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Hint: 186 = 31 * 6.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0166, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7310585786300049 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q2.3", "topic": "Modular arithmetic", "kind": null, "prompt": "5 divides 4^n - 1 for all even natural numbers n.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.012, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9525741268224334 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q2.4", "topic": "Modular arithmetic", "kind": null, "prompt": "What is the last digit (ones place) of 3^47?", "options": { "A": "2", "B": "9", "C": "1", "D": "3", "E": "5", "F": "6", "G": "4", "H": "8", "I": "0", "J": "7" }, "expected": [ "J" ], "jev_selected": [ "J" ], "jev_correct": true, "laya_selected": [ "E" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0999, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.19452830926747472 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q2.5", "topic": "Modular arithmetic", "kind": null, "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?", "options": { "A": "15", "B": "27", "C": "18", "D": "6" }, "expected": [ "C" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0226, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3177952409322493 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q2.6", "topic": "Modular arithmetic", "kind": null, "prompt": "Find (6^7 + 7^6) mod 91. Hint: use CRT; 91=7*13.", "options": { "A": "13", "B": "6", "C": "0", "D": "78" }, "expected": [ "B" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0099, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.3568632977686014 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q3.1", "topic": "Polynomials", "kind": null, "prompt": "Working modulo 6, how many distinct formal polynomials of degree exactly two are there?", "options": { "A": "125", "B": "180", "C": "216", "D": "150" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0352, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.40148861260281904 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q3.2", "topic": "Polynomials", "kind": null, "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.", "options": { "A": "g(x)=x^6", "B": "g(x)=0", "C": "g(x)=1", "D": "g(x)=x" }, "expected": [ "D" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0635, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.39344467898354535 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q3.3", "topic": "Polynomials", "kind": null, "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?", "options": { "A": "k", "B": "n+2k", "C": "n", "D": "1" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.059, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.45961514132177633 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q4.1", "topic": "Graphs", "kind": null, "prompt": "K_4 is planar.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.272, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8670357598021706 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q4.2", "topic": "Graphs", "kind": null, "prompt": "K_5 has an Eulerian tour.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2446, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8354835371034369 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q4.3", "topic": "Graphs", "kind": null, "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?", "options": { "A": "18", "B": "24", "C": "32", "D": "30" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0082, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.683070015215755 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q4.4", "topic": "Graphs", "kind": null, "prompt": "Which argument proves that, if a finite undirected graph has one odd-degree vertex, it must have another odd-degree vertex?", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.009, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9988212946444768 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q5.1", "topic": "Countability", "kind": null, "prompt": "The set of all complex numbers.", "options": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0007, "default_truncated": false, "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8986080102671171 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q5.2", "topic": "Countability", "kind": null, "prompt": "The set of all prime numbers.", "options": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0006, "default_truncated": false, "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6074815620620584 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q5.3", "topic": "Countability", "kind": null, "prompt": "The real interval [0.001,0.1].", "options": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0007, "default_truncated": false, "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8391011589485126 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q5.4", "topic": "Countability", "kind": null, "prompt": "The complex numbers a+bi where a,b are integers.", "options": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0015, "default_truncated": false, "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6849372534752349 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q5.5", "topic": "Countability", "kind": null, "prompt": "The set of edges of the complete bipartite graph K_(100,100).", "options": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0012, "default_truncated": false, "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7174510311561372 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q5.6", "topic": "Countability", "kind": null, "prompt": "The set of all integer powers of 2.", "options": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0019, "default_truncated": false, "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7042046051774811 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q5.7", "topic": "Countability", "kind": null, "prompt": "The set of real polynomials of degree at most two passing through (1,1) and (2,2).", "options": { "A": "Finite", "B": "Countably Infinite", "C": "Uncountably Infinite" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0011, "default_truncated": false, "context": "State whether each set is Finite, Countably Infinite, or Uncountably Infinite.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6220058829452726 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q6", "topic": "Computability", "kind": null, "prompt": "Choose the ordered tuple filling blanks (1),(2),(3),(4).", "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)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0116, "default_truncated": false, "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)----)", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4114753005233112 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q7.1", "topic": "Counting", "kind": null, "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?", "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!" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0293, "default_truncated": true, "context": "Answers may be unsimplified (binomial coefficients, factorials, exponents), but do not use summation or product notation. C(n,k) denotes a binomial coefficient.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.773681918275408 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q7.2", "topic": "Counting", "kind": null, "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?", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0661, "default_truncated": true, "context": "Answers may be unsimplified (binomial coefficients, factorials, exponents), but do not use summation or product notation. C(n,k) denotes a binomial coefficient.", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5670628324083504 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q7.3", "topic": "Counting", "kind": null, "prompt": "With no carrying limit, how many distributions of the 40 cookies are possible?", "options": { "A": "C(40,2)", "B": "3^40", "C": "C(42,2)", "D": "C(42,3)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0259, "default_truncated": true, "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.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6616658010904157 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q7.4", "topic": "Counting", "kind": null, "prompt": "How many distributions exceed Bob's limit? Other minions may also exceed their limits.", "options": { "A": "C(26,2)", "B": "C(27,2)", "C": "C(42,2)-C(26,2)", "D": "C(24,2)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0099, "default_truncated": true, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.2926661932032274 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q7.5", "topic": "Counting", "kind": null, "prompt": "How many distributions exceed both Bob's and Kevin's limits?", "options": { "A": "C(12,2)", "B": "C(10,2)", "C": "C(8,2)", "D": "C(26,2)^2" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0512, "default_truncated": true, "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.", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.39181905048556065 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q7.6", "topic": "Counting", "kind": null, "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.", "options": { "A": "a-b+c", "B": "a-3*b+6*c", "C": "a-3*b+3*c", "D": "a-3*b-3*c" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0241, "default_truncated": true, "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.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5825616614053231 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q8", "topic": "Conditional probability", "kind": null, "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.", "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." }, "expected": [ "D" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.011, "default_truncated": true, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4392175765597299 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q9.2", "topic": "Discrete distributions", "kind": null, "prompt": "Distribution of the number of heads in the first 70 flips?", "options": { "A": "Poisson(70*p)", "B": "Geometric(p)", "C": "Binomial(70,1-p)", "D": "Binomial(70,p)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0134, "default_truncated": true, "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,...).", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9844596867933282 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q9.3", "topic": "Discrete distributions", "kind": null, "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?", "options": { "A": "Geometric(p)", "B": "Uniform({1,2,...,20})", "C": "Binomial(20,p)", "D": "Uniform({0,1,...,19})" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.009, "default_truncated": true, "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,...).", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4118436823710276 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q9.4", "topic": "Discrete distributions", "kind": null, "prompt": "Distribution of the gap between the first two tails, counting intervening heads? THHHT has gap 3. Hint: a standard random variable minus 1.", "options": { "A": "Geometric(p)-1", "B": "Geometric(1-p)-1", "C": "Binomial(2,p)-1", "D": "Geometric(1-p)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0165, "default_truncated": true, "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,...).", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.695408563375147 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q9.5", "topic": "Discrete distributions", "kind": null, "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.", "options": { "A": "1+Geometric(1/2)", "B": "1+Binomial(2,1/2)", "C": "Geometric(1/2)", "D": "1+Geometric(1/4)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0151, "default_truncated": true, "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,...).", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5320794834992436 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q9.6", "topic": "Discrete distributions", "kind": null, "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.", "options": { "A": "1+Geometric(1/2)", "B": "1+Binomial(50,1/2)", "C": "1+Binomial(49,1/4)", "D": "1+Binomial(49,1/2)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0022, "default_truncated": true, "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,...).", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8303531443536248 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q10", "topic": "Recursive expectation", "kind": null, "prompt": "Let X be the final score. What is E[X]?", "options": { "A": "0", "B": "1", "C": "2", "D": "3", "E": "4", "F": "infinity" }, "expected": [ "C" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "E" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.3045, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.515845292316364 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q11.1", "topic": "Sums of random variables", "kind": null, "prompt": "Independent X~Normal(1,2) and Y~Normal(2,1). What is the distribution of Z=2X-Y+1?", "options": { "A": "Normal(1,5)", "B": "Normal(1,9)", "C": "Normal(1,3)", "D": "Normal(5,9)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.091, "default_truncated": false, "context": "Normal(mu,v) denotes a normal distribution with mean mu and variance v. Poisson parameters are positive.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5190768592430574 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q11.2", "topic": "Sums of random variables", "kind": null, "prompt": "Independent X,Y~Poisson(lambda). Let Z=X+Y. Select all true statements.", "options": { "A": "Z~Poisson(2*lambda)", "B": "Z has the distribution of 2P, where P~Poisson(lambda)", "C": "None of the above" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B", "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": null, "default_truncated": false, "context": "Normal(mu,v) denotes a normal distribution with mean mu and variance v. Poisson parameters are positive.", "semif_selected": [ "A", "B", "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": null }, { "exam": "final_fa25", "sample": "discovery", "id": "Q11.3", "topic": "Sums of random variables", "kind": null, "prompt": "Let X_1,...,X_n be independent Poisson(lambda) variables and S=sum_i X_i. Select all true statements.", "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" }, "expected": [ "A", "B" ], "jev_selected": [ "A", "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": null, "default_truncated": false, "context": "Normal(mu,v) denotes a normal distribution with mean mu and variance v. Poisson parameters are positive.", "semif_selected": [ "A", "B", "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": null }, { "exam": "final_fa25", "sample": "discovery", "id": "Q12.1", "topic": "Waiting times", "kind": null, "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?", "options": { "A": "1-exp(-5/3)", "B": "1-exp(-1/3)", "C": "exp(-1/3)", "D": "1-exp(-2)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0224, "default_truncated": false, "context": "Student arrivals are modeled as a homogeneous Poisson process at a rate of one student per 30 minutes.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8823538376461362 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q12.2", "topic": "Waiting times", "kind": null, "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?", "options": { "A": "30", "B": "2", "C": "1/2", "D": "1-exp(-2)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0166, "default_truncated": false, "context": "Student arrivals are modeled as a homogeneous Poisson process at a rate of one student per 30 minutes.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5590617435155356 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q12.3", "topic": "Waiting times", "kind": null, "prompt": "Ishan arrives at a random time. What is his expected wait until a train departs?", "options": { "A": "1/15 minute", "B": "15 minutes", "C": "7.5 minutes", "D": "30 minutes" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0071, "default_truncated": false, "context": "Time between BART train departures is exponential, with a mean of 15 minutes.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4827288274904063 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q12.4", "topic": "Waiting times", "kind": null, "prompt": "The station agent says the last train left 5 minutes ago. What is Ishan's expected remaining wait?", "options": { "A": "15 minutes", "B": "20 minutes", "C": "10 minutes", "D": "5 minutes" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0018, "default_truncated": false, "context": "Time between BART train departures is exponential, with a mean of 15 minutes.", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3704758839685164 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q12.5", "topic": "Waiting times", "kind": null, "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?", "options": { "A": "1-(1-p^5)^12", "B": "min(1,12*p^5)", "C": "min(1,12*p^10)", "D": "min(1,p^5)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0117, "default_truncated": true, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9926763544269318 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q13.6", "topic": "Bingo & variance", "kind": null, "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?", "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)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.051, "default_truncated": true, "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.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6980665689331522 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q14", "topic": "Covariance", "kind": null, "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)?", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0113, "default_truncated": true, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4478421610882494 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q15.1", "topic": "Continuous variables", "kind": null, "prompt": "Let X~Gaussian(3,6) and Y~Exponential(3) be independent. What is P(X=Y)?", "options": { "A": "0", "B": "Cannot be determined from the stated information", "C": "1", "D": "1/2" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0454, "default_truncated": true, "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9920422803928451 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q15.2", "topic": "Continuous variables", "kind": null, "prompt": "For X~Gaussian(3,6), which statement is true about P(X<4)?", "options": { "A": "<0.5", "B": "=0.5", "C": ">0.5", "D": "Not enough information" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0051, "default_truncated": true, "context": "Gaussian(mu,v) denotes a Gaussian distribution with mean mu and variance v.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5996737521525733 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q15.3", "topic": "Continuous variables", "kind": null, "prompt": "What is P(X=1 | A,B), in terms of A and B?", "options": { "A": "1-abs(A-B)/12", "B": "abs(A-B)/12", "C": "(A-B)/12", "D": "abs(A-B)/144" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0061, "default_truncated": true, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.767421987773428 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q15.4", "topic": "Continuous variables", "kind": null, "prompt": "Without conditioning on A,B, what is P(X=1)? Hint: there is a way without integrals.", "options": { "A": "1/2", "B": "1/6", "C": "2/3", "D": "1/3" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0192, "default_truncated": true, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6286402096508223 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q15.5", "topic": "Continuous variables", "kind": null, "prompt": "What is E[abs(A-B)]? Hint: use the previous two parts to avoid integrals. The original exam labels this particularly challenging.", "options": { "A": "0", "B": "4", "C": "8", "D": "6" }, "expected": [ "B" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0098, "default_truncated": true, "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).", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.37354501646787847 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q16.1", "topic": "Markov chains", "kind": null, "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).", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0275, "default_truncated": true, "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).", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7375911150590562 }, { "exam": "final_fa25", "sample": "discovery", "id": "Q16.2", "topic": "Markov chains", "kind": null, "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.", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0067, "default_truncated": true, "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).", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.789015488042761 }, { "exam": "final_fa23", "sample": "follow_up", "id": "2.1a", "topic": "logic", "kind": "recognition", "prompt": "P AND NOT P is equivalent to True.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0907, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5312093733737563 }, { "exam": "final_fa23", "sample": "follow_up", "id": "2.1b", "topic": "logic", "kind": "recognition", "prompt": "Simplify (P implies R) AND (P implies NOT R).", "options": { "A": "P", "B": "NOT R", "C": "NOT P", "D": "True" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1398, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.744604975769651 }, { "exam": "final_fa23", "sample": "follow_up", "id": "2.1c", "topic": "logic", "kind": "recognition", "prompt": "(P AND Q implies NOT R) is equivalent to NOT R OR NOT P OR NOT Q.