "B": "abs(A-B)/12", "C": "(A-B)/12", "D": "abs(A-B)/144" } }, "batch_ms": 377.990958000737, "batch_items": 5, "topic": "Continuous variables", "source_page": 28, "exam": "final_fa25", "sample": "discovery" }, { "id": "Q15.4", "section": 15, "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.\n\nA, B and C are independent Uniform(0,12) random variables. X indicates that C lies between A and B.", "prompt": "Without conditioning on A,B, what is P(X=1)? Hint: there is a way without integrals.", "response_mode": "single_choice", "options": { "A": "1/2", "B": "1/6", "C": "2/3", "D": "1/3" }, "correct": true, "valid_response": true, "selected": [ "D" ], "expected": [ "D" ], "adapted_weight": 2, "repairs": [], "reference_text": "D. 1/3", "robot_text": "D. 1/3", "output": { "type": "choice", "choice": "D", "confidence": 0.96, "probabilities": { "B": 0.01, "D": 0.96, "C": 0.01, "A": 0.02 } }, "request_question": { "type": "choice", "instructions": "Answer the problem in `problems[3]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "1/2", "B": "1/6", "C": "2/3", "D": "1/3" } }, "batch_ms": 377.990958000737, "batch_items": 5, "topic": "Continuous variables", "source_page": 29, "exam": "final_fa25", "sample": "discovery" }, { "id": "Q15.5", "section": 15, "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.\n\nA, B and C are independent Uniform(0,12) random variables. X indicates that C lies between A and B. Q15.3 asks for P(X=1 | A,B); Q15.4 asks for P(X=1).", "prompt": "What is E[abs(A-B)]? Hint: use the previous two parts to avoid integrals. The original exam labels this particularly challenging.", "response_mode": "single_choice", "options": { "A": "0", "B": "4", "C": "8", "D": "6" }, "correct": false, "valid_response": true, "selected": [ "C" ], "expected": [ "B" ], "adapted_weight": 5, "repairs": [], "reference_text": "B. 4", "robot_text": "C. 8", "output": { "type": "choice", "choice": "C", "confidence": 0.35, "probabilities": { "B": 0.45, "D": 0.03, "C": 0.52, "A": 0.0 } }, "request_question": { "type": "choice", "instructions": "Answer the problem in `problems[4]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "0", "B": "4", "C": "8", "D": "6" } }, "batch_ms": 377.990958000737, "batch_items": 5, "topic": "Continuous variables", "source_page": 29, "exam": "final_fa25", "sample": "discovery" }, { "id": "Q16.1", "section": 16, "context": "Tom travels from Chicago (CHI) to Miami (MIA). Each outgoing train route is equally likely. The diagram, transcribed without solving: CHI -> SFO (1/3), DC (1/3), NYC (1/3); SFO -> CHI (1); DC -> SFO (1/3), NYC (1/3), MIA (1/3); NYC -> CHI (1/2), DC (1/2); MIA -> MIA (1).", "prompt": "Find the probability of reaching MIA from CHI before visiting NYC. Select a system of equations, without solving it. h(s) denotes the probability of reaching MIA before NYC starting at s; the requested probability is h(CHI).", "response_mode": "single_choice", "options": { "A": "h(MIA)=1; h(NYC)=0; h(SFO)=h(CHI); h(DC)=(h(SFO)+h(NYC)+h(MIA))/3; h(CHI)=(h(SFO)+h(DC))/2.", "B": "h(MIA)=1; h(NYC)=0; h(SFO)=h(CHI); h(DC)=(h(NYC)+h(MIA))/2; h(CHI)=(h(SFO)+h(DC)+h(NYC))/3.", "C": "h(MIA)=0; h(NYC)=1; h(SFO)=h(CHI); h(DC)=(h(SFO)+h(NYC)+h(MIA))/3; h(CHI)=(h(SFO)+h(DC)+h(NYC))/3.", "D": "h(MIA)=1; h(NYC)=0; h(SFO)=h(CHI); h(DC)=(h(SFO)+h(NYC)+h(MIA))/3; h(CHI)=(h(SFO)+h(DC)+h(NYC))/3." }, "correct": true, "valid_response": true, "selected": [ "D" ], "expected": [ "D" ], "adapted_weight": 3, "repairs": [], "reference_text": "D. h(MIA)=1; h(NYC)=0; h(SFO)=h(CHI); h(DC)=(h(SFO)+h(NYC)+h(MIA))/3; h(CHI)=(h(SFO)+h(DC)+h(NYC))/3.", "robot_text": "D. h(MIA)=1; h(NYC)=0; h(SFO)=h(CHI); h(DC)=(h(SFO)+h(NYC)+h(MIA))/3; h(CHI)=(h(SFO)+h(DC)+h(NYC))/3.", "output": { "type": "choice", "choice": "D", "confidence": 0.98, "probabilities": { "B": 0.0, "D": 0.99, "C": 0.0, "A": 0.01 } }, "request_question": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "h(MIA)=1; h(NYC)=0; h(SFO)=h(CHI); h(DC)=(h(SFO)+h(NYC)+h(MIA))/3; h(CHI)=(h(SFO)+h(DC))/2.", "B": "h(MIA)=1; h(NYC)=0; h(SFO)=h(CHI); h(DC)=(h(NYC)+h(MIA))/2; h(CHI)=(h(SFO)+h(DC)+h(NYC))/3.", "C": "h(MIA)=0; h(NYC)=1; h(SFO)=h(CHI); h(DC)=(h(SFO)+h(NYC)+h(MIA))/3; h(CHI)=(h(SFO)+h(DC)+h(NYC))/3.", "D": "h(MIA)=1; h(NYC)=0; h(SFO)=h(CHI); h(DC)=(h(SFO)+h(NYC)+h(MIA))/3; h(CHI)=(h(SFO)+h(DC)+h(NYC))/3." } }, "batch_ms": 336.481874999663, "batch_items": 2, "topic": "Markov chains", "source_page": 30, "exam": "final_fa25", "sample": "discovery" }, { "id": "Q16.2", "section": 16, "context": "Tom travels from Chicago (CHI) to Miami (MIA). Each outgoing train route is equally likely. The diagram, transcribed without solving: CHI -> SFO (1/3), DC (1/3), NYC (1/3); SFO -> CHI (1); DC -> SFO (1/3), NYC (1/3), MIA (1/3); NYC -> CHI (1/2), DC (1/2); MIA -> MIA (1).", "prompt": "Tom spends 3 days on each visit to NYC and 1 day in every other city before departing. Travel takes zero days. Select equations for the expected days to first arrival at MIA, starting in CHI and including its initial day. Do not solve. t(s) denotes the remaining expected days starting at s.", "response_mode": "single_choice", "options": { "A": "t(MIA)=0; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(NYC)+t(MIA))/3; t(NYC)=3+(t(CHI)+t(DC))/2.", "B": "t(MIA)=0; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(SFO)+t(NYC)+t(MIA))/3; t(NYC)=3+(t(CHI)+t(DC))/2.", "C": "t(MIA)=0; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(SFO)+t(NYC)+t(MIA))/3; t(NYC)=1+(t(CHI)+t(DC))/2.", "D": "t(MIA)=1; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(SFO)+t(NYC)+t(MIA))/3; t(NYC)=3+(t(CHI)+t(DC))/2." }, "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 3, "repairs": [ "dc_edge_in_key" ], "reference_text": "B. t(MIA)=0; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(SFO)+t(NYC)+t(MIA))/3; t(NYC)=3+(t(CHI)+t(DC))/2.", "robot_text": "B. t(MIA)=0; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(SFO)+t(NYC)+t(MIA))/3; t(NYC)=3+(t(CHI)+t(DC))/2.", "output": { "type": "choice", "choice": "B", "confidence": 0.97, "probabilities": { "B": 0.97, "D": 0.02, "A": 0.01, "C": 0.0 } }, "request_question": { "type": "choice", "instructions": "Answer the problem in `problems[1]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "t(MIA)=0; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(NYC)+t(MIA))/3; t(NYC)=3+(t(CHI)+t(DC))/2.", "B": "t(MIA)=0; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(SFO)+t(NYC)+t(MIA))/3; t(NYC)=3+(t(CHI)+t(DC))/2.", "C": "t(MIA)=0; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(SFO)+t(NYC)+t(MIA))/3; t(NYC)=1+(t(CHI)+t(DC))/2.", "D": "t(MIA)=1; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(SFO)+t(NYC)+t(MIA))/3; t(NYC)=3+(t(CHI)+t(DC))/2." } }, "batch_ms": 336.481874999663, "batch_items": 2, "topic": "Markov chains", "source_page": 31, "exam": "final_fa25", "sample": "discovery" } ], "freeze_sha256": "b2ac3ee0d79407ddbaa1e3d711c49a9b392927b6e97279207f5db72350c8c96f", "run_file": "runs/20260918T081158.317869Z/responses.json" }, "discovery_evidence": { "protocol": { "exam.json": "f189ea0fc59630a271a4455128917b65329661aff00d3b8efa82361dbcd91608", "exam.md": "7e3770f853599fe647fa1c9882cc8b864151b2b249e2b2a83115eea7f5240031", "requests.json": "db1d5220d099c4e7dc43b462b3be5ab2c6a1045e4a2909b4a865d74fc57a5024" }, "exam": { "version": "1.0", "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "items": [ { "id": "Q1.1", "section": 1, "context": "", "prompt": "Is ((P AND Q) IMPLIES R) equivalent to (P IMPLIES (Q IMPLIES R))?", "response_mode": "single_choice", "options": { "A": "Always True", "B": "Sometimes False" } }, { "id": "Q1.2", "section": 1, "context": "", "prompt": "Is [forall x, (P(x) IMPLIES Q(x))] equivalent to [(forall x, NOT P(x)) OR (forall x, Q(x))]?", "response_mode": "single_choice", "options": { "A": "Always True", "B": "Sometimes False" } }, { "id": "Q2.1", "section": 2, "context": "", "prompt": "For integers a,b, let d=gcd(a,b). Which argument proves that d divides ax+by for every pair of integers x,y?", "response_mode": "single_choice", "options": { "A": "Since d divides a and b, Bezout gives ax+by=d for every pair of integers x,y. Therefore ax+by is always equal to d and hence divisible by d.", "B": "Since d divides a and b, write d=ak and d=bj for integers k,j. Substituting gives ax+by=d(x/k+y/j), whose parenthesized term must be an integer.", "C": "Since d divides a and b, write a=dk and b=dj for integers k,j. Then ax+by=d(kx+jy), and kx+jy is an integer, establishing divisibility.", "D": "Since d is the greatest common divisor, every integer smaller than d divides both a and b. In particular x and y divide both, so d divides ax+by." } }, { "id": "Q2.2", "section": 2, "context": "", "prompt": "7 divides 4^186 - 1. Hint: 186 = 31 * 6.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" } }, { "id": "Q2.3", "section": 2, "context": "", "prompt": "5 divides 4^n - 1 for all even natural numbers n.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" } }, { "id": "Q2.4", "section": 2, "context": "", "prompt": "What is the last digit (ones place) of 3^47?", "response_mode": "single_choice", "options": { "A": "2", "B": "9", "C": "1", "D": "3", "E": "5", "F": "6", "G": "4", "H": "8", "I": "0", "J": "7" } }, { "id": "Q2.5", "section": 2, "context": "", "prompt": "Alice implements RSA correctly with N=33 and e=7 and sends ciphertext y=6. Given 33=3*11, what was the original message x?", "response_mode": "single_choice", "options": { "A": "15", "B": "27", "C": "18", "D": "6" } }, { "id": "Q2.6", "section": 2, "context": "", "prompt": "Find (6^7 + 7^6) mod 91. Hint: use CRT; 91=7*13.", "response_mode": "single_choice", "options": { "A": "13", "B": "6", "C": "0", "D": "78" } }, { "id": "Q3.1", "section": 3, "context": "", "prompt": "Working modulo 6, how many distinct formal polynomials of degree exactly two are there?", "response_mode": "single_choice", "options": { "A": "125", "B": "180", "C": "216", "D": "150" } }, { "id": "Q3.2", "section": 3, "context": "", "prompt": "Over GF(7), find g(x) of degree strictly less than 7, with coefficients in GF(7), giving the same output as f(x)=9x^66+7x^58+5x^55+5x^24+10x^19 for every x in GF(7). Hint: the solution is a simple polynomial.", "response_mode": "single_choice", "options": { "A": "g(x)=x^6", "B": "g(x)=0", "C": "g(x)=1", "D": "g(x)=x" } }, { "id": "Q3.3", "section": 3, "context": "", "prompt": "You receive n+2k points from a polynomial of degree n-1, with up to k corruptions. How many messages can you recover using Berlekamp-Welch?", "response_mode": "single_choice", "options": { "A": "k", "B": "n+2k", "C": "n", "D": "1" } }, { "id": "Q4.1", "section": 4, "context": "", "prompt": "K_4 is planar.