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.439, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9626731126558706 }, { "exam": "final_fa23", "sample": "follow_up", "id": "2.2a", "topic": "logic", "kind": "recognition", "prompt": "NOT forall x exists y Q(x,y) is equivalent to exists x forall y NOT Q(x,y).", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0014, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9840936082881853 }, { "exam": "final_fa23", "sample": "follow_up", "id": "2.2b", "topic": "logic", "kind": "recognition", "prompt": "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 __.", "options": { "A": "True", "B": "Undetermined", "C": "False" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0996, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4630367419657945 }, { "exam": "final_fa23", "sample": "follow_up", "id": "3.1", "topic": "proof", "kind": "multi_step", "prompt": "For natural a,b,c, prove a|b and b|c imply a|c.", "options": { "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1513, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9787134404938486 }, { "exam": "final_fa23", "sample": "follow_up", "id": "3.2", "topic": "proof", "kind": "multi_step", "prompt": "Prove an integer whose decimal digit sum is divisible by nine is divisible by nine.", "options": { "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0452, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9783862358673109 }, { "exam": "final_fa23", "sample": "follow_up", "id": "4.1", "topic": "matching", "kind": "recognition", "prompt": "A job receiving its first-choice candidate in some stable matching must receive that candidate in every stable matching.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0601, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5 }, { "exam": "final_fa23", "sample": "follow_up", "id": "4.2", "topic": "matching", "kind": "recognition", "prompt": "A job ranked last by every candidate must be rejected n−1 times in job-proposing deferred acceptance.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0141, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5 }, { "exam": "final_fa23", "sample": "follow_up", "id": "4.3", "topic": "matching", "kind": "recognition", "prompt": "If some job proposes to its final-ranked candidate during synchronous job-proposing deferred acceptance, the algorithm terminates that day.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.003, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8807970779778823 }, { "exam": "final_fa23", "sample": "follow_up", "id": "5.1", "topic": "graphs", "kind": "recognition", "prompt": "Adding a new edge either creates a cycle containing that edge or reduces the component count.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0554, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9579122720843811 }, { "exam": "final_fa23", "sample": "follow_up", "id": "5.2a", "topic": "graphs", "kind": "recognition", "prompt": "Edges in a tree on n>0 vertices?", "options": { "A": "2n−2", "B": "n+1", "C": "n", "D": "n−1" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0056, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9899627451088997 }, { "exam": "final_fa23", "sample": "follow_up", "id": "5.2b", "topic": "graphs", "kind": "recognition", "prompt": "Delete k edges from a tree, keeping all vertices. Component count?", "options": { "A": "k", "B": "k−1", "C": "2k", "D": "k+1" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1718, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8774366511421361 }, { "exam": "final_fa23", "sample": "follow_up", "id": "5.2c", "topic": "graphs", "kind": "multi_step", "prompt": "A tree has exactly two degree-five vertices. Sharp minimum number of leaves?", "options": { "A": "10", "B": "6", "C": "5", "D": "8" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0039, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.40101791712441026 }, { "exam": "final_fa23", "sample": "follow_up", "id": "5.3a", "topic": "graphs", "kind": "recognition", "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.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0798, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6513548646660542 }, { "exam": "final_fa23", "sample": "follow_up", "id": "5.3b", "topic": "graphs", "kind": "multi_step", "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?", "options": { "A": "c", "B": "c+1", "C": "max(3,c)", "D": "2c" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.097, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.34478590299221606 }, { "exam": "final_fa23", "sample": "follow_up", "id": "5.4", "topic": "graphs", "kind": "recognition", "prompt": "The four-dimensional hypercube has an Eulerian tour.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1503, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7772998611746911 }, { "exam": "final_fa23", "sample": "follow_up", "id": "5.5", "topic": "graphs", "kind": "recognition", "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?", "options": { "A": "k", "B": "2k", "C": "k+1", "D": "k²" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0957, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.3806017701107147 }, { "exam": "final_fa23", "sample": "follow_up", "id": "5.6a", "topic": "graphs", "kind": "calculation", "prompt": "Connected planar embedding: n vertices,2n edges. Faces?", "options": { "A": "n", "B": "2n", "C": "n−2", "D": "n+2" }, "expected": [ "D" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0563, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4006796757977684 }, { "exam": "final_fa23", "sample": "follow_up", "id": "5.6b", "topic": "graphs", "kind": "calculation", "prompt": "Connected planar embedding: n vertices, average face boundary length five. Edges?", "options": { "A": "2n−4", "B": "3n−6", "C": "5(n−2)/3", "D": "5n/2" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0062, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5469986186804172 }, { "exam": "final_fa23", "sample": "follow_up", "id": "6.1", "topic": "number_theory", "kind": "multi_step", "prompt": "Choose integers (a,b) solving 47a+49b=1.", "options": { "A": "(−24,23)", "B": "(24,−23)", "C": "(23,−22)", "D": "(49,−47)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.038, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6879905371299883 }, { "exam": "final_fa23", "sample": "follow_up", "id": "6.2", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "2^30 modulo 35?", "options": { "A": "8", "B": "29", "C": "1", "D": "16" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0415, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4980398096880152 }, { "exam": "final_fa23", "sample": "follow_up", "id": "6.3a", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "Distinct primes p,q. Simplify q^p mod pq.", "options": { "A": "p", "B": "1", "C": "q", "D": "0" }, "expected": [ "C" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0435, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4124578775795746 }, { "exam": "final_fa23", "sample": "follow_up", "id": "6.3b", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "There are pq−__ invertible residues modulo pq for distinct primes p,q. Fill the blank.", "options": { "A": "p+q−1", "B": "p+q", "C": "p+q+1", "D": "pq−1" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2013, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.47905138671315767 }, { "exam": "final_fa23", "sample": "follow_up", "id": "7.1", "topic": "proof", "kind": "multi_step", "prompt": "Prime p: why is no a in 2..p−2 its own inverse?", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0724, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9984928348665174 }, { "exam": "final_fa23", "sample": "follow_up", "id": "7.2", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "Product of all residues 1,...,p−1 modulo prime p?", "options": { "A": "0", "B": "p−2", "C": "p−1", "D": "1" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1319, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5156969743011823 }, { "exam": "final_fa23", "sample": "follow_up", "id": "7.3", "topic": "proof", "kind": "multi_step", "prompt": "Distinct odd primes p,q: construct a nontrivial self-inverse unit modulo pq.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0464, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9798748439801382 }, { "exam": "final_fa23", "sample": "follow_up", "id": "8.1", "topic": "polynomials", "kind": "recognition", "prompt": "Give a degree-two polynomial over GF(5) with roots 0 and1.", "options": { "A": "x(x−1)", "B": "(x−1)²", "C": "x²+1", "D": "x(x+1)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1882, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9719354308519272 }, { "exam": "final_fa23", "sample": "follow_up", "id": "8.2", "topic": "polynomials", "kind": "recognition", "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?", "options": { "A": "min(dP,dQ)", "B": "dP+dQ", "C": "max(dP,dQ)", "D": "p" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.01, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.33482349867891054 }, { "exam": "final_fa23", "sample": "follow_up", "id": "8.3a", "topic": "secret_sharing", "kind": "recognition", "prompt": "Any two children can reconstruct b.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.3528, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8670357598021706 }, { "exam": "final_fa23", "sample": "follow_up", "id": "8.3b", "topic": "secret_sharing", "kind": "recognition", "prompt": "Recover b given m and P(1).", "options": { "A": "P(1)+m modp", "B": "P(1)−m modp", "C": "P(1)/m modp", "D": "m−P(1) modp" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0812, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8579118060948767 }, { "exam": "final_fa23", "sample": "follow_up", "id": "9.1", "topic": "countability", "kind": "recognition", "prompt": "The set of all subsets of the primes is countable.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0202, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9241418199787566 }, { "exam": "final_fa23", "sample": "follow_up", "id": "9.2", "topic": "countability", "kind": "recognition", "prompt": "For sets A1,...,An, let D={i in 1..n: i is not in Ai}. Then D differs from every Ai.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1958, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8175744761936437 }, { "exam": "final_fa23", "sample": "follow_up", "id": "9.3", "topic": "countability", "kind": "recognition", "prompt": "Programs that halt on input1 form a countable set.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0185, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5621765008857981 }, { "exam": "final_fa23", "sample": "follow_up", "id": "9.4", "topic": "computability", "kind": "recognition", "prompt": "A total algorithm can decide for any P,x and natural n whether P(x) runs for more than n steps.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0497, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6513548646660542 }, { "exam": "final_fa23", "sample": "follow_up", "id": "10.1", "topic": "counting", "kind": "calculation", "prompt": "Distinct arrangements of CAT?", "options": { "A": "6", "B": "1", "C": "9", "D": "3" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0227, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.411811260039643 }, { "exam": "final_fa23", "sample": "follow_up", "id": "10.2", "topic": "counting", "kind": "calculation", "prompt": "Histograms with k named bins and n observations: nonnegative counts sum to n. Count?", "options": { "A": "k^n", "B": "C(n+k−1,k−1)", "C": "C(n−1,k−1)", "D": "C(n,k)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0274, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9433640615525498 }, { "exam": "final_fa23", "sample": "follow_up", "id": "10.3a", "topic": "counting", "kind": "calculation", "prompt": "A staircase has k steps of heights one or two and total height n,k<=n<=2k. Number of height-two steps?", "options": { "A": "n−k", "B": "2k−n", "C": "k", "D": "n−2k" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0037, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9094308807765233 }, { "exam": "final_fa23", "sample": "follow_up", "id": "10.3b", "topic": "counting", "kind": "calculation", "prompt": "Number of ordered staircases of k steps, each height one or two, total n?", "options": { "A": "C(k,n−k)", "B": "C(n−k,k)", "C": "C(n,k)", "D": "2^k" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0776, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.34040633907578005 }, { "exam": "final_fa23", "sample": "follow_up", "id": "11.1a", "topic": "probability", "kind": "calculation", "prompt": "P(E given A)?", "options": { "A": "1/2", "B": "5/14", "C": "3/14", "D": "1/4" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0604, "default_truncated": false, "context": "Uniformly choose bag A:4 red,4 blue or bag B:3 red,5 blue. Draw two marbles without replacement. E:both blue.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6409876613450489 }, { "exam": "final_fa23", "sample": "follow_up", "id": "11.1b", "topic": "bayes", "kind": "multi_step", "prompt": "P(A given E)?", "options": { "A": "3/8", "B": "5/8", "C": "1/2", "D": "3/14" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.002, "default_truncated": false, "context": "Uniformly choose bag A:4 red,4 blue or bag B:3 red,5 blue. Draw two marbles without replacement. E:both blue.", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.31911523504877376 }, { "exam": "final_fa23", "sample": "follow_up", "id": "11.2a", "topic": "probability", "kind": "multi_step", "prompt": "P(X=1)?", "options": { "A": "1/4", "B": "1/3", "C": "5/18", "D": "1/6" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.012, "default_truncated": false, "context": "Roll two independent fair six-sided dice. X=sum modulo3,Y=sum modulo4.", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3469506810908233 }, { "exam": "final_fa23", "sample": "follow_up", "id": "11.2b", "topic": "probability", "kind": "multi_step", "prompt": "P(Y=3)?", "options": { "A": "1/4", "B": "1/6", "C": "5/18", "D": "1/3" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0466, "default_truncated": false, "context": "Roll two independent fair six-sided dice. X=sum modulo3,Y=sum modulo4.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.32205625344145955 }, { "exam": "final_fa23", "sample": "follow_up", "id": "11.2c", "topic": "probability", "kind": "multi_step", "prompt": "P(X=1 and Y=3)?", "options": { "A": "1/3", "B": "1/6", "C": "5/54", "D": "1/12" }, "expected": [ "B" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0727, "default_truncated": false, "context": "Roll two independent fair six-sided dice. X=sum modulo3,Y=sum modulo4.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4181874540337694 }, { "exam": "final_fa23", "sample": "follow_up", "id": "11.2d", "topic": "probability", "kind": "multi_step", "prompt": "P(X=1 or Y=3)?", "options": { "A": "11/18", "B": "4/9", "C": "5/54", "D": "1/2" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0302, "default_truncated": false, "context": "Roll two independent fair six-sided dice. X=sum modulo3,Y=sum modulo4.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4041551194358978 }, { "exam": "final_fa23", "sample": "follow_up", "id": "11.3a", "topic": "probability", "kind": "calculation", "prompt": "P(A)?", "options": { "A": "2/3", "B": "1/2", "C": "1", "D": "1/3" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0045, "default_truncated": false, "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.", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4220412305175926 }, { "exam": "final_fa23", "sample": "follow_up", "id": "11.3b", "topic": "bayes", "kind": "multi_step", "prompt": "P(B given A)?", "options": { "A": "1/3", "B": "2/3", "C": "1", "D": "1/2" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0027, "default_truncated": false, "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.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.504540483486473 }, { "exam": "final_fa23", "sample": "follow_up", "id": "12.1a", "topic": "probability", "kind": "calculation", "prompt": "P(X=0)?", "options": { "A": "exp(−n)", "B": "(1−1/n)^n", "C": "(1/n)^n", "D": "1−1/n" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0356, "default_truncated": false, "context": "n distinct balls independently choose uniform bins among n named bins. X counts balls in bin1.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9911510799992242 }, { "exam": "final_fa23", "sample": "follow_up", "id": "12.1b", "topic": "distributions", "kind": "recognition", "prompt": "Distribution of X?", "options": { "A": "Poisson(n)", "B": "Uniform{0,...,n}", "C": "Geom(1/n)", "D": "Bin(n,1/n)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0185, "default_truncated": false, "context": "n distinct balls independently choose uniform bins among n named bins. X counts balls in bin1.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9703103977181254 }, { "exam": "final_fa23", "sample": "follow_up", "id": "12.1c", "topic": "distributions", "kind": "recognition", "prompt": "Limit of P(X=10) as n grows?", "options": { "A": "1/10", "B": "0", "C": "exp(−1)/10!", "D": "exp(−10)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.17, "default_truncated": false, "context": "n distinct balls independently choose uniform bins among n named bins. X counts balls in bin1.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4305119116660897 }, { "exam": "final_fa23", "sample": "follow_up", "id": "12.2a", "topic": "probability", "kind": "calculation", "prompt": "P(X=0)?", "options": { "A": "1/2", "B": "C(100,0)/2^100", "C": "1/101", "D": "C(100,50)/2^100" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.076, "default_truncated": false, "context": "100 independent fair flips. X=number of heads minus number of tails; Y=X².", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.832392225513925 }, { "exam": "final_fa23", "sample": "follow_up", "id": "12.2b", "topic": "expectation", "kind": "recognition", "prompt": "E[X]?", "options": { "A": "50", "B": "0", "C": "1/2", "D": "100" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0409, "default_truncated": false, "context": "100 independent fair flips. X=number of heads minus number of tails; Y=X².", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4642658375914037 }, { "exam": "final_fa23", "sample": "follow_up", "id": "12.2c", "topic": "proof", "kind": "multi_step", "prompt": "Are X,Y independent? Justify.", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2025, "default_truncated": false, "context": "100 independent fair flips. X=number of heads minus number of tails; Y=X².", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8604033727910784 }, { "exam": "final_fa23", "sample": "follow_up", "id": "12.2d", "topic": "moments", "kind": "multi_step", "prompt": "Cov(X,Y)?", "options": { "A": "0", "B": "100", "C": "−100", "D": "10000" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2124, "default_truncated": false, "context": "100 independent fair flips. X=number of heads minus number of tails; Y=X².", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9128367627408548 }, { "exam": "final_fa23", "sample": "follow_up", "id": "13.1a", "topic": "probability", "kind": "calculation", "prompt": "Probability pair1 is adjacent?", "options": { "A": "1/n", "B": "1/(2n−1)", "C": "2/n", "D": "2/(2n−1)" }, "expected": [ "A" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0615, "default_truncated": false, "context": "Uniform random linear permutation of 2n distinct socks, n>=2 labeled left-right matching pairs. X counts pairs whose two socks are adjacent.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8118758930348103 }, { "exam": "final_fa23", "sample": "follow_up", "id": "13.1b", "topic": "expectation", "kind": "calculation", "prompt": "E[X]?", "options": { "A": "1", "B": "n", "C": "n/2", "D": "2" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0654, "default_truncated": false, "context": "Uniform random linear permutation of 2n distinct socks, n>=2 labeled left-right matching pairs. X counts pairs whose two socks are adjacent.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6193568137171783 }, { "exam": "final_fa23", "sample": "follow_up", "id": "13.1c", "topic": "probability", "kind": "multi_step", "prompt": "Probability both pairs1 and2 are adjacent?", "options": { "A": "2/((2n)(2n−1))", "B": "4/((2n−1)(2n−2))", "C": "1/n²", "D": "4/((2n)(2n−1))" }, "expected": [ "D" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0587, "default_truncated": false, "context": "Uniform random linear permutation of 2n distinct socks, n>=2 labeled left-right matching pairs. X counts pairs whose two socks are adjacent.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.525722406449243 }, { "exam": "final_fa23", "sample": "follow_up", "id": "13.1d", "topic": "moments", "kind": "multi_step", "prompt": "Let p=P(pair1 adjacent),q=P(pairs1,2 adjacent). Var(X)?", "options": { "A": "nq−n²p²", "B": "np(1−p)", "C": "np+n(n−1)q−n²p²", "D": "np+n(n−1)q" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0264, "default_truncated": false, "context": "Uniform random linear permutation of 2n distinct socks, n>=2 labeled left-right matching pairs. X counts pairs whose two socks are adjacent.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6594021680882453 }, { "exam": "final_fa23", "sample": "follow_up", "id": "13.2a", "topic": "expectation", "kind": "calculation", "prompt": "Expected unordered pairs sharing a birthday?", "options": { "A": "C(k,2)/365", "B": "C(k,2)/365²", "C": "365/C(k,2)", "D": "k/365" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0176, "default_truncated": false, "context": "k students have independent uniform birthdays on 365 days.