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" } }, { "id": "Q4.2", "section": 4, "context": "", "prompt": "K_5 has an Eulerian tour.", "response_mode": "single_choice", "options": { "A": "True", "B": "False" } }, { "id": "Q4.3", "section": 4, "context": "", "prompt": "A 12-sided die is a planar graph wrapped around a sphere, with 12 faces and 20 vertices. How many edges does it have?", "response_mode": "single_choice", "options": { "A": "18", "B": "24", "C": "32", "D": "30" } }, { "id": "Q4.4", "section": 4, "context": "", "prompt": "Which argument proves that, if a finite undirected graph has one odd-degree vertex, it must have another odd-degree vertex?", "response_mode": "single_choice", "options": { "A": "The sum of degrees equals twice the number of edges, so it is even. If exactly one degree were odd and all others even, that sum would be odd, a contradiction.", "B": "Removing any odd-degree vertex leaves all remaining degrees even. Restoring it changes every remaining degree by one, so every other vertex then has odd degree.", "C": "An edge incident to the odd-degree vertex has another endpoint. That endpoint has odd degree because the common edge contributes one to both endpoint degrees.", "D": "The sum of degrees equals the number of edges, so it is even. If exactly one degree were odd and all others even, that sum would be odd, a contradiction." } }, { "id": "Q5.1", "section": 5, "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "prompt": "The set of all complex numbers.", "response_mode": "single_choice", "options": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" } }, { "id": "Q5.2", "section": 5, "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "prompt": "The set of all prime numbers.", "response_mode": "single_choice", "options": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" } }, { "id": "Q5.3", "section": 5, "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "prompt": "The real interval [0.001,0.1].", "response_mode": "single_choice", "options": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" } }, { "id": "Q5.4", "section": 5, "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "prompt": "The complex numbers a+bi where a,b are integers.", "response_mode": "single_choice", "options": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" } }, { "id": "Q5.5", "section": 5, "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "prompt": "The set of edges of the complete bipartite graph K_(100,100).", "response_mode": "single_choice", "options": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" } }, { "id": "Q5.6", "section": 5, "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "prompt": "The set of all integer powers of 2.", "response_mode": "single_choice", "options": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" } }, { "id": "Q5.7", "section": 5, "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "prompt": "The set of real polynomials of degree at most two passing through (1,1) and (2,2).", "response_mode": "single_choice", "options": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" } }, { "id": "Q6", "section": 6, "context": "Two programs F and G are equivalent if, for every input, both halt with the same output or both do not halt. Suppose EQUIVALENT(F,G) always halts and returns TRUE iff F and G are equivalent. Complete HALT(P,x) to decide whether P halts on x:\ndef HALT(P,x):\n def F(y):\n ----(1)----\n ----(2)----\n def G(y):\n return 0\n return EQUIVALENT(----(3)----, ----(4)----)", "prompt": "Choose the ordered tuple filling blanks (1),(2),(3),(4).", "response_mode": "single_choice", "options": { "A": "(return 0; P(x); F; G)", "B": "(P(x); return 0; F; G)", "C": "(P(x); return 0; F; F)", "D": "(P(y); return 0; F; G)" } }, { "id": "Q7.1", "section": 7, "context": "Answers may be unsimplified (binomial coefficients, factorials, exponents), but do not use summation or product notation. C(n,k) denotes a binomial coefficient.", "prompt": "A deck has 52 cards, 4 suits and 13 ranks per suit. A hand is an unordered set of five cards. How many hands contain exactly one 6 and exactly one 7?", "response_mode": "single_choice", "options": { "A": "C(4,2)*C(44,3)", "B": "4*4*C(50,3)", "C": "4*4*C(44,3)", "D": "4*4*C(44,3)*5!" } }, { "id": "Q7.2", "section": 7, "context": "Answers may be unsimplified (binomial coefficients, factorials, exponents), but do not use summation or product notation. C(n,k) denotes a binomial coefficient.", "prompt": "Draw 3 cards without replacement. A means at least one ace. C_i means exactly i aces, for i=1,2,3. F,S,T mean respectively that the first, second or third card is an ace. Which representation of P(A) is easier, and why?", "response_mode": "single_choice", "options": { "A": "Use C_1,C_2,C_3: they are independent and their intersection is A, so their probabilities can be multiplied.", "B": "Use F,S,T: they are independent and their intersection is A, so their probabilities can be multiplied.", "C": "Use F,S,T: they are mutually exclusive and their union is A, so their probabilities can be added.", "D": "Use C_1,C_2,C_3: they are mutually exclusive and their union is A, so their probabilities can be added." } }, { "id": "Q7.3", "section": 7, "context": "Answers may be unsimplified (binomial coefficients, factorials, exponents), but do not use summation or product notation. C(n,k) denotes a binomial coefficient.\n\nAgnes distributes all 40 indistinguishable cookies among three distinct minions: Bob, Kevin and Stuart. Each has a carrying limit of 15 cookies.", "prompt": "With no carrying limit, how many distributions of the 40 cookies are possible?", "response_mode": "single_choice", "options": { "A": "C(40,2)", "B": "3^40", "C": "C(42,2)", "D": "C(42,3)" } }, { "id": "Q7.4", "section": 7, "context": "Answers may be unsimplified (binomial coefficients, factorials, exponents), but do not use summation or product notation. C(n,k) denotes a binomial coefficient.\n\nAgnes distributes all 40 indistinguishable cookies among three distinct minions: Bob, Kevin and Stuart. Each has a carrying limit of 15 cookies.", "prompt": "How many distributions exceed Bob's limit? Other minions may also exceed their limits.", "response_mode": "single_choice", "options": { "A": "C(26,2)", "B": "C(27,2)", "C": "C(42,2)-C(26,2)", "D": "C(24,2)" } }, { "id": "Q7.5", "section": 7, "context": "Answers may be unsimplified (binomial coefficients, factorials, exponents), but do not use summation or product notation. C(n,k) denotes a binomial coefficient.\n\nAgnes distributes all 40 indistinguishable cookies among three distinct minions: Bob, Kevin and Stuart. Each has a carrying limit of 15 cookies.", "prompt": "How many distributions exceed both Bob's and Kevin's limits?", "response_mode": "single_choice", "options": { "A": "C(12,2)", "B": "C(10,2)", "C": "C(8,2)", "D": "C(26,2)^2" } }, { "id": "Q7.6", "section": 7, "context": "Answers may be unsimplified (binomial coefficients, factorials, exponents), but do not use summation or product notation. C(n,k) denotes a binomial coefficient.\n\nAgnes distributes all 40 indistinguishable cookies among three distinct minions: Bob, Kevin and Stuart. Each has a carrying limit of 15 cookies. Here a counts unrestricted distributions; b counts distributions exceeding one specified limit; c counts distributions exceeding two specified limits.", "prompt": "Let a,b,c denote the answers to Q7.3,Q7.4,Q7.5 respectively. Express the number of distributions respecting all three limits in terms of a,b,c.", "response_mode": "single_choice", "options": { "A": "a-b+c", "B": "a-3*b+6*c", "C": "a-3*b+3*c", "D": "a-3*b-3*c" } }, { "id": "Q8", "section": 8, "context": "", "prompt": "In a town, 3/4 of residents have black hair and 2/3 are female, independently. Two people are selected randomly with replacement. Given that at least one is a black-haired female, what is the probability both have black hair? Select the answer with valid supporting work. Hint: the answer is not 9/16.", "response_mode": "single_choice", "options": { "A": "1/2: a randomly selected person is a black-haired female with probability (3/4)*(2/3)=1/2; replacement makes the requested conditional probability the same.", "B": "8/9: conditional on both people having black hair, the chance at least one is female is 1-(1/3)^2=8/9; this is the requested conditional probability.", "C": "3/4: if A means at least one black-haired female and B means both have black hair, then P(B|A)=P(B)/P(A)=(9/16)/(1-(1/2)^2)=3/4.", "D": "2/3: if A means at least one black-haired female and B means both have black hair, then P(B|A)=(9/16)*(1-(1/3)^2)/(1-(1/2)^2)=2/3." } }, { "id": "Q9.2", "section": 9, "context": "Flip a biased coin independently infinitely many times, with heads probability p and tails probability 1-p. Give the distribution and parameters. The common discrete distributions named on the exam are Bernoulli, Binomial, Geometric, Uniform and Poisson. Original worked example Q9.1 (0 points): the indicator that the first three flips are heads is Bernoulli(p^3). Geometric(q) counts trials up to and including the first success (support 1,2,...).", "prompt": "Distribution of the number of heads in the first 70 flips?", "response_mode": "single_choice", "options": { "A": "Poisson(70*p)", "B": "Geometric(p)", "C": "Binomial(70,1-p)", "D": "Binomial(70,p)" } }, { "id": "Q9.3", "section": 9, "context": "Flip a biased coin independently infinitely many times, with heads probability p and tails probability 1-p. Give the distribution and parameters. The common discrete distributions named on the exam are Bernoulli, Binomial, Geometric, Uniform and Poisson. Original worked example Q9.1 (0 points): the indicator that the first three flips are heads is Bernoulli(p^3). Geometric(q) counts trials up to and including the first success (support 1,2,...).", "prompt": "Conditional on exactly one head in the first 20 flips, what is the distribution of the flip number on which the first head occurs?", "response_mode": "single_choice", "options": { "A": "Geometric(p)", "B": "Uniform({1,2,...,20})", "C": "Binomial(20,p)", "D": "Uniform({0,1,...,19})" } }, { "id": "Q9.4", "section": 9, "context": "Flip a biased coin independently infinitely many times, with heads probability p and tails probability 1-p. Give the distribution and parameters. The common discrete distributions named on the exam are Bernoulli, Binomial, Geometric, Uniform and Poisson. Original worked example Q9.1 (0 points): the indicator that the first three flips are heads is Bernoulli(p^3). Geometric(q) counts trials up to and including the first success (support 1,2,...).", "prompt": "Distribution of the gap between the first two tails, counting intervening heads? THHHT has gap 3. Hint: a standard random variable minus 1.", "response_mode": "single_choice", "options": { "A": "Geometric(p)-1", "B": "Geometric(1-p)-1", "C": "Binomial(2,p)-1", "D": "Geometric(1-p)" } }, { "id": "Q9.5", "section": 9, "context": "Flip a biased coin independently infinitely many times, with heads probability p and tails probability 1-p. Give the distribution and parameters. The common discrete distributions named on the exam are Bernoulli, Binomial, Geometric, Uniform and Poisson. Original worked example Q9.1 (0 points): the indicator that the first three flips are heads is Bernoulli(p^3). Geometric(q) counts trials up to and including the first success (support 1,2,...).", "prompt": "Let p=1/2. Distribution of the number of flips until the same outcome occurs twice consecutively? Hint: 1 plus a standard random variable.", "response_mode": "single_choice", "options": { "A": "1+Geometric(1/2)", "B": "1+Binomial(2,1/2)", "C": "Geometric(1/2)", "D": "1+Geometric(1/4)" } }, { "id": "Q9.6", "section": 9, "context": "Flip a biased coin independently infinitely many times, with heads probability p and tails probability 1-p. Give the distribution and parameters. The common discrete distributions named on the exam are Bernoulli, Binomial, Geometric, Uniform and Poisson. Original worked example Q9.1 (0 points): the indicator that the first three flips are heads is Bernoulli(p^3). Geometric(q) counts trials up to and including the first success (support 1,2,...).", "prompt": "Let p=1/2. Distribution of the number of runs in the first 50 flips? A run is an uninterrupted sequence of identical outcomes, e.g. HHHHHTTTHTTH begins with runs HHHHH, TTT, H, TT. Hint: 1 plus a standard random variable.", "response_mode": "single_choice", "options": { "A": "1+Geometric(1/2)", "B": "1+Binomial(50,1/2)", "C": "1+Binomial(49,1/4)", "D": "1+Binomial(49,1/2)" } }, { "id": "Q10", "section": 10, "context": "Each spin has four equally likely outcomes: alpha earns 1 point; beta earns 2 points; gamma gives two additional spins; delta loses 1 point and gives one additional spin. Start with one spin. Additional spins use the same rules. The score is the sum of points earned or lost. Example: alpha gives score 1. Example sequence delta,gamma,gamma,alpha,beta,alpha gives score -1+1+2+1=3.", "prompt": "Let X be the final score. What is E[X]?", "response_mode": "single_choice", "options": { "A": "0", "B": "1", "C": "2", "D": "3", "E": "4", "F": "infinity" } }, { "id": "Q11.1", "section": 11, "context": "Normal(mu,v) denotes a normal distribution with mean mu and variance v. Poisson parameters are positive.", "prompt": "Independent X~Normal(1,2) and Y~Normal(2,1). What is the distribution of Z=2X-Y+1?", "response_mode": "single_choice", "options": { "A": "Normal(1,5)", "B": "Normal(1,9)", "C": "Normal(1,3)", "D": "Normal(5,9)" } }, { "id": "Q11.2", "section": 11, "context": "Normal(mu,v) denotes a normal distribution with mean mu and variance v. Poisson parameters are positive.", "prompt": "Independent X,Y~Poisson(lambda). Let Z=X+Y. Select all true statements.", "response_mode": "select_all", "options": { "A": "Z~Poisson(2*lambda)", "B": "Z has the distribution of 2P, where P~Poisson(lambda)", "C": "None of the above" } }, { "id": "Q11.3", "section": 11, "context": "Normal(mu,v) denotes a normal distribution with mean mu and variance v. Poisson parameters are positive.", "prompt": "Let X_1,...,X_n be independent Poisson(lambda) variables and S=sum_i X_i. Select all true statements.", "response_mode": "select_all", "options": { "A": "For every n, S~Poisson(lambda*n)", "B": "As n tends to infinity, (S-lambda*n)/sqrt(lambda*n) converges in distribution to Normal(0,1)", "C": "None of the above" } }, { "id": "Q12.1", "section": 12, "context": "Student arrivals are modeled as a homogeneous Poisson process at a rate of one student per 30 minutes.", "prompt": "Nobody arrives during the first 50 minutes. What is the probability at least one student arrives during the final 10 minutes of the hour?", "response_mode": "single_choice", "options": { "A": "1-exp(-5/3)", "B": "1-exp(-1/3)", "C": "exp(-1/3)", "D": "1-exp(-2)" } }, { "id": "Q12.2", "section": 12, "context": "Student arrivals are modeled as a homogeneous Poisson process at a rate of one student per 30 minutes.", "prompt": "At the same arrival rate, students never leave once they arrive. Starting from an empty room, what is the expected number at the end of one hour?", "response_mode": "single_choice", "options": { "A": "30", "B": "2", "C": "1/2", "D": "1-exp(-2)" } }, { "id": "Q12.3", "section": 12, "context": "Time between BART train departures is exponential, with a mean of 15 minutes.", "prompt": "Ishan arrives at a random time. What is his expected wait until a train departs?", "response_mode": "single_choice", "options": { "A": "1/15 minute", "B": "15 minutes", "C": "7.5 minutes", "D": "30 minutes" } }, { "id": "Q12.4", "section": 12, "context": "Time between BART train departures is exponential, with a mean of 15 minutes.", "prompt": "The station agent says the last train left 5 minutes ago. What is Ishan's expected remaining wait?", "response_mode": "single_choice", "options": { "A": "15 minutes", "B": "20 minutes", "C": "10 minutes", "D": "5 minutes" } }, { "id": "Q12.5", "section": 12, "context": "", "prompt": "Trains and passengers have first arrival times A_t and A_p. The joint PDF is f(a_p,a_t)=lambda_p*lambda_t*exp(-lambda_p*a_p-lambda_t*a_t) for a_p,a_t>=0 and zero otherwise. Give an integral for P(A_t=1), clipped at 1?", "response_mode": "single_choice", "options": { "A": "1-(1-p^5)^12", "B": "min(1,12*p^5)", "C": "min(1,12*p^10)", "D": "min(1,p^5)" } }, { "id": "Q13.6", "section": 13, "context": "A 5 by 5 bingo card represents 25 booths. Melody visits each booth independently with probability p; its square is marked exactly when visited. There is no free square. The 12 possible bingo lines are 5 rows, 5 columns and 2 diagonals. X is the number of completely marked lines.", "prompt": "Assume E[X]<2. Which is the standard two-sided Chebyshev upper-bound expression applied to P(X>=2), in terms of E[X] and Var(X), clipped at 1?", "response_mode": "single_choice", "options": { "A": "min(1,(2-E[X])^2/Var(X))", "B": "min(1,Var(X)/(2-E[X]))", "C": "min(1,Var(X)/(2-E[X])^2)", "D": "min(1,Var(X)/(2+E[X])^2)" } }, { "id": "Q14", "section": 14, "context": "", "prompt": "Events A,B have nonzero probability and indicators I_A,I_B. Which argument proves that Cov(I_A,I_B)>0 implies P(A|B)>P(A)?", "response_mode": "single_choice", "options": { "A": "Cov(I_A,I_B)=P(A union B)-P(A)P(B)=P(B)[P(A|B)-P(A)]. Because P(B)>0, positive covariance forces the bracket to be positive.", "B": "Cov(I_A,I_B)=P(A intersection B)-P(A)P(B)=P(A)[P(A|B)-P(A)]. Because P(A)>0, positive covariance forces the bracket to be positive.", "C": "Cov(I_A,I_B)=P(A intersection B)-P(A)P(B)=P(B)[P(A|B)-P(A)]. Because P(B)>0, positive covariance forces the bracket to be positive.", "D": "Cov(I_A,I_B)=P(A)P(B)-P(A intersection B)=P(B)[P(A|B)-P(A)]. Because P(B)>0, positive covariance forces the bracket to be positive." } }, { "id": "Q15.1", "section": 15, "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.", "prompt": "Let X~Gaussian(3,6) and Y~Exponential(3) be independent. What is P(X=Y)?", "response_mode": "single_choice", "options": { "A": "0", "B": "Cannot be determined from the stated information", "C": "1", "D": "1/2" } }, { "id": "Q15.2", "section": 15, "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.", "prompt": "For X~Gaussian(3,6), which statement is true about P(X<4)?", "response_mode": "single_choice", "options": { "A": "<0.5", "B": "=0.5", "C": ">0.5", "D": "Not enough information" } }, { "id": "Q15.3", "section": 15, "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.\n\nA, B and C are independent Uniform(0,12) random variables. X indicates that C lies between A and B.", "prompt": "What is P(X=1 | A,B), in terms of A and B?", "response_mode": "single_choice", "options": { "A": "1-abs(A-B)/12", "B": "abs(A-B)/12", "C": "(A-B)/12", "D": "abs(A-B)/144" } }, { "id": "Q15.4", "section": 15, "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.\n\nA, B and C are independent Uniform(0,12) random variables. X indicates that C lies between A and B.", "prompt": "Without conditioning on A,B, what is P(X=1)? Hint: there is a way without integrals.", "response_mode": "single_choice", "options": { "A": "1/2", "B": "1/6", "C": "2/3", "D": "1/3" } }, { "id": "Q15.5", "section": 15, "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.\n\nA, B and C are independent Uniform(0,12) random variables. X indicates that C lies between A and B. Q15.3 asks for P(X=1 | A,B); Q15.4 asks for P(X=1).", "prompt": "What is E[abs(A-B)]? Hint: use the previous two parts to avoid integrals. The original exam labels this particularly challenging.", "response_mode": "single_choice", "options": { "A": "0", "B": "4", "C": "8", "D": "6" } }, { "id": "Q16.1", "section": 16, "context": "Tom travels from Chicago (CHI) to Miami (MIA). Each outgoing train route is equally likely. The diagram, transcribed without solving: CHI -> SFO (1/3), DC (1/3), NYC (1/3); SFO -> CHI (1); DC -> SFO (1/3), NYC (1/3), MIA (1/3); NYC -> CHI (1/2), DC (1/2); MIA -> MIA (1).", "prompt": "Find the probability of reaching MIA from CHI before visiting NYC. Select a system of equations, without solving it. h(s) denotes the probability of reaching MIA before NYC starting at s; the requested probability is h(CHI).", "response_mode": "single_choice", "options": { "A": "h(MIA)=1; h(NYC)=0; h(SFO)=h(CHI); h(DC)=(h(SFO)+h(NYC)+h(MIA))/3; h(CHI)=(h(SFO)+h(DC))/2.", "B": "h(MIA)=1; h(NYC)=0; h(SFO)=h(CHI); h(DC)=(h(NYC)+h(MIA))/2; h(CHI)=(h(SFO)+h(DC)+h(NYC))/3.", "C": "h(MIA)=0; h(NYC)=1; h(SFO)=h(CHI); h(DC)=(h(SFO)+h(NYC)+h(MIA))/3; h(CHI)=(h(SFO)+h(DC)+h(NYC))/3.", "D": "h(MIA)=1; h(NYC)=0; h(SFO)=h(CHI); h(DC)=(h(SFO)+h(NYC)+h(MIA))/3; h(CHI)=(h(SFO)+h(DC)+h(NYC))/3." } }, { "id": "Q16.2", "section": 16, "context": "Tom travels from Chicago (CHI) to Miami (MIA). Each outgoing train route is equally likely. The diagram, transcribed without solving: CHI -> SFO (1/3), DC (1/3), NYC (1/3); SFO -> CHI (1); DC -> SFO (1/3), NYC (1/3), MIA (1/3); NYC -> CHI (1/2), DC (1/2); MIA -> MIA (1).", "prompt": "Tom spends 3 days on each visit to NYC and 1 day in every other city before departing. Travel takes zero days. Select equations for the expected days to first arrival at MIA, starting in CHI and including its initial day. Do not solve. t(s) denotes the remaining expected days starting at s.", "response_mode": "single_choice", "options": { "A": "t(MIA)=0; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(NYC)+t(MIA))/3; t(NYC)=3+(t(CHI)+t(DC))/2.", "B": "t(MIA)=0; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(SFO)+t(NYC)+t(MIA))/3; t(NYC)=3+(t(CHI)+t(DC))/2.", "C": "t(MIA)=0; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(SFO)+t(NYC)+t(MIA))/3; t(NYC)=1+(t(CHI)+t(DC))/2.", "D": "t(MIA)=1; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(SFO)+t(NYC)+t(MIA))/3; t(NYC)=3+(t(CHI)+t(DC))/2." } } ] }, "run": { "scope": "full", "request_manifest_sha256": "db1d5220d099c4e7dc43b462b3be5ab2c6a1045e4a2909b4a865d74fc57a5024", "started_at": "2026-09-18T06:51:42.380907+00:00", "sections": [ { "section": 1, "request": { "model": "jev-1.13.0", "state": { "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "problems": [ { "id": "Q1.1", "context": "", "prompt": "Is ((P AND Q) IMPLIES R) equivalent to (P IMPLIES (Q IMPLIES R))?" }, { "id": "Q1.2", "context": "", "prompt": "Is [forall x, (P(x) IMPLIES Q(x))] equivalent to [(forall x, NOT P(x)) OR (forall x, Q(x))]?" } ] }, "questions": { "Q1.1": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Always True", "B": "Sometimes False" } }, "Q1.2": { "type": "choice", "instructions": "Answer the problem in `problems[1]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Always True", "B": "Sometimes False" } } } }, "attempts": [ { "status": 200, "full_response_ms": 427.4284170005558, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"Q1.