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.981010616851495 }, { "exam": "final_fa23", "sample": "follow_up", "id": "13.2b", "topic": "expectation", "kind": "calculation", "prompt": "Expected unordered triples sharing a birthday?", "options": { "A": "C(k,3)/365³", "B": "C(k,3)/365²", "C": "k/365²", "D": "C(k,3)/365" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0451, "default_truncated": false, "context": "k students have independent uniform birthdays on 365 days.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5977709964227157 }, { "exam": "final_fa23", "sample": "follow_up", "id": "13.2c", "topic": "moments", "kind": "multi_step", "prompt": "Variance of number of matching-birthday pairs?", "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)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.03, "default_truncated": false, "context": "k students have independent uniform birthdays on 365 days.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5053181644785447 }, { "exam": "final_fa23", "sample": "follow_up", "id": "14.1", "topic": "proof", "kind": "multi_step", "prompt": "Why is Y unbiased for 1/p?", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0168, "default_truncated": false, "context": "Independent biased coin flips with heads probability p>0. X counts flips through the kth head, k>=1; Y=X/k.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9673436692548533 }, { "exam": "final_fa23", "sample": "follow_up", "id": "14.2", "topic": "moments", "kind": "calculation", "prompt": "Var(Y)?", "options": { "A": "k*(1−p)/p²", "B": "(1−p)/(k²*p²)", "C": "(1−p)/p²", "D": "(1−p)/(k*p²)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0569, "default_truncated": false, "context": "Independent biased coin flips with heads probability p>0. X counts flips through the kth head, k>=1; Y=X/k.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3690058809048272 }, { "exam": "final_fa23", "sample": "follow_up", "id": "14.3", "topic": "bounds", "kind": "multi_step", "prompt": "Chebyshev bound for P(|Y−1/p|>=epsilon/p), clipped at1?", "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))" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0096, "default_truncated": false, "context": "Independent biased coin flips with heads probability p>0. X counts flips through the kth head, k>=1; Y=X/k.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4298150052877617 }, { "exam": "final_fa23", "sample": "follow_up", "id": "14.4", "topic": "bounds", "kind": "multi_step", "prompt": "Only p>=1/2 is known. Smallest k certified by that Chebyshev bound for relative error strictly below10% with probability at least95%?", "options": { "A": "100", "B": "2000", "C": "500", "D": "1000" }, "expected": [ "D" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0096, "default_truncated": false, "context": "Independent biased coin flips with heads probability p>0. X counts flips through the kth head, k>=1; Y=X/k.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4209626922472216 }, { "exam": "final_fa23", "sample": "follow_up", "id": "15.1", "topic": "expectation", "kind": "multi_step", "prompt": "Drive 360 miles at one speed S chosen uniformly in [40,60] mph. Expected hours? Give a valid integral and value.", "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)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0278, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8212328360370813 }, { "exam": "final_fa23", "sample": "follow_up", "id": "15.2a", "topic": "distributions", "kind": "recognition", "prompt": "CDF of max(X,Y,Z) on [0,1]?", "options": { "A": "t³", "B": "1−(1−t)³", "C": "3t²", "D": "t" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.058, "default_truncated": false, "context": "Independent X,Y,Z~Uniform[0,1].", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7477245243536614 }, { "exam": "final_fa23", "sample": "follow_up", "id": "15.2b", "topic": "distributions", "kind": "calculation", "prompt": "PDF of max(X,Y,Z) on [0,1]?", "options": { "A": "3(1−t)²", "B": "3t²", "C": "1", "D": "t³" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.022, "default_truncated": false, "context": "Independent X,Y,Z~Uniform[0,1].", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9817936132897981 }, { "exam": "final_fa23", "sample": "follow_up", "id": "15.2c", "topic": "expectation", "kind": "calculation", "prompt": "E[max(X,Y,Z)]?", "options": { "A": "3/4", "B": "1/2", "C": "1/4", "D": "2/3" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0085, "default_truncated": false, "context": "Independent X,Y,Z~Uniform[0,1].", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.767130006865499 }, { "exam": "final_fa23", "sample": "follow_up", "id": "15.2d", "topic": "distributions", "kind": "recognition", "prompt": "PDF of min(X,Y,Z) on [0,1]?", "options": { "A": "3t²", "B": "1−t³", "C": "(1−t)³", "D": "3(1−t)²" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0098, "default_truncated": false, "context": "Independent X,Y,Z~Uniform[0,1].", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.47484966053225 }, { "exam": "final_fa23", "sample": "follow_up", "id": "15.2e", "topic": "expectation", "kind": "recognition", "prompt": "Expected median of X,Y,Z?", "options": { "A": "1/4", "B": "1/2", "C": "2/3", "D": "3/4" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0891, "default_truncated": false, "context": "Independent X,Y,Z~Uniform[0,1].", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.48343605907106213 }, { "exam": "final_fa23", "sample": "follow_up", "id": "15.3a", "topic": "distributions", "kind": "calculation", "prompt": "Distribution of TS+TJ, second parameter variance?", "options": { "A": "N(8,0.13)", "B": "N(4,0.5)", "C": "N(8,0.1)", "D": "N(8,0.5)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0127, "default_truncated": false, "context": "Independent TS~N(mean4,variance0.2),TJ~N(mean4,variance0.3).", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6271612123476552 }, { "exam": "final_fa23", "sample": "follow_up", "id": "15.3b", "topic": "distributions", "kind": "calculation", "prompt": "Distribution of TS−TJ, second parameter variance?", "options": { "A": "N(0,0.5)", "B": "N(0,0.1)", "C": "N(8,0.5)", "D": "N(0,−0.1)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0263, "default_truncated": false, "context": "Independent TS~N(mean4,variance0.2),TJ~N(mean4,variance0.3).", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9304467083307674 }, { "exam": "final_fa23", "sample": "follow_up", "id": "15.3c", "topic": "probability", "kind": "recognition", "prompt": "P(TS>TJ)?", "options": { "A": "2/5", "B": "0", "C": "1/2", "D": "3/5" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.007, "default_truncated": false, "context": "Independent TS~N(mean4,variance0.2),TJ~N(mean4,variance0.3).", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9338905864773901 }, { "exam": "final_fa23", "sample": "follow_up", "id": "16.1a", "topic": "expectation", "kind": "multi_step", "prompt": "Expected distance to center?", "options": { "A": "(a+b+c)/6", "B": "(a+b+c)/3", "C": "2(a²+b²+c²)/9", "D": "2(a+b+c)/9" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0516, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3017444864199176 }, { "exam": "final_fa23", "sample": "follow_up", "id": "16.1b", "topic": "proof", "kind": "multi_step", "prompt": "Justify the expected distance.", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0444, "default_truncated": false, "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.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9279680687744243 }, { "exam": "final_fa23", "sample": "follow_up", "id": "16.2a", "topic": "regression", "kind": "recognition", "prompt": "Given Y=y, give the minimum-MSE estimate of X and its conditional MSE.", "options": { "A": "(y+3,7)", "B": "(y+7,2)", "C": "(y+3,5)", "D": "(10,7)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1084, "default_truncated": false, "context": "Y has mean7,variance2. Independent Z~N(mean3,variance5). X=Y+Z.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7897335809744824 }, { "exam": "final_fa23", "sample": "follow_up", "id": "16.2b", "topic": "regression", "kind": "recognition", "prompt": "Best affine estimator of X from Y?", "options": { "A": "(2/7)Y+8", "B": "Y+7", "C": "10", "D": "Y+3" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0302, "default_truncated": false, "context": "Y has mean7,variance2. Independent Z~N(mean3,variance5). X=Y+Z.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.651274481825269 }, { "exam": "final_fa23", "sample": "follow_up", "id": "16.3", "topic": "regression", "kind": "multi_step", "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?", "options": { "A": "Z1", "B": "69+(4/3)(Z1−69)", "C": "69+(3/4)(Z1−69)", "D": "69+(1/4)(Z1−69)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0303, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.756844061569278 }, { "exam": "final_fa23", "sample": "follow_up", "id": "17.1a", "topic": "markov", "kind": "recognition", "prompt": "p=1/3. Choose stationary equations plus normalization.", "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" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0056, "default_truncated": true, "context": "Chain states0,1,2. Row0:[1−p,p,0]; row1:[1−p,0,p]; row2:[0,1−p,p].", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3596729290910387 }, { "exam": "final_fa23", "sample": "follow_up", "id": "17.1b", "topic": "regression", "kind": "multi_step", "prompt": "E[X given Y]?", "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" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0005, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5682517316796698 }, { "exam": "final_fa23", "sample": "follow_up", "id": "17.1c", "topic": "regression", "kind": "calculation", "prompt": "Best affine estimate of X as aY+b?", "options": { "A": "(5/4)Y+3/4", "B": "(3/4)Y+5/4", "C": "(18/13)Y+3/13", "D": "2Y" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0034, "default_truncated": false, "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.", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3466892061295826 }, { "exam": "final_fa23", "sample": "follow_up", "id": "17.2a", "topic": "markov", "kind": "multi_step", "prompt": "Give (PAA,PAB,PBA,PBB,PBC,PCC).", "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)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0171, "default_truncated": true, "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.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4345817613168092 }, { "exam": "final_fa23", "sample": "follow_up", "id": "17.2b", "topic": "markov", "kind": "recognition", "prompt": "This stopped chain is irreducible.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1053, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9149009549929797 }, { "exam": "final_fa23", "sample": "follow_up", "id": "17.2c", "topic": "markov", "kind": "recognition", "prompt": "Every state of this chain has period one.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0091, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5926665999540697 }, { "exam": "final_fa23", "sample": "follow_up", "id": "17.2d", "topic": "markov", "kind": "recognition", "prompt": "Let a0=PAA,a1=PAB,b1=PBA,b0=PBB,b2=PBC,c0=PCC. Equations for expected remaining rolls beta to hit C?", "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" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0201, "default_truncated": true, "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.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6331435241147898 }, { "exam": "final_fa24", "sample": "follow_up", "id": "2.1a", "topic": "logic", "kind": "recognition", "prompt": "Truth status of P OR Q OR R OR NOT R?", "options": { "A": "False", "B": "Could be either", "C": "True" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0261, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.994915977040172 }, { "exam": "final_fa24", "sample": "follow_up", "id": "2.1b", "topic": "logic", "kind": "recognition", "prompt": "Truth status of (NOT P implies (R AND NOT R)) implies P?", "options": { "A": "True", "B": "False", "C": "Could be either" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0686, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.48418985050779795 }, { "exam": "final_fa24", "sample": "follow_up", "id": "2.1c", "topic": "logic", "kind": "recognition", "prompt": "Truth status of (R AND NOT R) OR (Q AND P)?", "options": { "A": "Could be either", "B": "False", "C": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0325, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8978523870324252 }, { "exam": "final_fa24", "sample": "follow_up", "id": "2.2a", "topic": "logic", "kind": "recognition", "prompt": "P(x) is true and all R(x,y) false. Truth status of Q(x)?", "options": { "A": "Could be either", "B": "True", "C": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0084, "default_truncated": false, "context": "For all x in nonempty U, P(x) implies [Q(x) OR exists y R(x,y)].", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.49221309558396015 }, { "exam": "final_fa24", "sample": "follow_up", "id": "2.2b", "topic": "logic", "kind": "recognition", "prompt": "Q(x) is false and all R(x,y) false. Truth status of P(x)?", "options": { "A": "True", "B": "Could be either", "C": "False" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0881, "default_truncated": false, "context": "For all x in nonempty U, P(x) implies [Q(x) OR exists y R(x,y)].", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6775206578363818 }, { "exam": "final_fa24", "sample": "follow_up", "id": "2.2c", "topic": "logic", "kind": "recognition", "prompt": "P(x) is true and some R(x,y) true. Truth status of Q(x)?", "options": { "A": "False", "B": "True", "C": "Could be either" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0162, "default_truncated": false, "context": "For all x in nonempty U, P(x) implies [Q(x) OR exists y R(x,y)].", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7893977985726682 }, { "exam": "final_fa24", "sample": "follow_up", "id": "2.3", "topic": "algebra", "kind": "recognition", "prompt": "Express |a−b| using only min(a,b) and max(a,b).", "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)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0119, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9990120063679834 }, { "exam": "final_fa24", "sample": "follow_up", "id": "3.2", "topic": "logic", "kind": "recognition", "prompt": "Complete: forall n [P(n)=>P(n+1)] is equivalent to NOT exists n [P(n) AND __].", "options": { "A": "NOT P(n+1)", "B": "P(n+1)", "C": "P(n)", "D": "NOT P(n)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.3688, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3755444450071601 }, { "exam": "final_fa24", "sample": "follow_up", "id": "3.3a", "topic": "recurrences", "kind": "multi_step", "prompt": "Largest real c>=1 such that Fi>=c^(i−1) for every i>=1?", "options": { "A": "3/2", "B": "2", "C": "(1+sqrt(5))/2", "D": "sqrt(2)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0925, "default_truncated": false, "context": "F0=F1=1 and Fi=F(i−1)+F(i−2).", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.889741945458233 }, { "exam": "final_fa24", "sample": "follow_up", "id": "3.3b", "topic": "proof", "kind": "multi_step", "prompt": "Which proves the optimal exponential lower bound?", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0735, "default_truncated": true, "context": "F0=F1=1 and Fi=F(i−1)+F(i−2).", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9910242675471052 }, { "exam": "final_fa24", "sample": "follow_up", "id": "4.1a", "topic": "matching", "kind": "multi_step", "prompt": "Job-optimal matching?", "options": { "A": "A–2,B–1", "B": "A–1,B–2" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0976, "default_truncated": false, "context": "Jobs A:1>2,B:2>1; candidates 1:B>A,2:A>B.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5312093733737563 }, { "exam": "final_fa24", "sample": "follow_up", "id": "4.1b", "topic": "matching", "kind": "multi_step", "prompt": "Candidate-optimal matching?", "options": { "A": "A–1,B–2", "B": "A–2,B–1" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0663, "default_truncated": false, "context": "Jobs A:1>2,B:2>1; candidates 1:B>A,2:A>B.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5926665999540697 }, { "exam": "final_fa24", "sample": "follow_up", "id": "4.2a", "topic": "matching", "kind": "multi_step", "prompt": "Number of stable matchings?", "options": { "A": "1", "B": "2", "C": "8", "D": "4" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0408, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3512536987011174 }, { "exam": "final_fa24", "sample": "follow_up", "id": "4.2b", "topic": "matching", "kind": "multi_step", "prompt": "Choose a stable matching optimal for neither entire side.", "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" }, "expected": [ "C" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0081, "default_truncated": false, "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.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.3160424181481998 }, { "exam": "final_fa24", "sample": "follow_up", "id": "5.1a", "topic": "graphs", "kind": "recognition", "prompt": "Graph G has k connected components. Sum of vertex degrees?", "options": { "A": "2|V|", "B": "|E|", "C": "2|E|", "D": "|V|−k" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.027, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9842634019440519 }, { "exam": "final_fa24", "sample": "follow_up", "id": "5.1b", "topic": "graphs", "kind": "multi_step", "prompt": "Maximum edges in a simple graph with v vertices and k components,1<=k<=v?", "options": { "A": "C(v,2)−k", "B": "v−k", "C": "C(v−k+1,2)", "D": "k*C(v/k,2)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0154, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9250686191739236 }, { "exam": "final_fa24", "sample": "follow_up", "id": "5.1c", "topic": "graphs", "kind": "recognition", "prompt": "Minimum edges with v vertices and k components?", "options": { "A": "v+k", "B": "k−1", "C": "v−1", "D": "v−k" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0081, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9244749902502848 }, { "exam": "final_fa24", "sample": "follow_up", "id": "5.2a", "topic": "graphs", "kind": "recognition", "prompt": "Minimum vertex colors for a positive-dimensional hypercube?", "options": { "A": "log2(|V|)", "B": "log2(|V|)+1", "C": "2", "D": "1" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0094, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5792585299413737 }, { "exam": "final_fa24", "sample": "follow_up", "id": "5.2b", "topic": "graphs", "kind": "recognition", "prompt": "Minimum edge colors for a hypercube on |V| vertices?", "options": { "A": "2", "B": "log2(|V|)+1", "C": "log2(|V|)", "D": "|V|/2" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0128, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5296669737752049 }, { "exam": "final_fa24", "sample": "follow_up", "id": "5.3", "topic": "graphs", "kind": "recognition", "prompt": "Delete all edges of a simple k-cycle from a connected graph, retaining vertices. Sharp upper bound on component count?", "options": { "A": "k", "B": "2k", "C": "k−1", "D": "k+1" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0499, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.40059759587546095 }, { "exam": "final_fa24", "sample": "follow_up", "id": "6.1", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "2^63 modulo 77?", "options": { "A": "8", "B": "2", "C": "16", "D": "1" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0109, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4088038979914759 }, { "exam": "final_fa24", "sample": "follow_up", "id": "6.2", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "1*2*3*4*5*6 modulo 7?", "options": { "A": "0", "B": "5", "C": "1", "D": "6" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0796, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4137189778658777 }, { "exam": "final_fa24", "sample": "follow_up", "id": "6.3", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "Inverse of 63 modulo 71 in 0..70?", "options": { "A": "62", "B": "8", "C": "63", "D": "9" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.014, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5323950919386197 }, { "exam": "final_fa24", "sample": "follow_up", "id": "6.4a", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Number of a coprime to p?", "options": { "A": "(p−1)q", "B": "pq−p", "C": "pq−1", "D": "(p−1)(q−1)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0098, "default_truncated": false, "context": "p,q are distinct primes; a,b range independently over residues 0..pq−1.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.40322669517609055 }, { "exam": "final_fa24", "sample": "follow_up", "id": "6.4b", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Number of a coprime to both p and q?", "options": { "A": "(p−1)(q−1)", "B": "pq−p−q", "C": "pq−1", "D": "p+q−2" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0093, "default_truncated": false, "context": "p,q are distinct primes; a,b range independently over residues 0..pq−1.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8913319274047625 }, { "exam": "final_fa24", "sample": "follow_up", "id": "6.4c", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "Number of pairs (a,b) for which ax=b mod pq has exactly one solution?", "options": { "A": "pq−1", "B": "(pq)²", "C": "pq(p−1)(q−1)", "D": "(p−1)(q−1)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0117, "default_truncated": false, "context": "p,q are distinct primes; a,b range independently over residues 0..pq−1.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6802154594559744 }, { "exam": "final_fa24", "sample": "follow_up", "id": "6.4d", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "gcd(a,pq)=p and x is a solution. Smallest positive shift t for which x+t is another solution?", "options": { "A": "pq", "B": "1", "C": "q", "D": "p" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1535, "default_truncated": false, "context": "p,q are distinct primes; a,b range independently over residues 0..pq−1.