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.97,\"probabilities\":{\"A\":0.98,\"B\":0.02}},\"Q1.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.99,\"probabilities\":{\"A\":0.0,\"B\":1.0}}},\"usage\":{\"input_tokens\":588,\"output_tokens\":65}}" } ], "response": { "model": "jev-1.13.0", "answers": { "Q1.1": { "type": "choice", "choice": "A", "confidence": 0.97, "probabilities": { "A": 0.98, "B": 0.02 } }, "Q1.2": { "type": "choice", "choice": "B", "confidence": 0.99, "probabilities": { "A": 0.0, "B": 1.0 } } }, "usage": { "input_tokens": 588, "output_tokens": 65 } } }, { "section": 2, "request": { "model": "jev-1.13.0", "state": { "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "problems": [ { "id": "Q2.1", "context": "", "prompt": "For integers a,b, let d=gcd(a,b). Which argument proves that d divides ax+by for every pair of integers x,y?" }, { "id": "Q2.2", "context": "", "prompt": "7 divides 4^186 - 1. Hint: 186 = 31 * 6." }, { "id": "Q2.3", "context": "", "prompt": "5 divides 4^n - 1 for all even natural numbers n." }, { "id": "Q2.4", "context": "", "prompt": "What is the last digit (ones place) of 3^47?" }, { "id": "Q2.5", "context": "", "prompt": "Alice implements RSA correctly with N=33 and e=7 and sends ciphertext y=6. Given 33=3*11, what was the original message x?" }, { "id": "Q2.6", "context": "", "prompt": "Find (6^7 + 7^6) mod 91. Hint: use CRT; 91=7*13." } ] }, "questions": { "Q2.1": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Since d divides a and b, Bezout gives ax+by=d for every pair of integers x,y. Therefore ax+by is always equal to d and hence divisible by d.", "B": "Since d divides a and b, write d=ak and d=bj for integers k,j. Substituting gives ax+by=d(x/k+y/j), whose parenthesized term must be an integer.", "C": "Since d divides a and b, write a=dk and b=dj for integers k,j. Then ax+by=d(kx+jy), and kx+jy is an integer, establishing divisibility.", "D": "Since d is the greatest common divisor, every integer smaller than d divides both a and b. In particular x and y divide both, so d divides ax+by." } }, "Q2.2": { "type": "choice", "instructions": "Answer the problem in `problems[1]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "True", "B": "False" } }, "Q2.3": { "type": "choice", "instructions": "Answer the problem in `problems[2]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "True", "B": "False" } }, "Q2.4": { "type": "choice", "instructions": "Answer the problem in `problems[3]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "2", "B": "9", "C": "1", "D": "3", "E": "5", "F": "6", "G": "4", "H": "8", "I": "0", "J": "7" } }, "Q2.5": { "type": "choice", "instructions": "Answer the problem in `problems[4]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "15", "B": "27", "C": "18", "D": "6" } }, "Q2.6": { "type": "choice", "instructions": "Answer the problem in `problems[5]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "13", "B": "6", "C": "0", "D": "78" } } } }, "attempts": [ { "status": 200, "full_response_ms": 449.84275000024354, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"Q2.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"B\":0.0,\"A\":0.0,\"C\":1.0}},\"Q2.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.76,\"probabilities\":{\"B\":0.12,\"A\":0.88}},\"Q2.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.66,\"probabilities\":{\"B\":0.17,\"A\":0.83}},\"Q2.4\":{\"type\":\"choice\",\"choice\":\"J\",\"confidence\":0.57,\"probabilities\":{\"D\":0.06,\"E\":0.02,\"G\":0.03,\"A\":0.11,\"C\":0.02,\"I\":0.0,\"J\":0.61,\"F\":0.03,\"B\":0.06,\"H\":0.06}},\"Q2.5\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.13,\"probabilities\":{\"D\":0.35,\"B\":0.19,\"A\":0.19,\"C\":0.27}},\"Q2.6\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.16,\"probabilities\":{\"D\":0.37,\"B\":0.26,\"A\":0.28,\"C\":0.09}}},\"usage\":{\"input_tokens\":1377,\"output_tokens\":287}}" } ], "response": { "model": "jev-1.13.0", "answers": { "Q2.1": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "D": 0.0, "B": 0.0, "A": 0.0, "C": 1.0 } }, "Q2.2": { "type": "choice", "choice": "A", "confidence": 0.76, "probabilities": { "B": 0.12, "A": 0.88 } }, "Q2.3": { "type": "choice", "choice": "A", "confidence": 0.66, "probabilities": { "B": 0.17, "A": 0.83 } }, "Q2.4": { "type": "choice", "choice": "J", "confidence": 0.57, "probabilities": { "D": 0.06, "E": 0.02, "G": 0.03, "A": 0.11, "C": 0.02, "I": 0.0, "J": 0.61, "F": 0.03, "B": 0.06, "H": 0.06 } }, "Q2.5": { "type": "choice", "choice": "D", "confidence": 0.13, "probabilities": { "D": 0.35, "B": 0.19, "A": 0.19, "C": 0.27 } }, "Q2.6": { "type": "choice", "choice": "D", "confidence": 0.16, "probabilities": { "D": 0.37, "B": 0.26, "A": 0.28, "C": 0.09 } } }, "usage": { "input_tokens": 1377, "output_tokens": 287 } } }, { "section": 3, "request": { "model": "jev-1.13.0", "state": { "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "problems": [ { "id": "Q3.1", "context": "", "prompt": "Working modulo 6, how many distinct formal polynomials of degree exactly two are there?" }, { "id": "Q3.2", "context": "", "prompt": "Over GF(7), find g(x) of degree strictly less than 7, with coefficients in GF(7), giving the same output as f(x)=9x^66+7x^58+5x^55+5x^24+10x^19 for every x in GF(7). Hint: the solution is a simple polynomial." }, { "id": "Q3.3", "context": "", "prompt": "You receive n+2k points from a polynomial of degree n-1, with up to k corruptions. How many messages can you recover using Berlekamp-Welch?" } ] }, "questions": { "Q3.1": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "125", "B": "180", "C": "216", "D": "150" } }, "Q3.2": { "type": "choice", "instructions": "Answer the problem in `problems[1]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "g(x)=x^6", "B": "g(x)=0", "C": "g(x)=1", "D": "g(x)=x" } }, "Q3.3": { "type": "choice", "instructions": "Answer the problem in `problems[2]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "k", "B": "n+2k", "C": "n", "D": "1" } } } }, "attempts": [ { "status": 200, "full_response_ms": 424.9754579996079, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"Q3.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.28,\"probabilities\":{\"B\":0.45,\"C\":0.14,\"D\":0.34,\"A\":0.07}},\"Q3.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.18,\"probabilities\":{\"B\":0.03,\"C\":0.22,\"D\":0.37,\"A\":0.38}},\"Q3.3\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.83,\"probabilities\":{\"B\":0.03,\"C\":0.03,\"D\":0.88,\"A\":0.06}}},\"usage\":{\"input_tokens\":849,\"output_tokens\":138}}" } ], "response": { "model": "jev-1.13.0", "answers": { "Q3.1": { "type": "choice", "choice": "B", "confidence": 0.28, "probabilities": { "B": 0.45, "C": 0.14, "D": 0.34, "A": 0.07 } }, "Q3.2": { "type": "choice", "choice": "A", "confidence": 0.18, "probabilities": { "B": 0.03, "C": 0.22, "D": 0.37, "A": 0.38 } }, "Q3.3": { "type": "choice", "choice": "D", "confidence": 0.83, "probabilities": { "B": 0.03, "C": 0.03, "D": 0.88, "A": 0.06 } } }, "usage": { "input_tokens": 849, "output_tokens": 138 } } }, { "section": 4, "request": { "model": "jev-1.13.0", "state": { "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "problems": [ { "id": "Q4.1", "context": "", "prompt": "K_4 is planar." }, { "id": "Q4.2", "context": "", "prompt": "K_5 has an Eulerian tour." }, { "id": "Q4.3", "context": "", "prompt": "A 12-sided die is a planar graph wrapped around a sphere, with 12 faces and 20 vertices. How many edges does it have?" }, { "id": "Q4.4", "context": "", "prompt": "Which argument proves that, if a finite undirected graph has one odd-degree vertex, it must have another odd-degree vertex?" } ] }, "questions": { "Q4.1": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "True", "B": "False" } }, "Q4.2": { "type": "choice", "instructions": "Answer the problem in `problems[1]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "True", "B": "False" } }, "Q4.3": { "type": "choice", "instructions": "Answer the problem in `problems[2]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "18", "B": "24", "C": "32", "D": "30" } }, "Q4.4": { "type": "choice", "instructions": "Answer the problem in `problems[3]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "The sum of degrees equals twice the number of edges, so it is even. If exactly one degree were odd and all others even, that sum would be odd, a contradiction.", "B": "Removing any odd-degree vertex leaves all remaining degrees even. Restoring it changes every remaining degree by one, so every other vertex then has odd degree.", "C": "An edge incident to the odd-degree vertex has another endpoint. That endpoint has odd degree because the common edge contributes one to both endpoint degrees.", "D": "The sum of degrees equals the number of edges, so it is even. If exactly one degree were odd and all others even, that sum would be odd, a contradiction." } } } }, "attempts": [ { "status": 200, "full_response_ms": 340.7250000000204, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"Q4.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.12,\"probabilities\":{\"A\":0.56,\"B\":0.44}},\"Q4.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.27,\"probabilities\":{\"A\":0.64,\"B\":0.36}},\"Q4.3\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.96,\"probabilities\":{\"D\":0.97,\"C\":0.03,\"A\":0.0,\"B\":0.0}},\"Q4.4\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":1.0,\"probabilities\":{\"D\":0.0,\"C\":0.0,\"A\":1.0,\"B\":0.0}}},\"usage\":{\"input_tokens\":965,\"output_tokens\":155}}" } ], "response": { "model": "jev-1.13.0", "answers": { "Q4.1": { "type": "choice", "choice": "A", "confidence": 0.12, "probabilities": { "A": 0.56, "B": 0.44 } }, "Q4.2": { "type": "choice", "choice": "A", "confidence": 0.27, "probabilities": { "A": 0.64, "B": 0.36 } }, "Q4.3": { "type": "choice", "choice": "D", "confidence": 0.96, "probabilities": { "D": 0.97, "C": 0.03, "A": 0.0, "B": 0.0 } }, "Q4.4": { "type": "choice", "choice": "A", "confidence": 1.0, "probabilities": { "D": 0.0, "C": 0.0, "A": 1.0, "B": 0.0 } } }, "usage": { "input_tokens": 965, "output_tokens": 155 } } }, { "section": 5, "request": { "model": "jev-1.13.0", "state": { "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "problems": [ { "id": "Q5.1", "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "prompt": "The set of all complex numbers." }, { "id": "Q5.2", "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "prompt": "The set of all prime numbers." }, { "id": "Q5.3", "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "prompt": "The real interval [0.001,0.1]." }, { "id": "Q5.4", "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "prompt": "The complex numbers a+bi where a,b are integers." }, { "id": "Q5.5", "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "prompt": "The set of edges of the complete bipartite graph K_(100,100)." }, { "id": "Q5.6", "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "prompt": "The set of all integer powers of 2." }, { "id": "Q5.7", "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "prompt": "The set of real polynomials of degree at most two passing through (1,1) and (2,2)." } ] }, "questions": { "Q5.1": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" } }, "Q5.2": { "type": "choice", "instructions": "Answer the problem in `problems[1]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" } }, "Q5.3": { "type": "choice", "instructions": "Answer the problem in `problems[2]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" } }, "Q5.4": { "type": "choice", "instructions": "Answer the problem in `problems[3]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" } }, "Q5.5": { "type": "choice", "instructions": "Answer the problem in `problems[4]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" } }, "Q5.6": { "type": "choice", "instructions": "Answer the problem in `problems[5]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" } }, "Q5.7": { "type": "choice", "instructions": "Answer the problem in `problems[6]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" } } } }, "attempts": [ { "status": 200, "full_response_ms": 288.56083400023635, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"Q5.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"C\":1.0,\"B\":0.0,\"A\":0.0}},\"Q5.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":1.0,\"A\":0.0}},\"Q5.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":1.0,\"probabilities\":{\"C\":1.0,\"B\":0.0,\"A\":0.0}},\"Q5.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":1.0,\"A\":0.0}},\"Q5.5\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.97,\"probabilities\":{\"C\":0.0,\"B\":0.01,\"A\":0.99}},\"Q5.6\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"B\":1.0,\"A\":0.0}},\"Q5.7\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.46,\"probabilities\":{\"C\":0.64,\"B\":0.34,\"A\":0.02}}},\"usage\":{\"input_tokens\":1357,\"output_tokens\":269}}" } ], "response": { "model": "jev-1.13.0", "answers": { "Q5.1": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "C": 1.0, "B": 0.0, "A": 0.0 } }, "Q5.2": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 1.0, "A": 0.0 } }, "Q5.3": { "type": "choice", "choice": "C", "confidence": 1.0, "probabilities": { "C": 1.0, "B": 0.0, "A": 0.0 } }, "Q5.4": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 1.0, "A": 0.0 } }, "Q5.5": { "type": "choice", "choice": "A", "confidence": 0.97, "probabilities": { "C": 0.0, "B": 0.01, "A": 0.99 } }, "Q5.6": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "C": 0.0, "B": 1.0, "A": 0.0 } }, "Q5.7": { "type": "choice", "choice": "C", "confidence": 0.46, "probabilities": { "C": 0.64, "B": 0.34, "A": 0.02 } } }, "usage": { "input_tokens": 1357, "output_tokens": 269 } } }, { "section": 6, "request": { "model": "jev-1.13.0", "state": { "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "problems": [ { "id": "Q6", "context": "Two programs F and G are equivalent if, for every input, both halt with the same output or both do not halt. Suppose EQUIVALENT(F,G) always halts and returns TRUE iff F and G are equivalent. Complete HALT(P,x) to decide whether P halts on x:\ndef HALT(P,x):\n def F(y):\n ----(1)----\n ----(2)----\n def G(y):\n return 0\n return EQUIVALENT(----(3)----, ----(4)----)", "prompt": "Choose the ordered tuple filling blanks (1),(2),(3),(4)." } ] }, "questions": { "Q6": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "(return 0; P(x); F; G)", "B": "(P(x); return 0; F; G)", "C": "(P(x); return 0; F; F)", "D": "(P(y); return 0; F; G)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 461.1822500010021, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"Q6\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.91,\"probabilities\":{\"B\":0.94,\"D\":0.04,\"A\":0.02,\"C\":0.0}}},\"usage\":{\"input_tokens\":620,\"output_tokens\":46}}" } ], "response": { "model": "jev-1.13.0", "answers": { "Q6": { "type": "choice", "choice": "B", "confidence": 0.91, "probabilities": { "B": 0.94, "D": 0.04, "A": 0.02, "C": 0.0 } } }, "usage": { "input_tokens": 620, "output_tokens": 46 } } }, { "section": 7, "request": { "model": "jev-1.13.0", "state": { "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "problems": [ { "id": "Q7.1", "context": "Answers may be unsimplified (binomial coefficients, factorials, exponents), but do not use summation or product notation. C(n,k) denotes a binomial coefficient.", "prompt": "A deck has 52 cards, 4 suits and 13 ranks per suit. A hand is an unordered set of five cards. How many hands contain exactly one 6 and exactly one 7?" }, { "id": "Q7.2", "context": "Answers may be unsimplified (binomial coefficients, factorials, exponents), but do not use summation or product notation. C(n,k) denotes a binomial coefficient.", "prompt": "Draw 3 cards without replacement. A means at least one ace. C_i means exactly i aces, for i=1,2,3. F,S,T mean respectively that the first, second or third card is an ace. Which representation of P(A) is easier, and why?" }, { "id": "Q7.3", "context": "Answers may be unsimplified (binomial coefficients, factorials, exponents), but do not use summation or product notation. C(n,k) denotes a binomial coefficient.\n\nAgnes distributes all 40 indistinguishable cookies among three distinct minions: Bob, Kevin and Stuart. Each has a carrying limit of 15 cookies.", "prompt": "With no carrying limit, how many distributions of the 40 cookies are possible?" }, { "id": "Q7.4", "context": "Answers may be unsimplified (binomial coefficients, factorials, exponents), but do not use summation or product notation. C(n,k) denotes a binomial coefficient.\n\nAgnes distributes all 40 indistinguishable cookies among three distinct minions: Bob, Kevin and Stuart. Each has a carrying limit of 15 cookies.", "prompt": "How many distributions exceed Bob's limit? Other minions may also exceed their limits." }, { "id": "Q7.5", "context": "Answers may be unsimplified (binomial coefficients, factorials, exponents), but do not use summation or product notation. C(n,k) denotes a binomial coefficient.\n\nAgnes distributes all 40 indistinguishable cookies among three distinct minions: Bob, Kevin and Stuart. Each has a carrying limit of 15 cookies.", "prompt": "How many distributions exceed both Bob's and Kevin's limits?" }, { "id": "Q7.6", "context": "Answers may be unsimplified (binomial coefficients, factorials, exponents), but do not use summation or product notation. C(n,k) denotes a binomial coefficient.\n\nAgnes distributes all 40 indistinguishable cookies among three distinct minions: Bob, Kevin and Stuart. Each has a carrying limit of 15 cookies. Here a counts unrestricted distributions; b counts distributions exceeding one specified limit; c counts distributions exceeding two specified limits.", "prompt": "Let a,b,c denote the answers to Q7.3,Q7.4,Q7.5 respectively. Express the number of distributions respecting all three limits in terms of a,b,c." } ] }, "questions": { "Q7.1": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "C(4,2)*C(44,3)", "B": "4*4*C(50,3)", "C": "4*4*C(44,3)", "D": "4*4*C(44,3)*5!" } }, "Q7.2": { "type": "choice", "instructions": "Answer the problem in `problems[1]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Use C_1,C_2,C_3: they are independent and their intersection is A, so their probabilities can be multiplied.", "B": "Use F,S,T: they are independent and their intersection is A, so their probabilities can be multiplied.", "C": "Use F,S,T: they are mutually exclusive and their union is A, so their probabilities can be added.", "D": "Use C_1,C_2,C_3: they are mutually exclusive and their union is A, so their probabilities can be added." } }, "Q7.3": { "type": "choice", "instructions": "Answer the problem in `problems[2]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "C(40,2)", "B": "3^40", "C": "C(42,2)", "D": "C(42,3)" } }, "Q7.4": { "type": "choice", "instructions": "Answer the problem in `problems[3]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "C(26,2)", "B": "C(27,2)", "C": "C(42,2)-C(26,2)", "D": "C(24,2)" } }, "Q7.5": { "type": "choice", "instructions": "Answer the problem in `problems[4]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "C(12,2)", "B": "C(10,2)", "C": "C(8,2)", "D": "C(26,2)^2" } }, "Q7.6": { "type": "choice", "instructions": "Answer the problem in `problems[5]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "a-b+c", "B": "a-3*b+6*c", "C": "a-3*b+3*c", "D": "a-3*b-3*c" } } } }, "attempts": [ { "status": 200, "full_response_ms": 317.04349999927217, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"Q7.1\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.93,\"probabilities\":{\"D\":0.0,\"B\":0.05,\"C\":0.95,\"A\":0.0}},\"Q7.2\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.99,\"probabilities\":{\"D\":1.0,\"B\":0.0,\"C\":0.0,\"A\":0.0}},\"Q7.3\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.98,\"probabilities\":{\"C\":0.99,\"B\":0.0,\"A\":0.01,\"D\":0.0}},\"Q7.4\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.37,\"probabilities\":{\"D\":0.16,\"A\":0.53,\"B\":0.27,\"C\":0.04}},\"Q7.