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.45272356704739214 }, { "exam": "final_fa24", "sample": "follow_up", "id": "6.4e", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "Number of pairs (a,b) with exactly p solutions?", "options": { "A": "q(q−1)", "B": "pq", "C": "q(p−1)", "D": "p(p−1)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.003, "default_truncated": false, "context": "p,q are distinct primes; a,b range independently over residues 0..pq−1.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.3500288944602904 }, { "exam": "final_fa24", "sample": "follow_up", "id": "6.4f", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "Number of pairs (a,b) with no solutions?", "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)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0938, "default_truncated": false, "context": "p,q are distinct primes; a,b range independently over residues 0..pq−1.", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.44746601790678886 }, { "exam": "final_fa24", "sample": "follow_up", "id": "6.5", "topic": "number_theory", "kind": "recognition", "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.", "options": { "A": "1", "B": "2", "C": "0", "D": "N−1" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0689, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9548794651471004 }, { "exam": "final_fa24", "sample": "follow_up", "id": "6.6", "topic": "proof", "kind": "multi_step", "prompt": "Use CRT to count units modulo pq for distinct primes p,q.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2326, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9995519248805593 }, { "exam": "final_fa24", "sample": "follow_up", "id": "7.1", "topic": "polynomials", "kind": "calculation", "prompt": "GF(5): normalized Lagrange Delta0 for x-coordinates 0,1,4?", "options": { "A": "−x²+4", "B": "−x²−4", "C": "x²−1", "D": "x²+4" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0197, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.38603627300517357 }, { "exam": "final_fa24", "sample": "follow_up", "id": "7.2a", "topic": "coding", "kind": "recognition", "prompt": "Original polynomial?", "options": { "A": "3", "B": "1", "C": "x+1", "D": "4" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0014, "default_truncated": false, "context": "GF(5), a constant message polynomial is encoded; at most one error. Received points (0,1),(2,1),(3,4).", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3898952993151046 }, { "exam": "final_fa24", "sample": "follow_up", "id": "7.2b", "topic": "coding", "kind": "recognition", "prompt": "With Q(x)=q1*x+q0 and E(x)=x+b0, write the BW equation for received (3,4).", "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" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0454, "default_truncated": false, "context": "GF(5), a constant message polynomial is encoded; at most one error. Received points (0,1),(2,1),(3,4).", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8723623266666565 }, { "exam": "final_fa24", "sample": "follow_up", "id": "7.3a", "topic": "polynomials", "kind": "recognition", "prompt": "Normalized Lagrange basis on distinct field coordinates: Delta_i(i)?", "options": { "A": "0", "B": "i", "C": "1", "D": "yi" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0704, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9210457725999988 }, { "exam": "final_fa24", "sample": "follow_up", "id": "7.3b", "topic": "polynomials", "kind": "recognition", "prompt": "Normalized Lagrange basis: Delta_i(j) for a different interpolation coordinate j?", "options": { "A": "i−j", "B": "yj", "C": "1", "D": "0" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.006, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.3688734908995027 }, { "exam": "final_fa24", "sample": "follow_up", "id": "7.3c", "topic": "polynomials", "kind": "recognition", "prompt": "Interpolate (0,y0),(1,y1),(2,y2) using normalized basis D0,D1,D2.", "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" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0422, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9781133500788708 }, { "exam": "final_fa24", "sample": "follow_up", "id": "8.1", "topic": "counting", "kind": "calculation", "prompt": "Distinct anagrams of BERKELEY?", "options": { "A": "7!/3!", "B": "8!/(2!2!)", "C": "8!", "D": "8!/3!" }, "expected": [ "D" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0108, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6692186932803378 }, { "exam": "final_fa24", "sample": "follow_up", "id": "8.2", "topic": "counting", "kind": "calculation", "prompt": "Assign n labeled balls to m labeled bins without restrictions. Count?", "options": { "A": "n^m", "B": "m^n", "C": "C(n+m−1,m−1)", "D": "n!" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0116, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9444856494637358 }, { "exam": "final_fa24", "sample": "follow_up", "id": "8.3a", "topic": "counting", "kind": "calculation", "prompt": "Number of unordered five-card hands all of one suit in a standard deck, including straight flushes?", "options": { "A": "4*13^5", "B": "C(52,5)", "C": "4*C(13,5)", "D": "C(13,5)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0247, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9745360185695097 }, { "exam": "final_fa24", "sample": "follow_up", "id": "8.3b", "topic": "counting", "kind": "multi_step", "prompt": "You hold two hearts. Choose five unordered cards from the remaining 50. Count selections with at least three additional hearts.", "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)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0463, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5482601385401136 }, { "exam": "final_fa24", "sample": "follow_up", "id": "8.4", "topic": "counting", "kind": "calculation", "prompt": "Count integer solutions sum(xi,i=1..k)=n, each xi>=−1, n>=0.", "options": { "A": "k^n", "B": "C(n+2k−1,k−1)", "C": "C(n+k−1,k−1)", "D": "C(n−1,k−1)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0439, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5325505691875027 }, { "exam": "final_fa24", "sample": "follow_up", "id": "9.1", "topic": "counting", "kind": "calculation", "prompt": "Number of paths A to B?", "options": { "A": "2^12", "B": "C(35,5)", "C": "C(12,5)", "D": "7!*5!" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0705, "default_truncated": false, "context": "Unit square grid from A=(0,0) to B=(7,5); only right and up steps. P=(3,2).", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9759820377842195 }, { "exam": "final_fa24", "sample": "follow_up", "id": "9.2", "topic": "counting", "kind": "multi_step", "prompt": "Number of paths A to B through P?", "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)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0236, "default_truncated": false, "context": "Unit square grid from A=(0,0) to B=(7,5); only right and up steps. P=(3,2).", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7540656903107702 }, { "exam": "final_fa24", "sample": "follow_up", "id": "9.3", "topic": "proof", "kind": "multi_step", "prompt": "Combinatorially prove C(12,7)=sum(i=0..5)C(3+i,i)C(8−i,5−i).", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0418, "default_truncated": true, "context": "Unit square grid from A=(0,0) to B=(7,5); only right and up steps. P=(3,2).", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9607512736176046 }, { "exam": "final_fa24", "sample": "follow_up", "id": "10.1", "topic": "countability", "kind": "recognition", "prompt": "Distinct real-valued functions represented by finite integer-coefficient polynomials?", "options": { "A": "Countably infinite", "B": "Uncountably infinite", "C": "Finite" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0987, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5293165764052077 }, { "exam": "final_fa24", "sample": "follow_up", "id": "10.2", "topic": "countability", "kind": "recognition", "prompt": "Distinct real-valued functions represented by finite real-coefficient polynomials?", "options": { "A": "Uncountably infinite", "B": "Countably infinite", "C": "Finite" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1054, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6652409557748218 }, { "exam": "final_fa24", "sample": "follow_up", "id": "10.3", "topic": "countability", "kind": "recognition", "prompt": "Distinct functions GF(p)->GF(p) represented by finite polynomials for fixed prime p?", "options": { "A": "Countably infinite", "B": "Finite", "C": "Uncountably infinite" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0019, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6753152576935494 }, { "exam": "final_fa24", "sample": "follow_up", "id": "11.1", "topic": "computability", "kind": "multi_step", "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.", "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)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0289, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6505757913603881 }, { "exam": "final_fa24", "sample": "follow_up", "id": "12.1a", "topic": "probability", "kind": "multi_step", "prompt": "P(A intersection B)?", "options": { "A": "1/6", "B": "1/3", "C": "2/3", "D": "1/9" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0432, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5837367234394677 }, { "exam": "final_fa24", "sample": "follow_up", "id": "12.1b", "topic": "probability", "kind": "multi_step", "prompt": "P(B given A)?", "options": { "A": "2/3", "B": "1/6", "C": "1/3", "D": "1/2" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0322, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.38253225505393756 }, { "exam": "final_fa24", "sample": "follow_up", "id": "12.1c", "topic": "probability", "kind": "multi_step", "prompt": "P(C intersection A)?", "options": { "A": "2/9", "B": "1/3", "C": "1/9", "D": "1/6" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.033, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.480955762938224 }, { "exam": "final_fa24", "sample": "follow_up", "id": "12.1d", "topic": "probability", "kind": "multi_step", "prompt": "P(C given A)?", "options": { "A": "2/9", "B": "1/9", "C": "1/3", "D": "2/3" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0153, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.46601035735609475 }, { "exam": "final_fa24", "sample": "follow_up", "id": "12.2a", "topic": "moments", "kind": "calculation", "prompt": "Cov(ID,IE)?", "options": { "A": "1/6", "B": "1/5", "C": "−1/30", "D": "1/30" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0076, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.38754811993034344 }, { "exam": "final_fa24", "sample": "follow_up", "id": "12.2bi", "topic": "regression", "kind": "recognition", "prompt": "Sign of a?", "options": { "A": "Positive", "B": "Negative", "C": "Zero" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1254, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5373784946053918 }, { "exam": "final_fa24", "sample": "follow_up", "id": "12.2bii", "topic": "regression", "kind": "multi_step", "prompt": "Value of a?", "options": { "A": "1/3", "B": "1/5", "C": "1/30", "D": "2/15" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.023, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.34040633907578005 }, { "exam": "final_fa24", "sample": "follow_up", "id": "12.2biii", "topic": "regression", "kind": "recognition", "prompt": "Value of b?", "options": { "A": "0", "B": "1/5", "C": "1/3", "D": "1/2" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0033, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.33544561004293627 }, { "exam": "final_fa24", "sample": "follow_up", "id": "12.3a", "topic": "moments", "kind": "calculation", "prompt": "E[XY]?", "options": { "A": "5", "B": "6", "C": "9", "D": "1" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0417, "default_truncated": false, "context": "Independent X,Y have E[X]=2,E[Y]=3,Var(X)=1,Var(Y)=2.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7932662229439467 }, { "exam": "final_fa24", "sample": "follow_up", "id": "12.3b", "topic": "moments", "kind": "calculation", "prompt": "E[X²]?", "options": { "A": "1", "B": "4", "C": "5", "D": "3" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0172, "default_truncated": false, "context": "Independent X,Y have E[X]=2,E[Y]=3,Var(X)=1,Var(Y)=2.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.513313078784462 }, { "exam": "final_fa24", "sample": "follow_up", "id": "12.3c", "topic": "moments", "kind": "multi_step", "prompt": "Var(XY)?", "options": { "A": "2", "B": "36", "C": "55", "D": "19" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0304, "default_truncated": false, "context": "Independent X,Y have E[X]=2,E[Y]=3,Var(X)=1,Var(Y)=2.", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.477197535671976 }, { "exam": "final_fa24", "sample": "follow_up", "id": "12.4a", "topic": "distributions", "kind": "calculation", "prompt": "Density f(x)=x on [0,c], zero otherwise. What is c>0?", "options": { "A": "1/2", "B": "2", "C": "sqrt(2)", "D": "1" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0256, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5323223140344983 }, { "exam": "final_fa24", "sample": "follow_up", "id": "12.4b", "topic": "distributions", "kind": "calculation", "prompt": "Density f(x)=x on [0,c], zero elsewhere, with c>0 chosen to normalize it. CDF on [0,c]?", "options": { "A": "x", "B": "x²", "C": "x²/2", "D": "2x" }, "expected": [ "C" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2125, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8848401312124456 }, { "exam": "final_fa24", "sample": "follow_up", "id": "13.1a", "topic": "moments", "kind": "recognition", "prompt": "Independent X~Bin(n,p),Y~Bin(n,q). Cov(X,Y)?", "options": { "A": "n(p+q)", "B": "np(1−q)", "C": "0", "D": "npq" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2037, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.545099048828039 }, { "exam": "final_fa24", "sample": "follow_up", "id": "13.1b", "topic": "moments", "kind": "calculation", "prompt": "Independent X~Bin(n,p),Y~Bin(n,q). Var(X+Y)?", "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)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0734, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7918520087011474 }, { "exam": "final_fa24", "sample": "follow_up", "id": "13.2", "topic": "expectation", "kind": "calculation", "prompt": "X~Geom(p) counts trials in {1,2,...}. For integer x>=1, compute E[X−x given X>=x].", "options": { "A": "x/p", "B": "0", "C": "(1−p)/p", "D": "1/p" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0078, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7034605250992085 }, { "exam": "final_fa24", "sample": "follow_up", "id": "13.3", "topic": "distributions", "kind": "recognition", "prompt": "X~Poisson(lambda). P(X>0)?", "options": { "A": "lambda*exp(−lambda)", "B": "exp(−lambda)", "C": "1/lambda", "D": "1−exp(−lambda)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2871, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9940675782096317 }, { "exam": "final_fa24", "sample": "follow_up", "id": "13.4", "topic": "distributions", "kind": "recognition", "prompt": "Independent Poisson(lambda),Poisson(mu). P(X+Y>0)?", "options": { "A": "exp(−lambda−mu)", "B": "1−exp(−lambda)*mu", "C": "(1−exp(−lambda))(1−exp(−mu))", "D": "1−exp(−lambda−mu)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0519, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8717234277288637 }, { "exam": "final_fa24", "sample": "follow_up", "id": "13.5", "topic": "distributions", "kind": "recognition", "prompt": "PDF of Exp(rate lambda>0) on x>=0?", "options": { "A": "exp(−x/lambda)/lambda²", "B": "exp(−lambda*x)", "C": "lambda*exp(−lambda*x)", "D": "lambda*exp(lambda*x)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0045, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9079095751118242 }, { "exam": "final_fa24", "sample": "follow_up", "id": "13.6", "topic": "distributions", "kind": "recognition", "prompt": "PDF of min(X,Y) for independent Exp(rate lambda) variables, z>=0?", "options": { "A": "2lambda*exp(−lambda*z)", "B": "lambda²*exp(−2lambda*z)", "C": "2lambda*exp(−2lambda*z)", "D": "lambda*exp(−lambda*z)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0477, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6346325122552334 }, { "exam": "final_fa24", "sample": "follow_up", "id": "13.7", "topic": "distributions", "kind": "multi_step", "prompt": "A dart is uniform over a disk of radius r>0. Density of distance x to center?", "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" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0172, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9041410513982376 }, { "exam": "final_fa24", "sample": "follow_up", "id": "14.1", "topic": "distributions", "kind": "recognition", "prompt": "CDF of max(X,Y)?", "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" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0382, "default_truncated": false, "context": "Independent X,Y~Uniform[0,r],r>0.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6214218219776368 }, { "exam": "final_fa24", "sample": "follow_up", "id": "14.2", "topic": "distributions", "kind": "calculation", "prompt": "PDF of max(X,Y) on [0,r], zero elsewhere?", "options": { "A": "2(r−z)/r²", "B": "2z/r²", "C": "z²/r²", "D": "1/r" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0345, "default_truncated": false, "context": "Independent X,Y~Uniform[0,r],r>0.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9161432310172357 }, { "exam": "final_fa24", "sample": "follow_up", "id": "14.3", "topic": "expectation", "kind": "calculation", "prompt": "E[max(X,Y)]?", "options": { "A": "2r/3", "B": "r", "C": "r/2", "D": "r/3" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0035, "default_truncated": false, "context": "Independent X,Y~Uniform[0,r],r>0.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9254778085918131 }, { "exam": "final_fa24", "sample": "follow_up", "id": "14.4", "topic": "expectation", "kind": "calculation", "prompt": "E[min(X,Y)]? Hint: avoid integration.", "options": { "A": "r/4", "B": "2r/3", "C": "r/3", "D": "r/2" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0233, "default_truncated": false, "context": "Independent X,Y~Uniform[0,r],r>0.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5982184082446659 }, { "exam": "final_fa24", "sample": "follow_up", "id": "14.5", "topic": "expectation", "kind": "multi_step", "prompt": "E[|X−Y|]? Hint: use min, max and linearity.", "options": { "A": "2r/3", "B": "0", "C": "r/3", "D": "r/2" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0391, "default_truncated": false, "context": "Independent X,Y~Uniform[0,r],r>0.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6168242810247948 }, { "exam": "final_fa24", "sample": "follow_up", "id": "15.1", "topic": "expectation", "kind": "multi_step", "prompt": "Expected number close to their original bin?", "options": { "A": "2", "B": "3", "C": "2.98", "D": "1" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0233, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.58191101071155 }, { "exam": "final_fa24", "sample": "follow_up", "id": "15.2a", "topic": "probability", "kind": "calculation", "prompt": "E[I2*I5]?", "options": { "A": "6/(100*99)", "B": "3/100", "C": "9/10000", "D": "9/(100*99)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0338, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4443287098421572 }, { "exam": "final_fa24", "sample": "follow_up", "id": "15.2b", "topic": "probability", "kind": "multi_step", "prompt": "E[I2*I4]?", "options": { "A": "8/(100*99)", "B": "9/10000", "C": "9/(100*99)", "D": "6/(100*99)" }, "expected": [ "A" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0557, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.44238634968545487 }, { "exam": "final_fa24", "sample": "follow_up", "id": "15.2c", "topic": "probability", "kind": "recognition", "prompt": "Do there exist distinct i,j for which Ii,Ij are independent?", "options": { "A": "Yes", "B": "No" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.4282, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6224593312018546 }, { "exam": "final_fa24", "sample": "follow_up", "id": "16.1", "topic": "bayes", "kind": "multi_step", "prompt": "Given first card red, probability the chosen deck is fair?", "options": { "A": "1/2", "B": "3/5", "C": "2/5", "D": "1/4" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0177, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5693270163255819 }, { "exam": "final_fa24", "sample": "follow_up", "id": "16.2", "topic": "bayes", "kind": "multi_step", "prompt": "Given first card red, probability the second is red?", "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)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0247, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8604668282989028 }, { "exam": "final_fa24", "sample": "follow_up", "id": "17.1", "topic": "proof", "kind": "multi_step", "prompt": "X is a sum of indicators, and p=P(X>=1). Prove p<=E[X].", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0131, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.982769828896265 }, { "exam": "final_fa24", "sample": "follow_up", "id": "17.2", "topic": "proof", "kind": "multi_step", "prompt": "W counts successes in n independent Bernoulli(p) trials. For k>0, prove P(W>=k)<=np/k.", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0515, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9783225914592865 }, { "exam": "final_fa24", "sample": "follow_up", "id": "17.3", "topic": "proof", "kind": "multi_step", "prompt": "Y~Poisson(lambda),lambda>0. Prove P(Y>=10lambda)<=1/(81lambda).", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.103, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7789718864114699 }, { "exam": "final_fa24", "sample": "follow_up", "id": "18.1", "topic": "probability", "kind": "recognition", "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).", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0499, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5312093733737563 }, { "exam": "final_fa24", "sample": "follow_up", "id": "18.2", "topic": "proof", "kind": "multi_step", "prompt": "Choose a counterexample and its four probabilities in the order: C|A union B, C|A, C|B, C|A intersection B.", "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)." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.3506, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.45489072058880226 }, { "exam": "final_fa24", "sample": "follow_up", "id": "19.1", "topic": "expectation", "kind": "recognition", "prompt": "E[An]?", "options": { "A": "lambda", "B": "n*lambda", "C": "lambda/n", "D": "1/lambda" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0623, "default_truncated": false, "context": "Independent Xi~Poisson(lambda),lambda>0; An=(X1+...+Xn)/n.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9974963780187172 }, { "exam": "final_fa24", "sample": "follow_up", "id": "19.2", "topic": "bounds", "kind": "multi_step", "prompt": "Smallest integer sample size guaranteed by Chebyshev to make P(|An−E[An]|>=0.1E[An])<=0.05?", "options": { "A": "ceil(2000*lambda)", "B": "ceil(400/lambda)", "C": "ceil(2000/lambda)", "D": "ceil(200/lambda)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0482, "default_truncated": false, "context": "Independent Xi~Poisson(lambda),lambda>0; An=(X1+...+Xn)/n.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.3984585186882418 }, { "exam": "final_fa24", "sample": "follow_up", "id": "19.3", "topic": "bounds", "kind": "multi_step", "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?", "options": { "A": "ceil(400*lambda)", "B": "ceil(40/lambda)", "C": "ceil(2000/lambda)", "D": "ceil(400/lambda)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.067, "default_truncated": false, "context": "Independent Xi~Poisson(lambda),lambda>0; An=(X1+...+Xn)/n.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4189648173687437 }, { "exam": "final_fa24", "sample": "follow_up", "id": "20.1", "topic": "markov", "kind": "recognition", "prompt": "Let b_i be expected future even rolls before first reaching state2 from state i. Choose the first-step equations.", "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" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0046, "default_truncated": true, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7782784268579436 }, { "exam": "final_fa24", "sample": "follow_up", "id": "20.2", "topic": "markov", "kind": "recognition", "prompt": "Continue rolling even after reaching state2. Is there a unique stationary distribution?", "options": { "A": "No", "B": "Yes" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1303, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5926665999540697 }, { "exam": "final_fa24", "sample": "follow_up", "id": "20.3", "topic": "markov", "kind": "recognition", "prompt": "For the continuing chain, choose stationary equations plus normalization.", "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" }, "expected": [ "C" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.069, "default_truncated": true, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5119916095309854 }, { "exam": "final_fa24", "sample": "follow_up", "id": "21.1", "topic": "markov", "kind": "multi_step", "prompt": "Give (A,B,C,D,E).", "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)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0543, "default_truncated": true, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.35628748125978893 }, { "exam": "final_fa24", "sample": "follow_up", "id": "21.2", "topic": "markov", "kind": "multi_step", "prompt": "Probability of winning from both ready?", "options": { "A": "8/9", "B": "2/3", "C": "4/9", "D": "4/5" }, "expected": [ "D" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0959, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.38297195412290774 }, { "exam": "final_fa24", "sample": "follow_up", "id": "21.3", "topic": "markov", "kind": "multi_step", "prompt": "Expected turns until win or loss from both ready?", "options": { "A": "4", "B": "9/5", "C": "5", "D": "3" }, "expected": [ "D" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0042, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4799561165848732 }, { "exam": "final_sp23", "sample": "follow_up", "id": "2.1", "topic": "logic", "kind": "recognition", "prompt": "False implies False.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1159, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9992903296008995 }, { "exam": "final_sp23", "sample": "follow_up", "id": "2.2", "topic": "logic", "kind": "recognition", "prompt": "False implies True.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.107, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9902915235185259 }, { "exam": "final_sp23", "sample": "follow_up", "id": "2.3", "topic": "logic", "kind": "recognition", "prompt": "[NOT exists x P(x)] implies [forall x (P(x) implies Q(x))].", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0273, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9324533088603709 }, { "exam": "final_sp23", "sample": "follow_up", "id": "2.4", "topic": "logic", "kind": "recognition", "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))].", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.114, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9706877692486436 }, { "exam": "final_sp23", "sample": "follow_up", "id": "2.5", "topic": "matching", "kind": "recognition", "prompt": "No stable matching can give every job its least-preferred candidate.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0004, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5312093733737563 }, { "exam": "final_sp23", "sample": "follow_up", "id": "2.6", "topic": "matching", "kind": "recognition", "prompt": "If job-proposing deferred acceptance finishes on day one, the result is optimal for both jobs and candidates.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1557, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6513548646660542 }, { "exam": "final_sp23", "sample": "follow_up", "id": "3.1", "topic": "proof", "kind": "multi_step", "prompt": "For natural x<=y, choose a valid proof that x<=y/2 or y−x<=y/2.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1235, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9637543993678308 }, { "exam": "final_sp23", "sample": "follow_up", "id": "3.2a", "topic": "number_theory", "kind": "multi_step", "prompt": "How many ordered natural pairs (a,b) satisfy b²=3a²? Here zero is natural.", "options": { "A": "2", "B": "1", "C": "0", "D": "Infinitely many" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.3841, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6094598715535207 }, { "exam": "final_sp23", "sample": "follow_up", "id": "3.2b", "topic": "proof", "kind": "multi_step", "prompt": "Choose a valid argument determining all natural solutions of b²=3a², allowing zero.", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0563, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9873515006024902 }, { "exam": "final_sp23", "sample": "follow_up", "id": "4.1", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "Prime p and nonzero a modulo p: how many distinct residues ax arise as x ranges over 0..p−1?", "options": { "A": "p−1", "B": "p", "C": "1", "D": "gcd(a,p)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0047, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.565631184724166 }, { "exam": "final_sp23", "sample": "follow_up", "id": "4.2", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "Let gcd(a,N)=d. How many distinct residues ax mod N arise for x=0..N−1?", "options": { "A": "N", "B": "d", "C": "N/d", "D": "N−d" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0108, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9800816611366729 }, { "exam": "final_sp23", "sample": "follow_up", "id": "4.3", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "For prime p, evaluate 1²·2²···(p−1)² modulo p.", "options": { "A": "1", "B": "2", "C": "0", "D": "p−1" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0788, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.36144170843296086 }, { "exam": "final_sp23", "sample": "follow_up", "id": "5.1", "topic": "bayes", "kind": "multi_step", "prompt": "Choose a fair four-sided or fair six-sided die with equal probability. Roll and observe3. Probability it was the four-sided die?", "options": { "A": "3/5", "B": "1/4", "C": "1/2", "D": "2/5" }, "expected": [ "A" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0074, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4167500479242407 }, { "exam": "final_sp23", "sample": "follow_up", "id": "6.1", "topic": "graphs", "kind": "calculation", "prompt": "How many friendship edges result?", "options": { "A": "C(n,2)", "B": "7n−21", "C": "7n", "D": "7n−28" }, "expected": [ "D" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0709, "default_truncated": false, "context": "People indexed0,...,n−1 arrive in order, n>=8. Person i chooses min(i,7) distinct earlier people as friends. Friendship is undirected.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.38453152591357326 }, { "exam": "final_sp23", "sample": "follow_up", "id": "6.2", "topic": "graphs", "kind": "recognition", "prompt": "Maximum possible degree of a person?", "options": { "A": "14", "B": "7", "C": "n−7", "D": "n−1" }, "expected": [ "D" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.045, "default_truncated": false, "context": "People indexed0,...,n−1 arrive in order, n>=8. Person i chooses min(i,7) distinct earlier people as friends. Friendship is undirected.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4482780905009404 }, { "exam": "final_sp23", "sample": "follow_up", "id": "6.3a", "topic": "graphs", "kind": "multi_step", "prompt": "Smallest number of colors sufficient for every possible resulting graph?", "options": { "A": "n", "B": "9", "C": "7", "D": "8" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0399, "default_truncated": false, "context": "People indexed0,...,n−1 arrive in order, n>=8. Person i chooses min(i,7) distinct earlier people as friends. Friendship is undirected.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6126103831107212 }, { "exam": "final_sp23", "sample": "follow_up", "id": "6.3b", "topic": "proof", "kind": "multi_step", "prompt": "Which proves the optimal universal coloring bound?", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0396, "default_truncated": false, "context": "People indexed0,...,n−1 arrive in order, n>=8. Person i chooses min(i,7) distinct earlier people as friends. Friendship is undirected.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.994690705754621 }, { "exam": "final_sp23", "sample": "follow_up", "id": "6.4", "topic": "expectation", "kind": "multi_step", "prompt": "If each person uniformly chooses the prescribed-size subset of earlier people, expected degree of person0?", "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)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0105, "default_truncated": false, "context": "People indexed0,...,n−1 arrive in order, n>=8. Person i chooses min(i,7) distinct earlier people as friends. Friendship is undirected.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.46975000146471585 }, { "exam": "final_sp23", "sample": "follow_up", "id": "7.1", "topic": "graphs", "kind": "recognition", "prompt": "The three-dimensional cube has more edges than K4.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2619, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6513548646660542 }, { "exam": "final_sp23", "sample": "follow_up", "id": "7.2", "topic": "graphs", "kind": "recognition", "prompt": "Sharp maximum number of vertices in a graph with e edges and c connected components?", "options": { "A": "2e+c", "B": "e−c", "C": "e+c−1", "D": "e+c" }, "expected": [ "D" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0434, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.46789016732250677 }, { "exam": "final_sp23", "sample": "follow_up", "id": "7.3", "topic": "graphs", "kind": "recognition", "prompt": "A finite undirected graph can have an odd number of odd-degree vertices.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0224, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9399133498259924 }, { "exam": "final_sp23", "sample": "follow_up", "id": "7.4", "topic": "graphs", "kind": "calculation", "prompt": "Average vertex degree in a tree with n vertices?", "options": { "A": "2", "B": "n−1", "C": "1−1/n", "D": "2−2/n" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0094, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8618906057571681 }, { "exam": "final_sp23", "sample": "follow_up", "id": "7.5", "topic": "proof", "kind": "multi_step", "prompt": "Connected planar embedding has v vertices and average face boundary length s>2. Choose edge count with valid justification.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0303, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8922402686756796 }, { "exam": "final_sp23", "sample": "follow_up", "id": "8.1a", "topic": "polynomials", "kind": "recognition", "prompt": "Remainder when P is divided by x−6?", "options": { "A": "2", "B": "6", "C": "1", "D": "0" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0867, "default_truncated": false, "context": "Over GF(7), degree-at-most-two P satisfies P(0)=0 and P(1)=P(6)=2.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5061803028803236 }, { "exam": "final_sp23", "sample": "follow_up", "id": "8.1b", "topic": "polynomials", "kind": "calculation", "prompt": "Find P(x).", "options": { "A": "x²+1", "B": "2x²+2", "C": "2x²", "D": "2x" }, "expected": [ "C" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0083, "default_truncated": false, "context": "Over GF(7), degree-at-most-two P satisfies P(0)=0 and P(1)=P(6)=2.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.34040633907578005 }, { "exam": "final_sp23", "sample": "follow_up", "id": "8.1c", "topic": "polynomials", "kind": "recognition", "prompt": "Which constant polynomial disagrees with exactly one of these three given values?", "options": { "A": "0", "B": "1", "C": "2", "D": "6" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0391, "default_truncated": false, "context": "Over GF(7), degree-at-most-two P satisfies P(0)=0 and P(1)=P(6)=2.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3501043862896871 }, { "exam": "final_sp23", "sample": "follow_up", "id": "8.2", "topic": "polynomials", "kind": "calculation", "prompt": "Over the reals, degree-at-most-two P has P(0)=0 and P(1)=P(−1)=A. Find P.", "options": { "A": "Ax", "B": "Ax²", "C": "A(x²+1)", "D": "A" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0288, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.33873922196938544 }, { "exam": "final_sp23", "sample": "follow_up", "id": "8.3", "topic": "polynomials", "kind": "calculation", "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?", "options": { "A": "p^(d−k)", "B": "p^(d+1−k)", "C": "1", "D": "p^k" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0096, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5705442005449559 }, { "exam": "final_sp23", "sample": "follow_up", "id": "9.1", "topic": "countability", "kind": "recognition", "prompt": "Between any two distinct reals in [0,1] there is a rational.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1871, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.991422514586288 }, { "exam": "final_sp23", "sample": "follow_up", "id": "9.2", "topic": "countability", "kind": "recognition", "prompt": "The rationals and [0,1]×[0,1] have the same cardinality.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2962, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9796676466573412 }, { "exam": "final_sp23", "sample": "follow_up", "id": "9.3", "topic": "countability", "kind": "recognition", "prompt": "The set of finite programs is countable.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1065, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.991422514586288 }, { "exam": "final_sp23", "sample": "follow_up", "id": "9.4", "topic": "countability", "kind": "recognition", "prompt": "All outputs generated by finite programs on finite inputs form a countable set, even allowing infinite output sequences.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0588, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9820137900379085 }, { "exam": "final_sp23", "sample": "follow_up", "id": "9.5", "topic": "computability", "kind": "recognition", "prompt": "There exists a program P(Q,x) that halts exactly when Q(x) does not halt.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1903, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6224593312018546 }, { "exam": "final_sp23", "sample": "follow_up", "id": "10.1", "topic": "counting", "kind": "recognition", "prompt": "Number of two-element subsets of an n-element set?", "options": { "A": "C(n,2)", "B": "n²", "C": "2^n", "D": "n(n−1)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.048, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.983108393392118 }, { "exam": "final_sp23", "sample": "follow_up", "id": "10.2", "topic": "counting", "kind": "recognition", "prompt": "Number of nonempty subsets of an n-element set?", "options": { "A": "n!", "B": "2^n", "C": "2^n−1", "D": "n−1" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0608, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9836616939815281 }, { "exam": "final_sp23", "sample": "follow_up", "id": "10.3", "topic": "counting", "kind": "recognition", "prompt": "Orderings of k distinct elements?", "options": { "A": "k^k", "B": "2^k", "C": "k", "D": "k!" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0465, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.942368516117686 }, { "exam": "final_sp23", "sample": "follow_up", "id": "10.4", "topic": "counting", "kind": "calculation", "prompt": "Hamiltonian paths in K_n, counted as ordered vertex sequences (reversals distinct), n>=3?", "options": { "A": "C(n,2)", "B": "(n−1)!", "C": "n!", "D": "n!/2" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.074, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6255106978886558 }, { "exam": "final_sp23", "sample": "follow_up", "id": "10.5", "topic": "counting", "kind": "calculation", "prompt": "Hamiltonian cycles in K_n, rotations identified but reverse orientations distinct, n>=3?", "options": { "A": "(n−1)!/2", "B": "(n−1)!", "C": "n!", "D": "2^n" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0085, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8898773583420013 }, { "exam": "final_sp23", "sample": "follow_up", "id": "11.1", "topic": "counting", "kind": "calculation", "prompt": "S(n,n−1), n>=2?", "options": { "A": "n!", "B": "n−1", "C": "C(n,2)", "D": "2^(n−1)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0636, "default_truncated": false, "context": "S(n,k) counts partitions of n labeled objects into k nonempty unlabeled blocks.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9113033292702085 }, { "exam": "final_sp23", "sample": "follow_up", "id": "11.2", "topic": "counting", "kind": "calculation", "prompt": "S(n,2), n>=2?", "options": { "A": "2^n−1", "B": "C(n,2)", "C": "2^(n−1)−1", "D": "2^(n−1)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0981, "default_truncated": false, "context": "S(n,k) counts partitions of n labeled objects into k nonempty unlabeled blocks.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4462595090362637 }, { "exam": "final_sp23", "sample": "follow_up", "id": "11.3", "topic": "proof", "kind": "multi_step", "prompt": "Choose the correct recurrence with combinatorial justification.", "options": { "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1383, "default_truncated": false, "context": "S(n,k) counts partitions of n labeled objects into k nonempty unlabeled blocks.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9944303628186559 }, { "exam": "final_sp23", "sample": "follow_up", "id": "11.4", "topic": "counting", "kind": "multi_step", "prompt": "Multisets of size n from values1..m containing exactly k distinct values,1<=k<=min(n,m)?", "options": { "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)" }, "expected": [ "A" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1581, "default_truncated": false, "context": "S(n,k) counts partitions of n labeled objects into k nonempty unlabeled blocks.", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7715117589249575 }, { "exam": "final_sp23", "sample": "follow_up", "id": "11.5", "topic": "counting", "kind": "multi_step", "prompt": "Ordered length-n sequences from1..m using exactly k distinct values?", "options": { "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)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0753, "default_truncated": false, "context": "S(n,k) counts partitions of n labeled objects into k nonempty unlabeled blocks.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7941775680286627 }, { "exam": "final_sp23", "sample": "follow_up", "id": "12.1", "topic": "probability", "kind": "calculation", "prompt": "Exact P(S>=20)?", "options": { "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)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1388, "default_truncated": false, "context": "Flip a fair coin independently100 times. S=number of heads minus number of tails.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.45740709791046325 }, { "exam": "final_sp23", "sample": "follow_up", "id": "12.2", "topic": "bounds", "kind": "multi_step", "prompt": "Apply Markov to the nonnegative number of heads to bound P(S>=20).", "options": { "A": "0", "B": "1/2", "C": "1/5", "D": "5/6" }, "expected": [ "D" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.005, "default_truncated": false, "context": "Flip a fair coin independently100 times. S=number of heads minus number of tails.