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.28,\"probabilities\":{\"C\":0.29,\"A\":0.24,\"D\":0.02,\"B\":0.45}},\"Q7.6\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.96,\"probabilities\":{\"C\":0.97,\"B\":0.02,\"A\":0.01,\"D\":0.0}}},\"usage\":{\"input_tokens\":1861,\"output_tokens\":273}}" } ], "response": { "model": "jev-1.13.0", "answers": { "Q7.1": { "type": "choice", "choice": "C", "confidence": 0.93, "probabilities": { "D": 0.0, "B": 0.05, "C": 0.95, "A": 0.0 } }, "Q7.2": { "type": "choice", "choice": "D", "confidence": 0.99, "probabilities": { "D": 1.0, "B": 0.0, "C": 0.0, "A": 0.0 } }, "Q7.3": { "type": "choice", "choice": "C", "confidence": 0.98, "probabilities": { "C": 0.99, "B": 0.0, "A": 0.01, "D": 0.0 } }, "Q7.4": { "type": "choice", "choice": "A", "confidence": 0.37, "probabilities": { "D": 0.16, "A": 0.53, "B": 0.27, "C": 0.04 } }, "Q7.5": { "type": "choice", "choice": "B", "confidence": 0.28, "probabilities": { "C": 0.29, "A": 0.24, "D": 0.02, "B": 0.45 } }, "Q7.6": { "type": "choice", "choice": "C", "confidence": 0.96, "probabilities": { "C": 0.97, "B": 0.02, "A": 0.01, "D": 0.0 } } }, "usage": { "input_tokens": 1861, "output_tokens": 273 } } }, { "section": 8, "request": { "model": "jev-1.13.0", "state": { "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "problems": [ { "id": "Q8", "context": "", "prompt": "In a town, 3/4 of residents have black hair and 2/3 are female, independently. Two people are selected randomly with replacement. Given that at least one is a black-haired female, what is the probability both have black hair? Select the answer with valid supporting work. Hint: the answer is not 9/16." } ] }, "questions": { "Q8": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "1/2: a randomly selected person is a black-haired female with probability (3/4)*(2/3)=1/2; replacement makes the requested conditional probability the same.", "B": "8/9: conditional on both people having black hair, the chance at least one is female is 1-(1/3)^2=8/9; this is the requested conditional probability.", "C": "3/4: if A means at least one black-haired female and B means both have black hair, then P(B|A)=P(B)/P(A)=(9/16)/(1-(1/2)^2)=3/4.", "D": "2/3: if A means at least one black-haired female and B means both have black hair, then P(B|A)=(9/16)*(1-(1/3)^2)/(1-(1/2)^2)=2/3." } } } }, "attempts": [ { "status": 200, "full_response_ms": 340.84749999965425, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"Q8\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.56,\"probabilities\":{\"B\":0.07,\"C\":0.67,\"D\":0.23,\"A\":0.03}}},\"usage\":{\"input_tokens\":700,\"output_tokens\":46}}" } ], "response": { "model": "jev-1.13.0", "answers": { "Q8": { "type": "choice", "choice": "C", "confidence": 0.56, "probabilities": { "B": 0.07, "C": 0.67, "D": 0.23, "A": 0.03 } } }, "usage": { "input_tokens": 700, "output_tokens": 46 } } }, { "section": 9, "request": { "model": "jev-1.13.0", "state": { "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "problems": [ { "id": "Q9.2", "context": "Flip a biased coin independently infinitely many times, with heads probability p and tails probability 1-p. Give the distribution and parameters. The common discrete distributions named on the exam are Bernoulli, Binomial, Geometric, Uniform and Poisson. Original worked example Q9.1 (0 points): the indicator that the first three flips are heads is Bernoulli(p^3). Geometric(q) counts trials up to and including the first success (support 1,2,...).", "prompt": "Distribution of the number of heads in the first 70 flips?" }, { "id": "Q9.3", "context": "Flip a biased coin independently infinitely many times, with heads probability p and tails probability 1-p. Give the distribution and parameters. The common discrete distributions named on the exam are Bernoulli, Binomial, Geometric, Uniform and Poisson. Original worked example Q9.1 (0 points): the indicator that the first three flips are heads is Bernoulli(p^3). Geometric(q) counts trials up to and including the first success (support 1,2,...).", "prompt": "Conditional on exactly one head in the first 20 flips, what is the distribution of the flip number on which the first head occurs?" }, { "id": "Q9.4", "context": "Flip a biased coin independently infinitely many times, with heads probability p and tails probability 1-p. Give the distribution and parameters. The common discrete distributions named on the exam are Bernoulli, Binomial, Geometric, Uniform and Poisson. Original worked example Q9.1 (0 points): the indicator that the first three flips are heads is Bernoulli(p^3). Geometric(q) counts trials up to and including the first success (support 1,2,...).", "prompt": "Distribution of the gap between the first two tails, counting intervening heads? THHHT has gap 3. Hint: a standard random variable minus 1." }, { "id": "Q9.5", "context": "Flip a biased coin independently infinitely many times, with heads probability p and tails probability 1-p. Give the distribution and parameters. The common discrete distributions named on the exam are Bernoulli, Binomial, Geometric, Uniform and Poisson. Original worked example Q9.1 (0 points): the indicator that the first three flips are heads is Bernoulli(p^3). Geometric(q) counts trials up to and including the first success (support 1,2,...).", "prompt": "Let p=1/2. Distribution of the number of flips until the same outcome occurs twice consecutively? Hint: 1 plus a standard random variable." }, { "id": "Q9.6", "context": "Flip a biased coin independently infinitely many times, with heads probability p and tails probability 1-p. Give the distribution and parameters. The common discrete distributions named on the exam are Bernoulli, Binomial, Geometric, Uniform and Poisson. Original worked example Q9.1 (0 points): the indicator that the first three flips are heads is Bernoulli(p^3). Geometric(q) counts trials up to and including the first success (support 1,2,...).", "prompt": "Let p=1/2. Distribution of the number of runs in the first 50 flips? A run is an uninterrupted sequence of identical outcomes, e.g. HHHHHTTTHTTH begins with runs HHHHH, TTT, H, TT. Hint: 1 plus a standard random variable." } ] }, "questions": { "Q9.2": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Poisson(70*p)", "B": "Geometric(p)", "C": "Binomial(70,1-p)", "D": "Binomial(70,p)" } }, "Q9.3": { "type": "choice", "instructions": "Answer the problem in `problems[1]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Geometric(p)", "B": "Uniform({1,2,...,20})", "C": "Binomial(20,p)", "D": "Uniform({0,1,...,19})" } }, "Q9.4": { "type": "choice", "instructions": "Answer the problem in `problems[2]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Geometric(p)-1", "B": "Geometric(1-p)-1", "C": "Binomial(2,p)-1", "D": "Geometric(1-p)" } }, "Q9.5": { "type": "choice", "instructions": "Answer the problem in `problems[3]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "1+Geometric(1/2)", "B": "1+Binomial(2,1/2)", "C": "Geometric(1/2)", "D": "1+Geometric(1/4)" } }, "Q9.6": { "type": "choice", "instructions": "Answer the problem in `problems[4]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "1+Geometric(1/2)", "B": "1+Binomial(50,1/2)", "C": "1+Binomial(49,1/4)", "D": "1+Binomial(49,1/2)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 447.7047920008772, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"Q9.2\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":1.0,\"probabilities\":{\"D\":1.0,\"C\":0.0,\"B\":0.0,\"A\":0.0}},\"Q9.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.96,\"probabilities\":{\"D\":0.02,\"C\":0.0,\"B\":0.98,\"A\":0.0}},\"Q9.4\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.62,\"probabilities\":{\"D\":0.01,\"B\":0.71,\"A\":0.28,\"C\":0.0}},\"Q9.5\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.73,\"probabilities\":{\"D\":0.09,\"C\":0.11,\"B\":0.0,\"A\":0.8}},\"Q9.6\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.89,\"probabilities\":{\"D\":0.92,\"C\":0.01,\"B\":0.0,\"A\":0.07}}},\"usage\":{\"input_tokens\":1760,\"output_tokens\":228}}" } ], "response": { "model": "jev-1.13.0", "answers": { "Q9.2": { "type": "choice", "choice": "D", "confidence": 1.0, "probabilities": { "D": 1.0, "C": 0.0, "B": 0.0, "A": 0.0 } }, "Q9.3": { "type": "choice", "choice": "B", "confidence": 0.96, "probabilities": { "D": 0.02, "C": 0.0, "B": 0.98, "A": 0.0 } }, "Q9.4": { "type": "choice", "choice": "B", "confidence": 0.62, "probabilities": { "D": 0.01, "B": 0.71, "A": 0.28, "C": 0.0 } }, "Q9.5": { "type": "choice", "choice": "A", "confidence": 0.73, "probabilities": { "D": 0.09, "C": 0.11, "B": 0.0, "A": 0.8 } }, "Q9.6": { "type": "choice", "choice": "D", "confidence": 0.89, "probabilities": { "D": 0.92, "C": 0.01, "B": 0.0, "A": 0.07 } } }, "usage": { "input_tokens": 1760, "output_tokens": 228 } } }, { "section": 10, "request": { "model": "jev-1.13.0", "state": { "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "problems": [ { "id": "Q10", "context": "Each spin has four equally likely outcomes: alpha earns 1 point; beta earns 2 points; gamma gives two additional spins; delta loses 1 point and gives one additional spin. Start with one spin. Additional spins use the same rules. The score is the sum of points earned or lost. Example: alpha gives score 1. Example sequence delta,gamma,gamma,alpha,beta,alpha gives score -1+1+2+1=3.", "prompt": "Let X be the final score. What is E[X]?" } ] }, "questions": { "Q10": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "0", "B": "1", "C": "2", "D": "3", "E": "4", "F": "infinity" } } } }, "attempts": [ { "status": 200, "full_response_ms": 410.32979100054945, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"Q10\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.19,\"probabilities\":{\"E\":0.04,\"F\":0.16,\"A\":0.31,\"B\":0.32,\"C\":0.12,\"D\":0.05}}},\"usage\":{\"input_tokens\":587,\"output_tokens\":61}}" } ], "response": { "model": "jev-1.13.0", "answers": { "Q10": { "type": "choice", "choice": "B", "confidence": 0.19, "probabilities": { "E": 0.04, "F": 0.16, "A": 0.31, "B": 0.32, "C": 0.12, "D": 0.05 } } }, "usage": { "input_tokens": 587, "output_tokens": 61 } } }, { "section": 11, "request": { "model": "jev-1.13.0", "state": { "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "problems": [ { "id": "Q11.1", "context": "Normal(mu,v) denotes a normal distribution with mean mu and variance v. Poisson parameters are positive.", "prompt": "Independent X~Normal(1,2) and Y~Normal(2,1). What is the distribution of Z=2X-Y+1?" }, { "id": "Q11.2", "context": "Normal(mu,v) denotes a normal distribution with mean mu and variance v. Poisson parameters are positive.", "prompt": "Independent X,Y~Poisson(lambda). Let Z=X+Y. Select all true statements." }, { "id": "Q11.3", "context": "Normal(mu,v) denotes a normal distribution with mean mu and variance v. Poisson parameters are positive.", "prompt": "Let X_1,...,X_n be independent Poisson(lambda) variables and S=sum_i X_i. Select all true statements." } ] }, "questions": { "Q11.1": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Normal(1,5)", "B": "Normal(1,9)", "C": "Normal(1,3)", "D": "Normal(5,9)" } }, "Q11.2__A": { "type": "noul", "instructions": "Answer the problem in `problems[1]` using its context