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.47905138671315767 }, { "exam": "final_sp23", "sample": "follow_up", "id": "12.3", "topic": "bounds", "kind": "calculation", "prompt": "What bound comes from P(S>=20)<=P(|S|>=20) followed directly by two-sided Chebyshev, without symmetry refinement?", "options": { "A": "1/2", "B": "1/20", "C": "1/4", "D": "1/8" }, "expected": [ "C" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0599, "default_truncated": false, "context": "Flip a fair coin independently100 times. S=number of heads minus number of tails.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.2821833983601388 }, { "exam": "final_sp23", "sample": "follow_up", "id": "12.4", "topic": "normal_approximation", "kind": "calculation", "prompt": "Use the CLT without continuity correction to approximate P(S>=20), writing Phi for standard Normal CDF.", "options": { "A": "Phi(2)", "B": "1−Phi(2)", "C": "1−Phi(0.2)", "D": "1−Phi(20)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0043, "default_truncated": false, "context": "Flip a fair coin independently100 times. S=number of heads minus number of tails.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.783149058767882 }, { "exam": "final_sp23", "sample": "follow_up", "id": "13.1", "topic": "expectation", "kind": "recognition", "prompt": "E[R]?", "options": { "A": "1/4", "B": "1/2", "C": "3/4", "D": "1" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0342, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.41172531682861635 }, { "exam": "final_sp23", "sample": "follow_up", "id": "13.2", "topic": "expectation", "kind": "multi_step", "prompt": "E[J]?", "options": { "A": "4 ln(3)", "B": "2", "C": "2 ln(3)", "D": "ln(3)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0037, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3444545218043904 }, { "exam": "final_sp23", "sample": "follow_up", "id": "13.3", "topic": "probability", "kind": "recognition", "prompt": "P(J>70 given R=r)?", "options": { "A": "(1−r)^70", "B": "r^70", "C": "(1−r)^69", "D": "1−(1−r)^70" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1123, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7513046241663918 }, { "exam": "final_sp23", "sample": "follow_up", "id": "13.4", "topic": "probability", "kind": "multi_step", "prompt": "P(J>70)?", "options": { "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" }, "expected": [ "A" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0277, "default_truncated": false, "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.", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.47484966053225 }, { "exam": "final_sp23", "sample": "follow_up", "id": "14.1", "topic": "expectation", "kind": "multi_step", "prompt": "N~Poisson(mu) is independent of iid lifetimes Xi~Exponential(rate lambda). T=sum(i=1..N)Xi, empty sum zero. E[T]?", "options": { "A": "mu*lambda", "B": "lambda/mu", "C": "mu/lambda", "D": "1/lambda" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0239, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5893012251445238 }, { "exam": "final_sp23", "sample": "follow_up", "id": "15.1", "topic": "expectation", "kind": "recognition", "prompt": "E[X]?", "options": { "A": "mu²", "B": "mu", "C": "0", "D": "sigma" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0469, "default_truncated": false, "context": "X~Normal(mu,sigma²), where sigma² denotes the variance.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9953074165206746 }, { "exam": "final_sp23", "sample": "follow_up", "id": "15.2", "topic": "expectation", "kind": "calculation", "prompt": "E[X²]?", "options": { "A": "sigma²", "B": "mu²+sigma²", "C": "mu²", "D": "mu+sigma²" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0514, "default_truncated": false, "context": "X~Normal(mu,sigma²), where sigma² denotes the variance.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9977744899538294 }, { "exam": "final_sp23", "sample": "follow_up", "id": "15.3", "topic": "expectation", "kind": "multi_step", "prompt": "E[X³]?", "options": { "A": "mu³+3mu²*sigma", "B": "mu³+sigma³", "C": "mu³+3mu*sigma²", "D": "mu³" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.089, "default_truncated": false, "context": "X~Normal(mu,sigma²), where sigma² denotes the variance.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8366882079045871 }, { "exam": "final_sp23", "sample": "follow_up", "id": "16.1", "topic": "probability", "kind": "calculation", "prompt": "Marginals (S=1,2,3) and (P=1,2,3)?", "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)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1877, "default_truncated": false, "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).", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.47925689068307464 }, { "exam": "final_sp23", "sample": "follow_up", "id": "16.2", "topic": "expectation", "kind": "calculation", "prompt": "(E[S],E[P])?", "options": { "A": "(2.40,2.35)", "B": "(.55,.55)", "C": "(2.35,2.40)", "D": "(2,2)" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1463, "default_truncated": false, "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).", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7065613113761032 }, { "exam": "final_sp23", "sample": "follow_up", "id": "16.3", "topic": "probability", "kind": "calculation", "prompt": "P(S=P)?", "options": { "A": "1/3", "B": "2/5", "C": "1/2", "D": "3/4" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0225, "default_truncated": false, "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).", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4668385510947485 }, { "exam": "final_sp23", "sample": "follow_up", "id": "16.4", "topic": "conditional_probability", "kind": "multi_step", "prompt": "P(S=3 given S>=P)?", "options": { "A": "4/5", "B": "11/20", "C": "3/5", "D": "11/15" }, "expected": [ "D" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0467, "default_truncated": false, "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).", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4725528059327327 }, { "exam": "final_sp23", "sample": "follow_up", "id": "17.1", "topic": "proof", "kind": "recognition", "prompt": "Why does E[I_A]=P(A)?", "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)." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0231, "default_truncated": false, "context": "I_A is1 when event A occurs and0 otherwise. Events need not be independent.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9977354405233358 }, { "exam": "final_sp23", "sample": "follow_up", "id": "17.2a", "topic": "proof", "kind": "recognition", "prompt": "Choose the valid pointwise identity and justification.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0039, "default_truncated": false, "context": "I_A is1 when event A occurs and0 otherwise. Events need not be independent.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9969362428195301 }, { "exam": "final_sp23", "sample": "follow_up", "id": "17.2b", "topic": "proof", "kind": "recognition", "prompt": "Choose the valid union identity and justification.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1523, "default_truncated": false, "context": "I_A is1 when event A occurs and0 otherwise. Events need not be independent.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9945848892933096 }, { "exam": "final_sp23", "sample": "follow_up", "id": "17.2c", "topic": "proof", "kind": "multi_step", "prompt": "Why are I_A,I_B uncorrelated exactly when A,B are independent?", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.079, "default_truncated": false, "context": "I_A is1 when event A occurs and0 otherwise. Events need not be independent.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9950601428144769 }, { "exam": "final_sp23", "sample": "follow_up", "id": "17.3", "topic": "proof", "kind": "multi_step", "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).", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0215, "default_truncated": false, "context": "I_A is1 when event A occurs and0 otherwise. Events need not be independent.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.999175427160517 }, { "exam": "final_sp23", "sample": "follow_up", "id": "18.1", "topic": "probability", "kind": "calculation", "prompt": "Normalization alpha?", "options": { "A": "1/[n(n+1)]", "B": "1/n", "C": "2/[n(n+1)]", "D": "6/[n(n+1)(2n+1)]" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1558, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8977076232361051 }, { "exam": "final_sp23", "sample": "follow_up", "id": "18.2", "topic": "expectation", "kind": "calculation", "prompt": "E[M], expressed in terms of alpha?", "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" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.015, "default_truncated": false, "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.", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4839056816586273 }, { "exam": "final_sp23", "sample": "follow_up", "id": "18.3", "topic": "probability", "kind": "calculation", "prompt": "CDF at integer m?", "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" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0026, "default_truncated": true, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9798420035310602 }, { "exam": "final_sp23", "sample": "follow_up", "id": "19.1", "topic": "transformations", "kind": "recognition", "prompt": "Y as a function of Theta?", "options": { "A": "tan(Theta)", "B": "cos(Theta)", "C": "sin(Theta)", "D": "1/tan(Theta)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0783, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.878840374749932 }, { "exam": "final_sp23", "sample": "follow_up", "id": "19.2", "topic": "transformations", "kind": "multi_step", "prompt": "CDF F_Y(y) for real y?", "options": { "A": "1/2+arctan(y)/pi", "B": "1/2+tan(y)/pi", "C": "arctan(y)/pi", "D": "1/[pi(1+y²)]" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0991, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7535943173531575 }, { "exam": "final_sp23", "sample": "follow_up", "id": "19.3", "topic": "transformations", "kind": "calculation", "prompt": "Density f_Y(y) for real y?", "options": { "A": "1/[pi(1+y²)]", "B": "1/2+arctan(y)/pi", "C": "exp(−y²/2)/sqrt(2pi)", "D": "1/[pi(1+y)]" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0927, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.906241703609325 }, { "exam": "final_sp23", "sample": "follow_up", "id": "20.1", "topic": "probability", "kind": "recognition", "prompt": "P(C_1)?", "options": { "A": "1/24", "B": "1/4", "C": "3/4", "D": "1/2" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0151, "default_truncated": false, "context": "Four guests receive their four distinct umbrellas in a uniformly random permutation. C_i means guest i receives their own umbrella.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7131835234204389 }, { "exam": "final_sp23", "sample": "follow_up", "id": "20.2", "topic": "proof", "kind": "recognition", "prompt": "Why is P(C_i)=P(C_j) for every i,j?", "options": { "A": "Symmetry", "B": "Transitivity", "C": "Independence", "D": "Disjointness" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1216, "default_truncated": false, "context": "Four guests receive their four distinct umbrellas in a uniformly random permutation. C_i means guest i receives their own umbrella.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9989665839028293 }, { "exam": "final_sp23", "sample": "follow_up", "id": "20.3", "topic": "counting", "kind": "calculation", "prompt": "Number of distinct pairwise-intersection terms in inclusion-exclusion?", "options": { "A": "16", "B": "12", "C": "6", "D": "4" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0081, "default_truncated": false, "context": "Four guests receive their four distinct umbrellas in a uniformly random permutation. C_i means guest i receives their own umbrella.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.41829521274664994 }, { "exam": "final_sp23", "sample": "follow_up", "id": "20.4", "topic": "probability", "kind": "multi_step", "prompt": "Probability at least one guest receives their own umbrella?", "options": { "A": "9/24", "B": "1−(3/4)^4", "C": "15/24", "D": "1/4" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0389, "default_truncated": false, "context": "Four guests receive their four distinct umbrellas in a uniformly random permutation. C_i means guest i receives their own umbrella.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.31911523504877376 }, { "exam": "final_sp23", "sample": "follow_up", "id": "21.1", "topic": "expectation", "kind": "multi_step", "prompt": "E[X]?", "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)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2554, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8548363271406563 }, { "exam": "final_sp23", "sample": "follow_up", "id": "21.2", "topic": "variance", "kind": "multi_step", "prompt": "Define a=((n−2)/n)^n, b=((n−3)/n)^n, c=((n−4)/n)^n. Var(X)?", "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" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0874, "default_truncated": true, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.394136452221266 }, { "exam": "final_sp24", "sample": "follow_up", "id": "1.a", "topic": "logic", "kind": "recognition", "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).", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1564, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8519528019683106 }, { "exam": "final_sp24", "sample": "follow_up", "id": "1.b", "topic": "logic", "kind": "calculation", "prompt": "Number of propositional forms on n variables up to logical equivalence, with AND, OR, NOT and no quantifiers?", "options": { "A": "n", "B": "2n", "C": "n²", "D": "2^n", "E": "2^(2^n)" }, "expected": [ "E" ], "jev_selected": [ "E" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0161, "default_truncated": false, "context": "", "semif_selected": [ "E" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9733314598629771 }, { "exam": "final_sp24", "sample": "follow_up", "id": "1.c", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "Odd prime p and integer n satisfy n=−n mod p. What residue must n have?", "options": { "A": "1", "B": "0", "C": "2", "D": "p/2", "E": "p−1" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0128, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8686829798623817 }, { "exam": "final_sp24", "sample": "follow_up", "id": "1.d", "topic": "computability", "kind": "recognition", "prompt": "A total program can decide for arbitrary P,input x and line number n whether that line is ever executed in P(x).", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1246, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6513548646660542 }, { "exam": "final_sp24", "sample": "follow_up", "id": "1.e", "topic": "countability", "kind": "recognition", "prompt": "For fixed nonnegative k, number of integer-coefficient polynomials of degree at most k?", "options": { "A": "Countably infinite", "B": "Uncountably infinite", "C": "Finite" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0677, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6775191155197939 }, { "exam": "final_sp24", "sample": "follow_up", "id": "1.f", "topic": "coding", "kind": "multi_step", "prompt": "Ten message symbols in 0..15, correct up to six corrupted transmitted symbols with Reed–Solomon. Smallest prime field size?", "options": { "A": "11", "B": "17", "C": "23", "D": "29" }, "expected": [ "C" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.003, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6554679826728957 }, { "exam": "final_sp24", "sample": "follow_up", "id": "1.g", "topic": "graphs", "kind": "recognition", "prompt": "A graph on n vertices is a disjoint union of k trees. Edge count?", "options": { "A": "k(n−1)", "B": "n−k+1", "C": "n−1", "D": "n−k", "E": "Not determined" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0209, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6380422292575714 }, { "exam": "final_sp24", "sample": "follow_up", "id": "1.h", "topic": "counting", "kind": "recognition", "prompt": "Positive n,m,k with k<=n,m. Evaluate sum(i=0..k) C(n,i)C(m,k−i).", "options": { "A": "C(n+m,k)", "B": "C(nm,k)", "C": "n!m!/(k!)²", "D": "2^(n+m)", "E": "None of these" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2615, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9961494665919228 }, { "exam": "final_sp24", "sample": "follow_up", "id": "1.i", "topic": "graphs", "kind": "multi_step", "prompt": "Shortest paths in the 7-cube from 0000000 to 1011001?", "options": { "A": "128", "B": "1", "C": "16", "D": "24", "E": "4" }, "expected": [ "D" ], "jev_selected": [ "E" ], "jev_correct": false, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0167, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.2610761978266843 }, { "exam": "final_sp24", "sample": "follow_up", "id": "1.j1", "topic": "graphs", "kind": "recognition", "prompt": "There exists a simple undirected graph with 7 vertices and 7 edges containing an Eulerian tour.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0121, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5621765008857981 }, { "exam": "final_sp24", "sample": "follow_up", "id": "1.j2", "topic": "graphs", "kind": "recognition", "prompt": "There exists a simple undirected graph with 4 vertices and 5 edges containing an Eulerian tour.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.3058, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6224593312018546 }, { "exam": "final_sp24", "sample": "follow_up", "id": "1.j3", "topic": "graphs", "kind": "recognition", "prompt": "There exists a simple undirected graph with 5 vertices and 7 edges containing an Eulerian tour.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0104, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5621765008857981 }, { "exam": "final_sp24", "sample": "follow_up", "id": "2.a", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "3^42 modulo 55?", "options": { "A": "27", "B": "1", "C": "9", "D": "3" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0075, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3690058809048272 }, { "exam": "final_sp24", "sample": "follow_up", "id": "2.b", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "Solve 17x=7 mod 42, x in 0..41.", "options": { "A": "7", "B": "35", "C": "17", "D": "5" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0439, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.58369784626725 }, { "exam": "final_sp24", "sample": "follow_up", "id": "2.c", "topic": "secret_sharing", "kind": "multi_step", "prompt": "GF(5), threshold three shares. P(1)=1,P(2)=1,P(4)=2; degree at most two. Secret P(0)?", "options": { "A": "1", "B": "2", "C": "4", "D": "3" }, "expected": [ "D" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.091, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.31712010892822357 }, { "exam": "final_sp24", "sample": "follow_up", "id": "2.d", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "Is f(x)=x² mod 7 a bijection on 0..6? Give a witness or inverse.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0154, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5577580730018756 }, { "exam": "final_sp24", "sample": "follow_up", "id": "3.a", "topic": "proof", "kind": "multi_step", "prompt": "For x≠1, prove (x^n−1)/(x−1)=sum(i=0..n−1)x^i by induction on n>=1.", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0454, "default_truncated": true, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9952134118323331 }, { "exam": "final_sp24", "sample": "follow_up", "id": "3.b", "topic": "proof", "kind": "multi_step", "prompt": "Prove 2^m−1 prime implies m prime for natural m. Hint: factor a geometric sum.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0055, "default_truncated": true, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9972748333496176 }, { "exam": "final_sp24", "sample": "follow_up", "id": "4.a", "topic": "matching", "kind": "multi_step", "prompt": "Classify (1,B),(2,C),(3,A),(4,D).", "options": { "A": "Stable and job-optimal", "B": "Unstable; blocking pair (1,A)", "C": "Unstable; blocking pair (4,C)", "D": "Stable and candidate-optimal" }, "expected": [ "D" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1184, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.41967679996925283 }, { "exam": "final_sp24", "sample": "follow_up", "id": "4.b", "topic": "matching", "kind": "multi_step", "prompt": "Classify (1,A),(2,B),(3,C),(4,D).", "options": { "A": "Unstable; blocking pair (4,C)", "B": "Stable and job-optimal", "C": "Stable and candidate-optimal", "D": "Unstable; blocking pair (1,B)" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1181, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5284271652417336 }, { "exam": "final_sp24", "sample": "follow_up", "id": "5.a", "topic": "bayes", "kind": "multi_step", "prompt": "Given first roll 3, probability second roll 2?", "options": { "A": "5/12", "B": "3/8", "C": "1/4", "D": "1/2" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0167, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4129089721781154 }, { "exam": "final_sp24", "sample": "follow_up", "id": "5.b", "topic": "bayes", "kind": "multi_step", "prompt": "Given first roll 2, probability second roll 2?", "options": { "A": "1/4", "B": "3/8", "C": "5/12", "D": "1/2" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1051, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4025265304015081 }, { "exam": "final_sp24", "sample": "follow_up", "id": "6.a", "topic": "probability", "kind": "calculation", "prompt": "Probability P(x)=Q(x) for every field element?", "options": { "A": "1/p^d", "B": "1/p^(d+1)", "C": "1/(d+1)", "D": "1/p" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0106, "default_truncated": false, "context": "P,Q have degree at most d with every coefficient independently uniform in GF(p), prime p>d.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.3872602052974305 }, { "exam": "final_sp24", "sample": "follow_up", "id": "6.b", "topic": "probability", "kind": "recognition", "prompt": "Probability P(0)=Q(0)?", "options": { "A": "1/p^(d+1)", "B": "1/p²", "C": "1/p", "D": "1/2" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.003, "default_truncated": false, "context": "P,Q have degree at most d with every coefficient independently uniform in GF(p), prime p>d.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4189648173687437 }, { "exam": "final_sp24", "sample": "follow_up", "id": "6.c", "topic": "probability", "kind": "multi_step", "prompt": "For 0<=k<=d−1, probability P(i)=Q(i) for every i=0,...,k?", "options": { "A": "(k+1)/p", "B": "1/p^k", "C": "1/p^(k+1)", "D": "1/p^(d−k)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0073, "default_truncated": false, "context": "P,Q have degree at most d with every coefficient independently uniform in GF(p), prime p>d.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.387932840448484 }, { "exam": "final_sp24", "sample": "follow_up", "id": "7.1", "topic": "probability", "kind": "multi_step", "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)?", "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" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1571, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.38104487026319805 }, { "exam": "final_sp24", "sample": "follow_up", "id": "8.a", "topic": "distributions", "kind": "calculation", "prompt": "Normalizing alpha?", "options": { "A": "1", "B": "1/2", "C": "2", "D": "3" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0682, "default_truncated": false, "context": "Density f(x)=alpha*x on [0,1], zero elsewhere; mu=E[X].", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6258811043410035 }, { "exam": "final_sp24", "sample": "follow_up", "id": "8.b", "topic": "probability", "kind": "calculation", "prompt": "P(X<=1/2), expressed in alpha?", "options": { "A": "alpha/8", "B": "alpha/4", "C": "alpha/2", "D": "alpha/16" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.014, "default_truncated": false, "context": "Density f(x)=alpha*x on [0,1], zero elsewhere; mu=E[X].", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.41331496088848524 }, { "exam": "final_sp24", "sample": "follow_up", "id": "8.c", "topic": "expectation", "kind": "calculation", "prompt": "E[X], expressed in alpha?", "options": { "A": "alpha/4", "B": "alpha/3", "C": "alpha/2", "D": "alpha²/3" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0061, "default_truncated": false, "context": "Density f(x)=alpha*x on [0,1], zero elsewhere; mu=E[X].", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5625620434340824 }, { "exam": "final_sp24", "sample": "follow_up", "id": "8.d", "topic": "moments", "kind": "calculation", "prompt": "Var(X), expressed in alpha and mu?", "options": { "A": "alpha/3−mu²", "B": "alpha/4−mu", "C": "alpha/4+mu²", "D": "alpha/4−mu²" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0019, "default_truncated": false, "context": "Density f(x)=alpha*x on [0,1], zero elsewhere; mu=E[X].", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5446947567423296 }, { "exam": "final_sp24", "sample": "follow_up", "id": "9.1", "topic": "moments", "kind": "recognition", "prompt": "Under the given assumptions, X and Z are independent.", "options": { "A": "Could be either", "B": "True", "C": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.3018, "default_truncated": false, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4942194415277362 }, { "exam": "final_sp24", "sample": "follow_up", "id": "9.2", "topic": "moments", "kind": "recognition", "prompt": "Under the given assumptions, E[X+2Y−Z]=0.", "options": { "A": "Could be either", "B": "True", "C": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0679, "default_truncated": false, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7009606824018237 }, { "exam": "final_sp24", "sample": "follow_up", "id": "9.3", "topic": "moments", "kind": "recognition", "prompt": "Under the given assumptions, P(Y>=10)<=1/10.", "options": { "A": "True", "B": "False", "C": "Could be either" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.5479, "default_truncated": false, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6094778270712142 }, { "exam": "final_sp24", "sample": "follow_up", "id": "9.4", "topic": "moments", "kind": "recognition", "prompt": "Under the given assumptions, E[XY]=2.", "options": { "A": "Could be either", "B": "False", "C": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.4783, "default_truncated": false, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7042046051774811 }, { "exam": "final_sp24", "sample": "follow_up", "id": "9.5", "topic": "moments", "kind": "recognition", "prompt": "Under the given assumptions, Var(X−Y)=20.", "options": { "A": "Could be either", "B": "True", "C": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2136, "default_truncated": false, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7295618969005627 }, { "exam": "final_sp24", "sample": "follow_up", "id": "9.6", "topic": "moments", "kind": "recognition", "prompt": "Under the given assumptions, Var(2X)=20.", "options": { "A": "True", "B": "Could be either", "C": "False" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1821, "default_truncated": false, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6541152022931706 }, { "exam": "final_sp24", "sample": "follow_up", "id": "9.7", "topic": "moments", "kind": "recognition", "prompt": "Under the given assumptions, P(|X|>=5)<=2/5.", "options": { "A": "True", "B": "False", "C": "Could be either" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.3244, "default_truncated": false, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5956031089205047 }, { "exam": "final_sp24", "sample": "follow_up", "id": "9.8", "topic": "moments", "kind": "recognition", "prompt": "Under the given assumptions, P(|X|>=sqrt(10))>0.", "options": { "A": "True", "B": "Could be either", "C": "False" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2503, "default_truncated": false, "context": "X,Y are independent; Y,Z are independent. E[X]=0,E[Y]=1,E[Z]=2. Each variance equals 10.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7869860421615985 }, { "exam": "final_sp24", "sample": "follow_up", "id": "10.1", "topic": "distributions", "kind": "multi_step", "prompt": "Xi are iid uniform on {1,2,3}; Sn=sum Xi. Choose a,b so (Sn−a)/b converges to N(0,1).", "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)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1084, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7312801107506337 }, { "exam": "final_sp24", "sample": "follow_up", "id": "11.a", "topic": "counting", "kind": "calculation", "prompt": "Five-key chords with exactly two white and three black?", "options": { "A": "C(88,5)", "B": "C(52,3)C(36,2)", "C": "52²*36³", "D": "C(52,2)C(36,3)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1188, "default_truncated": false, "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.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9651100186365297 }, { "exam": "final_sp24", "sample": "follow_up", "id": "11.b", "topic": "counting", "kind": "calculation", "prompt": "Chords from a fixed set of ten keys?", "options": { "A": "C(10,2)", "B": "2^10−11", "C": "2^10", "D": "2^10−1" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0643, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.45360944044158075 }, { "exam": "final_sp24", "sample": "follow_up", "id": "11.c", "topic": "counting", "kind": "calculation", "prompt": "Melodies of exactly twelve keys?", "options": { "A": "12^88", "B": "88!/76!", "C": "C(88,12)", "D": "88^12" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.005, "default_truncated": false, "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.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9178564625657176 }, { "exam": "final_sp24", "sample": "follow_up", "id": "11.d", "topic": "counting", "kind": "multi_step", "prompt": "Melodies with 40 white and 10 black notes, no adjacent black notes?", "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)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0173, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.3756998283262494 }, { "exam": "final_sp24", "sample": "follow_up", "id": "11.e", "topic": "counting", "kind": "calculation", "prompt": "Choose 200 keys as a multiset, unlimited repetition?", "options": { "A": "C(200,88)", "B": "C(287,87)", "C": "C(288,88)", "D": "88^200" }, "expected": [ "B" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0726, "default_truncated": false, "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.", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6319535882129181 }, { "exam": "final_sp24", "sample": "follow_up", "id": "11.f", "topic": "counting", "kind": "multi_step", "prompt": "Split a fixed 200-note melody into four consecutive parts, at least twenty notes each. Count?", "options": { "A": "C(199,3)", "B": "C(120,3)", "C": "C(123,3)", "D": "C(124,4)" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0138, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.33065623127838456 }, { "exam": "final_sp24", "sample": "follow_up", "id": "12.a", "topic": "probability", "kind": "calculation", "prompt": "Probability a fixed vertex is isolated?", "options": { "A": "1/8", "B": "1/4", "C": "1/n", "D": "1/2" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.007, "default_truncated": false, "context": "Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.49501294744843477 }, { "exam": "final_sp24", "sample": "follow_up", "id": "12.b", "topic": "expectation", "kind": "calculation", "prompt": "E[X]?", "options": { "A": "n/2", "B": "1/4", "C": "n/8", "D": "n/4" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0214, "default_truncated": false, "context": "Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.494531900087298 }, { "exam": "final_sp24", "sample": "follow_up", "id": "12.c", "topic": "probability", "kind": "calculation", "prompt": "Probability adjacent vertices are both isolated?", "options": { "A": "1/4", "B": "1/2", "C": "1/16", "D": "1/8" }, "expected": [ "D" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0105, "default_truncated": false, "context": "Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.40544191343510666 }, { "exam": "final_sp24", "sample": "follow_up", "id": "12.d", "topic": "probability", "kind": "calculation", "prompt": "For two nonadjacent vertices (when present), probability both isolated?", "options": { "A": "1/4", "B": "1/16", "C": "1/32", "D": "1/8" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0142, "default_truncated": false, "context": "Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3824574411254738 }, { "exam": "final_sp24", "sample": "follow_up", "id": "12.e", "topic": "moments", "kind": "multi_step", "prompt": "Var(X)?", "options": { "A": "5n/16", "B": "3n/16", "C": "n/4", "D": "n²/16" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0049, "default_truncated": false, "context": "Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4566302152970893 }, { "exam": "final_sp24", "sample": "follow_up", "id": "12.f", "topic": "bounds", "kind": "multi_step", "prompt": "Chebyshev upper bound on P(X>=n/2), clipped to probability range?", "options": { "A": "min(1,5/(4n))", "B": "min(1,5/n)", "C": "min(1,20/n)", "D": "min(1,3/n)" }, "expected": [ "B" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0026, "default_truncated": false, "context": "Cycle graph on n>=3 vertices. Each edge independently fails with probability 1/2 and is removed. X counts isolated vertices.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3395893495876965 }, { "exam": "final_sp24", "sample": "follow_up", "id": "13.a", "topic": "distributions", "kind": "recognition", "prompt": "Distribution of arrivals in one hour?", "options": { "A": "Poisson(3)", "B": "Poisson(20)", "C": "Bin(60,1/20)", "D": "Geom(1/20)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2393, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9250119760330004 }, { "exam": "final_sp24", "sample": "follow_up", "id": "13.b", "topic": "distributions", "kind": "recognition", "prompt": "Distribution of minutes a dolphin stays, counting its first minute?", "options": { "A": "Geom(p) on 1,2,...", "B": "Geom(q) on 1,2,...", "C": "Exp(rate q)", "D": "Poisson(1/q)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0624, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9172459880206503 }, { "exam": "final_sp24", "sample": "follow_up", "id": "13.c", "topic": "probability", "kind": "calculation", "prompt": "Probability a dolphin stays exactly five minutes?", "options": { "A": "(1−q)^4*q", "B": "(1−q)^4", "C": "(1−q)^5*q", "D": "q^4*(1−q)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0485, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5538108313577431 }, { "exam": "final_sp24", "sample": "follow_up", "id": "13.d", "topic": "expectation", "kind": "recognition", "prompt": "Expected minutes a dolphin stays?", "options": { "A": "q", "B": "1/p", "C": "1/q", "D": "(1−q)/q" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0121, "default_truncated": false, "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.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7516486191399615 }, { "exam": "final_sp24", "sample": "follow_up", "id": "13.e", "topic": "probability", "kind": "calculation", "prompt": "Given a twenty-minute stay, probability of exactly k tricks?", "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)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0234, "default_truncated": false, "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.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6012955320530238 }, { "exam": "final_sp24", "sample": "follow_up", "id": "13.f", "topic": "expectation", "kind": "calculation", "prompt": "Given a twenty-minute stay, expected tricks?", "options": { "A": "20q", "B": "20p", "C": "20/p", "D": "p/q" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0127, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7098956919259457 }, { "exam": "final_sp24", "sample": "follow_up", "id": "13.g", "topic": "expectation", "kind": "multi_step", "prompt": "Unconditional expected total tricks per dolphin?", "options": { "A": "1/(pq)", "B": "p/(1−q)", "C": "q/p", "D": "p/q" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0169, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7934068107443792 }, { "exam": "final_sp24", "sample": "follow_up", "id": "14.a", "topic": "markov", "kind": "multi_step", "prompt": "Is it irreducible? Justify.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.02, "default_truncated": false, "context": "Markov chain: 1->2 with probability1; 2->3 with1; 3->1 and 3->4 each1/2; 4->5 with1; 5->2 with1.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7863841160740562 }, { "exam": "final_sp24", "sample": "follow_up", "id": "14.b", "topic": "markov", "kind": "multi_step", "prompt": "Is it aperiodic? Justify.", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1743, "default_truncated": false, "context": "Markov chain: 1->2 with probability1; 2->3 with1; 3->1 and 3->4 each1/2; 4->5 with1; 5->2 with1.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8025063060147087 }, { "exam": "final_sp24", "sample": "follow_up", "id": "14.c", "topic": "markov", "kind": "calculation", "prompt": "Stationary pi1=pi5=1/7. Find pi2.", "options": { "A": "3/7", "B": "1/7", "C": "2/7", "D": "1/5" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0017, "default_truncated": false, "context": "Markov chain: 1->2 with probability1; 2->3 with1; 3->1 and 3->4 each1/2; 4->5 with1; 5->2 with1.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5533142800849036 }, { "exam": "final_sp24", "sample": "follow_up", "id": "14.d", "topic": "markov", "kind": "calculation", "prompt": "bi is expected hitting time of state1 from i. Given b3=5, find b5.", "options": { "A": "5", "B": "10", "C": "7", "D": "6" }, "expected": [ "C" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0241, "default_truncated": false, "context": "Markov chain: 1->2 with probability1; 2->3 with1; 3->1 and 3->4 each1/2; 4->5 with1; 5->2 with1.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.36617639865290497 }, { "exam": "final_sp24", "sample": "follow_up", "id": "14.e", "topic": "markov", "kind": "multi_step", "prompt": "Replace 4->5 by a probability-one self-loop at4. Expected hitting time of state1 from5?", "options": { "A": "14", "B": "Infinity", "C": "5", "D": "7" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0368, "default_truncated": false, "context": "Markov chain: 1->2 with probability1; 2->3 with1; 3->1 and 3->4 each1/2; 4->5 with1; 5->2 with1.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5725761621239776 }, { "exam": "final_sp25", "sample": "follow_up", "id": "2.1", "topic": "course_context", "kind": "recognition", "prompt": "Which TA has a birthday on May 16?", "options": { "A": "Andy", "B": "Malia", "C": "Evelyn", "D": "Jay" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.12, "default_truncated": false, "context": "The last question of this same exam states: it is Jay’s birthday today (May 16).", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.65053067527017 }, { "exam": "final_sp25", "sample": "follow_up", "id": "3.1a", "topic": "logic", "kind": "recognition", "prompt": "Complete: (P implies NOT Q) is equivalent to (NOT P OR __).", "options": { "A": "Q", "B": "P", "C": "NOT Q", "D": "True" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0842, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8954049873547066 }, { "exam": "final_sp25", "sample": "follow_up", "id": "3.1b", "topic": "logic", "kind": "recognition", "prompt": "P AND (Q OR R) is equivalent to (P OR Q) AND (P OR R).", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.289, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9971990730328789 }, { "exam": "final_sp25", "sample": "follow_up", "id": "3.1c", "topic": "logic", "kind": "recognition", "prompt": "Truth status of P AND (NOT P OR Q) AND (NOT P OR NOT Q)?", "options": { "A": "Could be either", "B": "False", "C": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1287, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.750611584184323 }, { "exam": "final_sp25", "sample": "follow_up", "id": "3.2ai", "topic": "logic", "kind": "recognition", "prompt": "P(n) holds for every even n.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2844, "default_truncated": false, "context": "N includes zero. For every n, P(n)=>Q(n+1) and Q(n)=>P(n+1). Also P(0) holds.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7057850278370112 }, { "exam": "final_sp25", "sample": "follow_up", "id": "3.2aii", "topic": "logic", "kind": "recognition", "prompt": "For every n, NOT Q(n+1) implies NOT P(n).", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0002, "default_truncated": false, "context": "N includes zero. For every n, P(n)=>Q(n+1) and Q(n)=>P(n+1). Also P(0) holds.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9149009549929797 }, { "exam": "final_sp25", "sample": "follow_up", "id": "3.2aiii", "topic": "logic", "kind": "recognition", "prompt": "Q(0) implies Q(n) for every n.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1034, "default_truncated": false, "context": "N includes zero. For every n, P(n)=>Q(n+1) and Q(n)=>P(n+1). Also P(0) holds.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8933094060543487 }, { "exam": "final_sp25", "sample": "follow_up", "id": "3.2b", "topic": "logic", "kind": "recognition", "prompt": "NOT forall n [P(n) OR Q(n)] implies exists n NOT P(n).", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1867, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8670357598021706 }, { "exam": "final_sp25", "sample": "follow_up", "id": "4.1", "topic": "proof", "kind": "multi_step", "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.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.014, "default_truncated": true, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9892741382289654 }, { "exam": "final_sp25", "sample": "follow_up", "id": "4.2a", "topic": "graphs", "kind": "recognition", "prompt": "Up to isomorphism, give all trees on four vertices as edge sets on {1,2,3,4}.", "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}" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0342, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8724986919054307 }, { "exam": "final_sp25", "sample": "follow_up", "id": "4.2b", "topic": "graphs", "kind": "recognition", "prompt": "Every three-vertex tree is a subgraph of K3 (not necessarily induced).", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1024, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8933094060543487 }, { "exam": "final_sp25", "sample": "follow_up", "id": "4.2c", "topic": "proof", "kind": "multi_step", "prompt": "Prove a simple graph of minimum degree d contains every k-vertex tree for 1<=k<=d.", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0714, "default_truncated": true, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9614836241444372 }, { "exam": "final_sp25", "sample": "follow_up", "id": "5.1", "topic": "matching", "kind": "recognition", "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?", "options": { "A": "n−1", "B": "n", "C": "1", "D": "n²" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0851, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.40295183295074993 }, { "exam": "final_sp25", "sample": "follow_up", "id": "5.2", "topic": "matching", "kind": "recognition", "prompt": "If all jobs have the same strict ranking, the stable matching is unique.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1148, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8933094060543487 }, { "exam": "final_sp25", "sample": "follow_up", "id": "5.3", "topic": "matching", "kind": "recognition", "prompt": "If all candidates have the same strict ranking, the stable matching is unique.