and the exam instructions. Is the following option true? Z~Poisson(2*lambda)" }, "Q11.2__B": { "type": "noul", "instructions": "Answer the problem in `problems[1]` using its context and the exam instructions. Is the following option true? Z has the distribution of 2P, where P~Poisson(lambda)" }, "Q11.2__C": { "type": "noul", "instructions": "Answer the problem in `problems[1]` using its context and the exam instructions. Is the following option true? None of the above Here \"None of the above\" means every substantive option is false. The substantive options are: {\"A\": \"Z~Poisson(2*lambda)\", \"B\": \"Z has the distribution of 2P, where P~Poisson(lambda)\"}" }, "Q11.3__A": { "type": "noul", "instructions": "Answer the problem in `problems[2]` using its context and the exam instructions. Is the following option true? For every n, S~Poisson(lambda*n)" }, "Q11.3__B": { "type": "noul", "instructions": "Answer the problem in `problems[2]` using its context and the exam instructions. Is the following option true? As n tends to infinity, (S-lambda*n)/sqrt(lambda*n) converges in distribution to Normal(0,1)" }, "Q11.3__C": { "type": "noul", "instructions": "Answer the problem in `problems[2]` using its context and the exam instructions. Is the following option true? None of the above Here \"None of the above\" means every substantive option is false. The substantive options are: {\"A\": \"For every n, S~Poisson(lambda*n)\", \"B\": \"As n tends to infinity, (S-lambda*n)/sqrt(lambda*n) converges in distribution to Normal(0,1)\"}" } } }, "attempts": [ { "status": 200, "full_response_ms": 400.96266599903174, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"Q11.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.23,\"probabilities\":{\"C\":0.0,\"A\":0.41,\"B\":0.42,\"D\":0.17}},\"Q11.2__A\":{\"type\":\"noul\",\"noul\":0.97},\"Q11.2__B\":{\"type\":\"noul\",\"noul\":0.05},\"Q11.2__C\":{\"type\":\"noul\",\"noul\":0.12},\"Q11.3__A\":{\"type\":\"noul\",\"noul\":0.96},\"Q11.3__B\":{\"type\":\"noul\",\"noul\":0.97},\"Q11.3__C\":{\"type\":\"noul\",\"noul\":0.03}},\"usage\":{\"input_tokens\":1072,\"output_tokens\":182}}" } ], "response": { "model": "jev-1.13.0", "answers": { "Q11.1": { "type": "choice", "choice": "B", "confidence": 0.23, "probabilities": { "C": 0.0, "A": 0.41, "B": 0.42, "D": 0.17 } }, "Q11.2__A": { "type": "noul", "noul": 0.97 }, "Q11.2__B": { "type": "noul", "noul": 0.05 }, "Q11.2__C": { "type": "noul", "noul": 0.12 }, "Q11.3__A": { "type": "noul", "noul": 0.96 }, "Q11.3__B": { "type": "noul", "noul": 0.97 }, "Q11.3__C": { "type": "noul", "noul": 0.03 } }, "usage": { "input_tokens": 1072, "output_tokens": 182 } } }, { "section": 12, "request": { "model": "jev-1.13.0", "state": { "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "problems": [ { "id": "Q12.1", "context": "Student arrivals are modeled as a homogeneous Poisson process at a rate of one student per 30 minutes.", "prompt": "Nobody arrives during the first 50 minutes. What is the probability at least one student arrives during the final 10 minutes of the hour?" }, { "id": "Q12.2", "context": "Student arrivals are modeled as a homogeneous Poisson process at a rate of one student per 30 minutes.", "prompt": "At the same arrival rate, students never leave once they arrive. Starting from an empty room, what is the expected number at the end of one hour?" }, { "id": "Q12.3", "context": "Time between BART train departures is exponential, with a mean of 15 minutes.", "prompt": "Ishan arrives at a random time. What is his expected wait until a train departs?" }, { "id": "Q12.4", "context": "Time between BART train departures is exponential, with a mean of 15 minutes.", "prompt": "The station agent says the last train left 5 minutes ago. What is Ishan's expected remaining wait?" }, { "id": "Q12.5", "context": "", "prompt": "Trains and passengers have first arrival times A_t and A_p. The joint PDF is f(a_p,a_t)=lambda_p*lambda_t*exp(-lambda_p*a_p-lambda_t*a_t) for a_p,a_t>=0 and zero otherwise. Give an integral for P(A_t=1), clipped at 1?" }, { "id": "Q13.6", "context": "A 5 by 5 bingo card represents 25 booths. Melody visits each booth independently with probability p; its square is marked exactly when visited. There is no free square. The 12 possible bingo lines are 5 rows, 5 columns and 2 diagonals. X is the number of completely marked lines.", "prompt": "Assume E[X]<2. Which is the standard two-sided Chebyshev upper-bound expression applied to P(X>=2), in terms of E[X] and Var(X), clipped at 1?" } ] }, "questions": { "Q13.1": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "p^25", "B": "12*p^5", "C": "12*p", "D": "25*p" } }, "Q13.2": { "type": "choice", "instructions": "Answer the problem in `problems[1]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "p^9", "B": "p^5", "C": "p^8", "D": "p^10" } }, "Q13.3": { "type": "choice", "instructions": "Answer the problem in `problems[2]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "p^10", "B": "p^9", "C": "p^25", "D": "2*p^5" } }, "Q13.4": { "type": "choice", "instructions": "Answer the problem in `problems[3]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "a+40*b+92*c-a^2", "B": "a+46*b+20*c-a^2", "C": "a+92*b+40*c", "D": "a+92*b+40*c-a^2" } }, "Q13.5": { "type": "choice", "instructions": "Answer the problem in `problems[4]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "1-(1-p^5)^12", "B": "min(1,12*p^5)", "C": "min(1,12*p^10)", "D": "min(1,p^5)" } }, "Q13.6": { "type": "choice", "instructions": "Answer the problem in `problems[5]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "min(1,(2-E[X])^2/Var(X))", "B": "min(1,Var(X)/(2-E[X]))", "C": "min(1,Var(X)/(2-E[X])^2)", "D": "min(1,Var(X)/(2+E[X])^2)" } } } }, "attempts": [ { "status": 200, "full_response_ms": 439.7859159998916, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"Q13.1\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"A\":0.0,\"B\":1.0,\"D\":0.0}},\"Q13.2\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"C\":0.0,\"A\":1.0,\"B\":0.0,\"D\":0.0}},\"Q13.3\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"C\":0.0,\"A\":0.99,\"B\":0.01,\"D\":0.0}},\"Q13.4\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.82,\"probabilities\":{\"C\":0.0,\"A\":0.1,\"B\":0.03,\"D\":0.87}},\"Q13.5\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":1.0,\"probabilities\":{\"C\":0.0,\"A\":0.0,\"B\":1.0,\"D\":0.0}},\"Q13.6\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.97,\"probabilities\":{\"C\":0.98,\"A\":0.01,\"B\":0.01,\"D\":0.0}}},\"usage\":{\"input_tokens\":1778,\"output_tokens\":279}}" } ], "response": { "model": "jev-1.13.0", "answers": { "Q13.1": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "C": 0.0, "A": 0.0, "B": 1.0, "D": 0.0 } }, "Q13.2": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "C": 0.0, "A": 1.0, "B": 0.0, "D": 0.0 } }, "Q13.3": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "C": 0.0, "A": 0.99, "B": 0.01, "D": 0.0 } }, "Q13.4": { "type": "choice", "choice": "D", "confidence": 0.82, "probabilities": { "C": 0.0, "A": 0.1, "B": 0.03, "D": 0.87 } }, "Q13.5": { "type": "choice", "choice": "B", "confidence": 1.0, "probabilities": { "C": 0.0, "A": 0.0, "B": 1.0, "D": 0.0 } }, "Q13.6": { "type": "choice", "choice": "C", "confidence": 0.97, "probabilities": { "C": 0.98, "A": 0.01, "B": 0.01, "D": 0.0 } } }, "usage": { "input_tokens": 1778, "output_tokens": 279 } } }, { "section": 14, "request": { "model": "jev-1.13.0", "state": { "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "problems": [ { "id": "Q14", "context": "", "prompt": "Events A,B have nonzero probability and indicators I_A,I_B. Which argument proves that Cov(I_A,I_B)>0 implies P(A|B)>P(A)?" } ] }, "questions": { "Q14": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "Cov(I_A,I_B)=P(A union B)-P(A)P(B)=P(B)[P(A|B)-P(A)]. Because P(B)>0, positive covariance forces the bracket to be positive.", "B": "Cov(I_A,I_B)=P(A intersection B)-P(A)P(B)=P(A)[P(A|B)-P(A)]. Because P(A)>0, positive covariance forces the bracket to be positive.", "C": "Cov(I_A,I_B)=P(A intersection B)-P(A)P(B)=P(B)[P(A|B)-P(A)]. Because P(B)>0, positive covariance forces the bracket to be positive.", "D": "Cov(I_A,I_B)=P(A)P(B)-P(A intersection B)=P(B)[P(A|B)-P(A)]. Because P(B)>0, positive covariance forces the bracket to be positive." } } } }, "attempts": [ { "status": 200, "full_response_ms": 384.9657089995162, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"Q14\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.97,\"probabilities\":{\"A\":0.02,\"B\":0.0,\"C\":0.98,\"D\":0.0}}},\"usage\":{\"input_tokens\":655,\"output_tokens\":47}}" } ], "response": { "model": "jev-1.13.0", "answers": { "Q14": { "type": "choice", "choice": "C", "confidence": 0.97, "probabilities": { "A": 0.02, "B": 0.0, "C": 0.98, "D": 0.0 } } }, "usage": { "input_tokens": 655, "output_tokens": 47 } } }, { "section": 15, "request": { "model": "jev-1.13.0", "state": { "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "problems": [ { "id": "Q15.1", "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.", "prompt": "Let X~Gaussian(3,6) and Y~Exponential(3) be independent. What is P(X=Y)?" }, { "id": "Q15.2", "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.", "prompt": "For X~Gaussian(3,6), which statement is true about P(X<4)?" }, { "id": "Q15.3", "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.\n\nA, B and C are independent Uniform(0,12) random variables. X indicates that C lies between A and B.", "prompt": "What is P(X=1 | A,B), in terms of A and B?" }, { "id": "Q15.4", "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.\n\nA, B and C are independent Uniform(0,12) random variables. X indicates that C lies between A and B.", "prompt": "Without conditioning on A,B, what is P(X=1)? Hint: there is a way without integrals." }, { "id": "Q15.5", "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.\n\nA, B and C are independent Uniform(0,12) random variables. X indicates that C lies between A and B. Q15.3 asks for P(X=1 | A,B); Q15.4 asks for P(X=1).", "prompt": "What is E[abs(A-B)]? Hint: use the previous two parts to avoid integrals. The original exam labels this particularly challenging." } ] }, "questions": { "Q15.1": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "0", "B": "Cannot be determined from the stated information", "C": "1", "D": "1/2" } }, "Q15.2": { "type": "choice", "instructions": "Answer the problem in `problems[1]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "<0.5", "B": "=0.5", "C": ">0.5", "D": "Not enough information" } }, "Q15.3": { "type": "choice", "instructions": "Answer the problem in `problems[2]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "1-abs(A-B)/12", "B": "abs(A-B)/12", "C": "(A-B)/12", "D": "abs(A-B)/144" } }, "Q15.4": { "type": "choice", "instructions": "Answer the problem in `problems[3]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "1/2", "B": "1/6", "C": "2/3", "D": "1/3" } }, "Q15.5": { "type": "choice", "instructions": "Answer the problem in `problems[4]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "0", "B": "4", "C": "8", "D": "6" } } } }, "attempts": [ { "status": 200, "full_response_ms": 377.990958000737, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"Q15.1\":{\"type\":\"choice\",\"choice\":\"A\",\"confidence\":0.99,\"probabilities\":{\"B\":0.0,\"D\":0.0,\"A\":1.0,\"C\":0.0}},\"Q15.2\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.38,\"probabilities\":{\"B\":0.02,\"D\":0.02,\"C\":0.54,\"A\":0.42}},\"Q15.3\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.96,\"probabilities\":{\"B\":0.97,\"D\":0.0,\"C\":0.0,\"A\":0.03}},\"Q15.4\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.96,\"probabilities\":{\"B\":0.01,\"D\":0.96,\"C\":0.01,\"A\":0.02}},\"Q15.5\":{\"type\":\"choice\",\"choice\":\"C\",\"confidence\":0.35,\"probabilities\":{\"B\":0.45,\"D\":0.03,\"C\":0.52,\"A\":0.0}}},\"usage\":{\"input_tokens\":1288,\"output_tokens\":233}}" } ], "response": { "model": "jev-1.13.0", "answers": { "Q15.1": { "type": "choice", "choice": "A", "confidence": 0.99, "probabilities": { "B": 0.0, "D": 0.0, "A": 1.0, "C": 0.0 } }, "Q15.2": { "type": "choice", "choice": "C", "confidence": 0.38, "probabilities": { "B": 0.02, "D": 0.02, "C": 0.54, "A": 0.42 } }, "Q15.3": { "type": "choice", "choice": "B", "confidence": 0.96, "probabilities": { "B": 0.97, "D": 0.0, "C": 0.0, "A": 0.03 } }, "Q15.4": { "type": "choice", "choice": "D", "confidence": 0.96, "probabilities": { "B": 0.01, "D": 0.96, "C": 0.01, "A": 0.02 } }, "Q15.5": { "type": "choice", "choice": "C", "confidence": 0.35, "probabilities": { "B": 0.45, "D": 0.03, "C": 0.52, "A": 0.0 } } }, "usage": { "input_tokens": 1288, "output_tokens": 233 } } }, { "section": 16, "request": { "model": "jev-1.13.0", "state": { "instructions": "Select the best answer for single-choice items and all correct options for select-all items. Mathematical expressions are exact; exp(x)=e^x. No calculator, external tools, retrieval, worked solutions or answer feedback. Answer each item from its stated context. All questions are independent API judgments; no previous correct answers are supplied.", "problems": [ { "id": "Q16.1", "context": "Tom travels from Chicago (CHI) to Miami (MIA). Each outgoing train route is equally likely. The diagram, transcribed without solving: CHI -> SFO (1/3), DC (1/3), NYC (1/3); SFO -> CHI (1); DC -> SFO (1/3), NYC (1/3), MIA (1/3); NYC -> CHI (1/2), DC (1/2); MIA -> MIA (1).", "prompt": "Find the probability of reaching MIA from CHI before visiting NYC. Select a system of equations, without solving it. h(s) denotes the probability of reaching MIA before NYC starting at s; the requested probability is h(CHI)." }, { "id": "Q16.2", "context": "Tom travels from Chicago (CHI) to Miami (MIA). Each outgoing train route is equally likely. The diagram, transcribed without solving: CHI -> SFO (1/3), DC (1/3), NYC (1/3); SFO -> CHI (1); DC -> SFO (1/3), NYC (1/3), MIA (1/3); NYC -> CHI (1/2), DC (1/2); MIA -> MIA (1).", "prompt": "Tom spends 3 days on each visit to NYC and 1 day in every other city before departing. Travel takes zero days. Select equations for the expected days to first arrival at MIA, starting in CHI and including its initial day. Do not solve. t(s) denotes the remaining expected days starting at s." } ] }, "questions": { "Q16.1": { "type": "choice", "instructions": "Answer the problem in `problems[0]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "h(MIA)=1; h(NYC)=0; h(SFO)=h(CHI); h(DC)=(h(SFO)+h(NYC)+h(MIA))/3; h(CHI)=(h(SFO)+h(DC))/2.", "B": "h(MIA)=1; h(NYC)=0; h(SFO)=h(CHI); h(DC)=(h(NYC)+h(MIA))/2; h(CHI)=(h(SFO)+h(DC)+h(NYC))/3.", "C": "h(MIA)=0; h(NYC)=1; h(SFO)=h(CHI); h(DC)=(h(SFO)+h(NYC)+h(MIA))/3; h(CHI)=(h(SFO)+h(DC)+h(NYC))/3.", "D": "h(MIA)=1; h(NYC)=0; h(SFO)=h(CHI); h(DC)=(h(SFO)+h(NYC)+h(MIA))/3; h(CHI)=(h(SFO)+h(DC)+h(NYC))/3." } }, "Q16.2": { "type": "choice", "instructions": "Answer the problem in `problems[1]` using its context and the exam instructions. Select the best complete answer.", "criteria": { "A": "t(MIA)=0; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(NYC)+t(MIA))/3; t(NYC)=3+(t(CHI)+t(DC))/2.", "B": "t(MIA)=0; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(SFO)+t(NYC)+t(MIA))/3; t(NYC)=3+(t(CHI)+t(DC))/2.", "C": "t(MIA)=0; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(SFO)+t(NYC)+t(MIA))/3; t(NYC)=1+(t(CHI)+t(DC))/2.", "D": "t(MIA)=1; t(SFO)=1+t(CHI); t(CHI)=1+(t(SFO)+t(DC)+t(NYC))/3; t(DC)=1+(t(SFO)+t(NYC)+t(MIA))/3; t(NYC)=3+(t(CHI)+t(DC))/2." } } } }, "attempts": [ { "status": 200, "full_response_ms": 336.481874999663, "raw_response": "{\"model\":\"jev-1.13.0\",\"answers\":{\"Q16.1\":{\"type\":\"choice\",\"choice\":\"D\",\"confidence\":0.98,\"probabilities\":{\"B\":0.0,\"D\":0.99,\"C\":0.0,\"A\":0.01}},\"Q16.2\":{\"type\":\"choice\",\"choice\":\"B\",\"confidence\":0.97,\"probabilities\":{\"B\":0.97,\"D\":0.02,\"A\":0.01,\"C\":0.0}}},\"usage\":{\"input_tokens\":1392,\"output_tokens\":95}}" } ], "response": { "model": "jev-1.13.0", "answers": { "Q16.1": { "type": "choice", "choice": "D", "confidence": 0.98, "probabilities": { "B": 0.0, "D": 0.99, "C": 0.0, "A": 0.01 } }, "Q16.2": { "type": "choice", "choice": "B", "confidence": 0.97, "probabilities": { "B": 0.97, "D": 0.02, "A": 0.01, "C": 0.0 } } }, "usage": { "input_tokens": 1392, "output_tokens": 95 } } } ], "elapsed_seconds": 6.179232874999798 }, "scores": { "correct_items": 52, "total_items": 58, "accuracy": 0.896551724137931, "invalid_or_missing_items": 0, "adapted_weighted_points": 129.0, "adapted_weighted_possible": 154.0, "weighting_note": "Q12 weights are 12/5 each by evaluator convention, not an original published subpart rubric. No partial credit. This is not an original-exam grade.", "source_format_preserved": { "correct": 16, "items": 17 }, "adapted_free_response": { "correct": 36, "items": 41 }, "repaired_items": { "correct": 7, "items": 7 }, "unrepaired_items": { "correct": 45, "items": 51 }, "items": [ { "id": "Q1.1", "correct": true, "valid_response": true, "selected": [ "A" ], "expected": [ "A" ], "adapted_weight": 2, "repairs": [] }, { "id": "Q1.2", "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 2, "repairs": [] }, { "id": "Q2.1", "correct": true, "valid_response": true, "selected": [ "C" ], "expected": [ "C" ], "adapted_weight": 5, "repairs": [] }, { "id": "Q2.2", "correct": true, "valid_response": true, "selected": [ "A" ], "expected": [ "A" ], "adapted_weight": 2, "repairs": [] }, { "id": "Q2.3", "correct": true, "valid_response": true, "selected": [ "A" ], "expected": [ "A" ], "adapted_weight": 2, "repairs": [] }, { "id": "Q2.4", "correct": true, "valid_response": 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"id": "Q7.5", "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 2, "repairs": [] }, { "id": "Q7.6", "correct": true, "valid_response": true, "selected": [ "C" ], "expected": [ "C" ], "adapted_weight": 4, "repairs": [] }, { "id": "Q8", "correct": false, "valid_response": true, "selected": [ "C" ], "expected": [ "D" ], "adapted_weight": 7, "repairs": [] }, { "id": "Q9.2", "correct": true, "valid_response": true, "selected": [ "D" ], "expected": [ "D" ], "adapted_weight": 2, "repairs": [] }, { "id": "Q9.3", "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 2, "repairs": [] }, { "id": "Q9.4", "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 3, "repairs": [] }, { "id": "Q9.5", "correct": true, "valid_response": true, "selected": [ "A" ], "expected": [ "A" ], "adapted_weight": 3, "repairs": [] }, { "id": "Q9.6", "correct": true, "valid_response": true, "selected": [ "D" ], "expected": [ "D" ], "adapted_weight": 3, "repairs": [] }, { "id": "Q10", "correct": false, "valid_response": true, "selected": [ "B" ], "expected": [ "C" ], "adapted_weight": 3, "repairs": [] }, { "id": "Q11.1", "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 3, "repairs": [] }, { "id": "Q11.2", "correct": true, "valid_response": true, "selected": [ "A" ], "expected": [ "A" ], "adapted_weight": 2, "repairs": [] }, { "id": "Q11.3", "correct": true, "valid_response": true, "selected": [ "A", "B" ], "expected": [ "A", "B" ], "adapted_weight": 2, "repairs": [ "positive_poisson_rate" ] }, { "id": "Q12.1", "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 2.4, "repairs": [ "poisson_process" ] }, { "id": "Q12.2", "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 2.4, "repairs": [ "poisson_process" ] }, { "id": "Q12.3", "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 2.4, "repairs": [] }, { "id": "Q12.4", "correct": true, "valid_response": true, "selected": [ "A" ], "expected": [ "A" ], "adapted_weight": 2.4, "repairs": [] }, { "id": "Q12.5", "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 2.4, "repairs": [] }, { "id": "Q13.1", "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 3, "repairs": [] }, { "id": "Q13.2", "correct": true, "valid_response": true, "selected": [ "A" ], "expected": [ "A" ], "adapted_weight": 2, "repairs": [] }, { "id": "Q13.3", "correct": true, "valid_response": true, "selected": [ "A" ], "expected": [ "A" ], "adapted_weight": 2, "repairs": [] }, { "id": "Q13.4", "correct": true, "valid_response": true, "selected": [ "D" ], "expected": [ "D" ], "adapted_weight": 5, "repairs": [] }, { "id": "Q13.5", "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 2, "repairs": [] }, { "id": "Q13.6", "correct": true, "valid_response": true, "selected": [ "C" ], "expected": [ "C" ], "adapted_weight": 3, "repairs": [ "chebyshev_condition" ] }, { "id": "Q14", "correct": true, "valid_response": true, "selected": [ "C" ], "expected": [ "C" ], "adapted_weight": 5, "repairs": [] }, { "id": "Q15.1", "correct": true, "valid_response": true, "selected": [ "A" ], "expected": [ "A" ], "adapted_weight": 2, "repairs": [ "independence_added" ] }, { "id": "Q15.2", "correct": true, "valid_response": true, "selected": [ "C" ], "expected": [ "C" ], "adapted_weight": 2, "repairs": [] }, { "id": "Q15.3", "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 2, "repairs": [] }, { "id": "Q15.4", "correct": true, "valid_response": true, "selected": [ "D" ], "expected": [ "D" ], "adapted_weight": 2, "repairs": [] }, { "id": "Q15.5", "correct": false, "valid_response": true, "selected": [ "C" ], "expected": [ "B" ], "adapted_weight": 5, "repairs": [] }, { "id": "Q16.1", "correct": true, "valid_response": true, "selected": [ "D" ], "expected": [ "D" ], "adapted_weight": 3, "repairs": [] }, { "id": "Q16.2", "correct": true, "valid_response": true, "selected": [ "B" ], "expected": [ "B" ], "adapted_weight": 3, "repairs": [ "dc_edge_in_key" ] } ] } } }