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1014, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9149009549929797 }, { "exam": "final_sp25", "sample": "follow_up", "id": "5.4a", "topic": "expectation", "kind": "multi_step", "prompt": "All preference lists are independently uniform permutations, n jobs and n candidates. Expected number of candidates receiving exactly i first-day proposals?", "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)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2263, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3287145937103622 }, { "exam": "final_sp25", "sample": "follow_up", "id": "5.4b", "topic": "probability", "kind": "multi_step", "prompt": "For n=2 with independently uniform strict preference lists, probability both possible matchings are stable?", "options": { "A": "1/16", "B": "1/4", "C": "1/8", "D": "1/2" }, "expected": [ "C" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0341, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4160513129390438 }, { "exam": "final_sp25", "sample": "follow_up", "id": "6.1", "topic": "graphs", "kind": "calculation", "prompt": "A connected simple planar embedding has n vertices and every face boundary has four edge-sides. Edge count?", "options": { "A": "2n−2", "B": "3n−6", "C": "n−1", "D": "2n−4" }, "expected": [ "D" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.055, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6274569912315332 }, { "exam": "final_sp25", "sample": "follow_up", "id": "6.2a", "topic": "graphs", "kind": "recognition", "prompt": "Number of vertices of F?", "options": { "A": "v+f+e", "B": "2v", "C": "v+f", "D": "v+e" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2802, "default_truncated": false, "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.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8139272079531032 }, { "exam": "final_sp25", "sample": "follow_up", "id": "6.2b", "topic": "graphs", "kind": "multi_step", "prompt": "Number of edges of F?", "options": { "A": "3(v+f)−6", "B": "3v−6", "C": "2e", "D": "e+f" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0359, "default_truncated": false, "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.", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5290611475209448 }, { "exam": "final_sp25", "sample": "follow_up", "id": "6.3", "topic": "graphs", "kind": "recognition", "prompt": "Remove one edge from a tree. How many connected components result?", "options": { "A": "2", "B": "1", "C": "3", "D": "0" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0606, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9846862635883732 }, { "exam": "final_sp25", "sample": "follow_up", "id": "6.4", "topic": "graphs", "kind": "calculation", "prompt": "Minimum edges removed from K(2n) to make it bipartite?", "options": { "A": "n−1", "B": "n²", "C": "n(n−1)", "D": "2n(n−1)" }, "expected": [ "C" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0091, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5140406332527597 }, { "exam": "final_sp25", "sample": "follow_up", "id": "6.5", "topic": "graphs", "kind": "recognition", "prompt": "Minimum vertex colors for a tree with at least one edge?", "options": { "A": "1", "B": "d", "C": "2", "D": "d+1" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0518, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.972380346410241 }, { "exam": "final_sp25", "sample": "follow_up", "id": "6.6", "topic": "graphs", "kind": "recognition", "prompt": "Minimum edge colors for a tree of maximum degree d?", "options": { "A": "2d", "B": "d", "C": "d+1", "D": "2" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0403, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5576165227439774 }, { "exam": "final_sp25", "sample": "follow_up", "id": "6.7", "topic": "graphs", "kind": "recognition", "prompt": "Maximum edges in a simple n-vertex graph, n>=3, where every cycle has length at least n?", "options": { "A": "2n−2", "B": "n", "C": "n−1", "D": "n+1" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0424, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6197193300843672 }, { "exam": "final_sp25", "sample": "follow_up", "id": "6.8", "topic": "graphs", "kind": "recognition", "prompt": "Length in edges of an Eulerian tour in a graph with n vertices,m edges?", "options": { "A": "n", "B": "n−1", "C": "m", "D": "2m" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0367, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9685453248995409 }, { "exam": "final_sp25", "sample": "follow_up", "id": "6.9a", "topic": "graphs", "kind": "recognition", "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?", "options": { "A": "k", "B": "1", "C": "k−1", "D": "2" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0509, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4279733272551463 }, { "exam": "final_sp25", "sample": "follow_up", "id": "6.9b", "topic": "graphs", "kind": "recognition", "prompt": "Delete the edges of a simple k-cycle from a connected graph, retaining all vertices. Maximum possible remaining component count?", "options": { "A": "k+1", "B": "2k", "C": "k", "D": "k−1" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0262, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6206149522767143 }, { "exam": "final_sp25", "sample": "follow_up", "id": "7.1", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "2^36 mod 7?", "options": { "A": "1", "B": "2", "C": "4", "D": "0" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0507, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4740698200688341 }, { "exam": "final_sp25", "sample": "follow_up", "id": "7.2", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "2^36 mod 35?", "options": { "A": "4", "B": "16", "C": "1", "D": "8" }, "expected": [ "C" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0568, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.511193498646212 }, { "exam": "final_sp25", "sample": "follow_up", "id": "7.5", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "gcd(385,70)?", "options": { "A": "5", "B": "35", "C": "70", "D": "7" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0318, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8570181706881451 }, { "exam": "final_sp25", "sample": "follow_up", "id": "7.7a", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Number of distinct square residues modulo 3, including zero?", "options": { "A": "2", "B": "3", "C": "1", "D": "0" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0279, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9418162722902653 }, { "exam": "final_sp25", "sample": "follow_up", "id": "7.7c", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Number of distinct square residues modulo 15, including zero?", "options": { "A": "6", "B": "8", "C": "4", "D": "7" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.016, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5858957832727165 }, { "exam": "final_sp25", "sample": "follow_up", "id": "7.3", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "If x=0 mod d, then bx−kd=0 mod d for all integers b,k.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.4941, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9859363729567544 }, { "exam": "final_sp25", "sample": "follow_up", "id": "7.4", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "If x=1 mod d, then bx−kd=1 mod d for all integers b,k.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0053, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5621765008857981 }, { "exam": "final_sp25", "sample": "follow_up", "id": "7.6a", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "Assume d>1. Give the least positive t, nonzero modulo m, with (2+t)a=b mod m.", "options": { "A": "m/d", "B": "d", "C": "a/d", "D": "m" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0068, "default_truncated": false, "context": "gcd(a,m)=d and 2a=b mod m, with positive modulus m.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5397401132359269 }, { "exam": "final_sp25", "sample": "follow_up", "id": "7.6b", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "b must be even.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.6273, "default_truncated": false, "context": "gcd(a,m)=d and 2a=b mod m, with positive modulus m.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6513548646660542 }, { "exam": "final_sp25", "sample": "follow_up", "id": "7.6c", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "d divides b.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.5898, "default_truncated": false, "context": "gcd(a,m)=d and 2a=b mod m, with positive modulus m.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9241418199787566 }, { "exam": "final_sp25", "sample": "follow_up", "id": "7.6d", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "How many distinct solutions modulo m does ax=b mod m have, given it is solvable?", "options": { "A": "d", "B": "m/d", "C": "1", "D": "m" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0567, "default_truncated": false, "context": "gcd(a,m)=d and 2a=b mod m, with positive modulus m.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9689740617230034 }, { "exam": "final_sp25", "sample": "follow_up", "id": "7.7b", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Prime p>2: count square residues including zero.", "options": { "A": "(p−1)/2", "B": "p", "C": "(p+1)/2", "D": "p−1" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1515, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9758563097952977 }, { "exam": "final_sp25", "sample": "follow_up", "id": "7.7d", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "Distinct primes p,q>2: count square residues modulo pq, including zero. Hint: CRT.", "options": { "A": "(p−1)(q−1)/4", "B": "(p+q)/2", "C": "(pq+1)/2", "D": "(p+1)(q+1)/4" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0234, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6764172107664038 }, { "exam": "final_sp25", "sample": "follow_up", "id": "8.1", "topic": "polynomials", "kind": "multi_step", "prompt": "GF(5) interpolation yields P=x²+4x+1. One normalized Lagrange basis polynomial is Delta=x²+3x+2. Recover the three interpolation points.", "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)}" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0316, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4282793553102726 }, { "exam": "final_sp25", "sample": "follow_up", "id": "8.2", "topic": "polynomials", "kind": "recognition", "prompt": "Every polynomial of degree exactly two is a bijection over every prime field.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.011, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9890130573694068 }, { "exam": "final_sp25", "sample": "follow_up", "id": "8.3", "topic": "polynomials", "kind": "recognition", "prompt": "Every polynomial of degree exactly one is a bijection over a prime field.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0022, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5621765008857981 }, { "exam": "final_sp25", "sample": "follow_up", "id": "8.4a", "topic": "polynomials", "kind": "calculation", "prompt": "Where do 2x+3 and 3x+2 intersect over GF(5)?", "options": { "A": "4", "B": "0", "C": "1", "D": "2" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.04, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.43363430882835424 }, { "exam": "final_sp25", "sample": "follow_up", "id": "8.4b", "topic": "polynomials", "kind": "recognition", "prompt": "Distinct formal polynomials of degrees dP,dQ over GF(p), p>max(dP,dQ): tight general upper bound on intersection coordinates?", "options": { "A": "dP+dQ", "B": "p", "C": "min(dP,dQ)", "D": "max(dP,dQ)" }, "expected": [ "D" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0273, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8605955069617338 }, { "exam": "final_sp25", "sample": "follow_up", "id": "8.4c", "topic": "polynomials", "kind": "calculation", "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).", "options": { "A": "(p−1)p", "B": "p²", "C": "1", "D": "p" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0046, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5167798295882955 }, { "exam": "final_sp25", "sample": "follow_up", "id": "8.5a", "topic": "coding", "kind": "recognition", "prompt": "Encode n message symbols as polynomial coefficients. Degree bound of message P?", "options": { "A": "At most n+2k", "B": "At most n−1", "C": "Exactly n", "D": "Exactly k" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0842, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7126901921857878 }, { "exam": "final_sp25", "sample": "follow_up", "id": "8.5b", "topic": "coding", "kind": "recognition", "prompt": "How many distinct evaluations suffice to correct k corruptions in n message symbols?", "options": { "A": "n+2k−1", "B": "2n+k", "C": "n+k", "D": "n+2k" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0223, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4641394801214649 }, { "exam": "final_sp25", "sample": "follow_up", "id": "8.5c", "topic": "proof", "kind": "multi_step", "prompt": "Why does E(i)(P(i)−ri)=0 at each transmitted coordinate, where E vanishes at error locations?", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0088, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9956089500028346 }, { "exam": "final_sp25", "sample": "follow_up", "id": "9.1", "topic": "counting", "kind": "calculation", "prompt": "Distinct arrangements of BAA?", "options": { "A": "1", "B": "3", "C": "6", "D": "2" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0914, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5589294450415401 }, { "exam": "final_sp25", "sample": "follow_up", "id": "9.2", "topic": "counting", "kind": "calculation", "prompt": "Distinct arrangements of 126?", "options": { "A": "1", "B": "3", "C": "9", "D": "6" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0231, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4658923272442273 }, { "exam": "final_sp25", "sample": "follow_up", "id": "9.3", "topic": "counting", "kind": "calculation", "prompt": "n distinct bears, k distinct children,n>k. Assign one bear each, without reuse. Count?", "options": { "A": "n^k", "B": "k^n", "C": "C(n,k)", "D": "n!/(n−k)!" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0317, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7932518446560699 }, { "exam": "final_sp25", "sample": "follow_up", "id": "9.4", "topic": "counting", "kind": "calculation", "prompt": "Bipartite labeled graph with fixed parts each of size n and exactly m edges. Count?", "options": { "A": "2^(n²)", "B": "C(n²,m)", "C": "n^(2m)", "D": "C(2n,m)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.03, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5290911484236372 }, { "exam": "final_sp25", "sample": "follow_up", "id": "9.5", "topic": "counting", "kind": "calculation", "prompt": "Number of functions from p residues to p residues?", "options": { "A": "p^p", "B": "p!", "C": "2^p", "D": "p²" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0501, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9992948676074792 }, { "exam": "final_sp25", "sample": "follow_up", "id": "9.6", "topic": "counting", "kind": "calculation", "prompt": "Number of bijections from p residues to themselves?", "options": { "A": "p²", "B": "p^p", "C": "p", "D": "p!" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0368, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8921936172883431 }, { "exam": "final_sp25", "sample": "follow_up", "id": "9.7", "topic": "counting", "kind": "calculation", "prompt": "Form three named teams of four distinct players from 15 people, no overlap. Count?", "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)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0093, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9774987170943725 }, { "exam": "final_sp25", "sample": "follow_up", "id": "10.1", "topic": "proof", "kind": "multi_step", "prompt": "Give a combinatorial proof of sum(i=0..n) C(n,i)C(n,n−i)=C(2n,n).", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0881, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9978928356402774 }, { "exam": "final_sp25", "sample": "follow_up", "id": "10.2", "topic": "counting", "kind": "recognition", "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?", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0003, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9806526398501125 }, { "exam": "final_sp25", "sample": "follow_up", "id": "11.1", "topic": "countability", "kind": "recognition", "prompt": "The power set of N has the same cardinality as [0,1].", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0907, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9579122720843811 }, { "exam": "final_sp25", "sample": "follow_up", "id": "11.2", "topic": "countability", "kind": "recognition", "prompt": "A finite Cartesian product of countable sets is countable.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1485, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9046505351008906 }, { "exam": "final_sp25", "sample": "follow_up", "id": "11.3a", "topic": "countability", "kind": "recognition", "prompt": "A is countable.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.4014, "default_truncated": false, "context": "A=Z×Z; f:A->C is onto and g:C->B is injective.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7057850278370112 }, { "exam": "final_sp25", "sample": "follow_up", "id": "11.3b", "topic": "countability", "kind": "recognition", "prompt": "These assumptions imply |A| is strictly smaller than |B|.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1849, "default_truncated": false, "context": "A=Z×Z; f:A->C is onto and g:C->B is injective.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8519528019683106 }, { "exam": "final_sp25", "sample": "follow_up", "id": "11.4", "topic": "functions", "kind": "recognition", "prompt": "Give a bijection (0,1]->[1,infinity).", "options": { "A": "f(x)=1/x", "B": "f(x)=1/(1−x)", "C": "f(x)=x+1", "D": "f(x)=−ln(x)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0108, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.63673481074447 }, { "exam": "final_sp25", "sample": "follow_up", "id": "12.1", "topic": "computability", "kind": "recognition", "prompt": "There is an algorithm deciding whether P(x) halts within |x|^k steps, for arbitrary P, nonempty finite x and nonnegative integer k.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0474, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5621765008857981 }, { "exam": "final_sp25", "sample": "follow_up", "id": "12.2", "topic": "computability", "kind": "multi_step", "prompt": "Hypothetical HaltsOnEmpty decides whether a program halts on empty input. Fill: Halt(P,x) defines inner(y): __1__; then returns HaltsOnEmpty(__2__).", "options": { "A": "(return True; inner)", "B": "(P(x); inner)", "C": "(P(x); x)", "D": "(P(y); P)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0002, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.408483547916507 }, { "exam": "final_sp25", "sample": "follow_up", "id": "13.1", "topic": "bounds", "kind": "recognition", "prompt": "Sharp upper bound on P(A union B) using only a=P(A),b=P(B)?", "options": { "A": "a+b", "B": "a*b", "C": "max(a,b)", "D": "min(1,a+b)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0591, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9955894184027475 }, { "exam": "final_sp25", "sample": "follow_up", "id": "13.2", "topic": "bounds", "kind": "recognition", "prompt": "Sharp lower bound on P(A union B) using only a=P(A),b=P(B)?", "options": { "A": "a+b", "B": "max(a,b)", "C": "min(a,b)", "D": "a*b" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.139, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7926142236535423 }, { "exam": "final_sp25", "sample": "follow_up", "id": "13.3", "topic": "probability", "kind": "recognition", "prompt": "A is a subset of B and both have positive probability. P(A given B)?", "options": { "A": "1", "B": "P(B)/P(A)", "C": "P(A)/P(B)", "D": "P(A)P(B)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0245, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5238779483998147 }, { "exam": "final_sp25", "sample": "follow_up", "id": "13.4", "topic": "probability", "kind": "recognition", "prompt": "A is a subset of B and P(A)>0. P(B given A)?", "options": { "A": "0", "B": "P(B)", "C": "P(A)/P(B)", "D": "1" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0274, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8112722501325894 }, { "exam": "final_sp25", "sample": "follow_up", "id": "13.5a", "topic": "bounds", "kind": "recognition", "prompt": "Sharp upper bound on P(A intersection C)?", "options": { "A": "a+c", "B": "min(a,b,c)", "C": "alpha", "D": "min(a,c)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.006, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.2866128117777278 }, { "exam": "final_sp25", "sample": "follow_up", "id": "13.5b", "topic": "bounds", "kind": "multi_step", "prompt": "Sharp lower bound on P(A intersection C) using all the given quantities?", "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" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0079, "default_truncated": false, "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.", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5057615653588341 }, { "exam": "final_sp25", "sample": "follow_up", "id": "13.5c", "topic": "proof", "kind": "multi_step", "prompt": "Justify a sharp lower bound using the partition into B and its complement.", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0938, "default_truncated": true, "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.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9919154972715989 }, { "exam": "final_sp25", "sample": "follow_up", "id": "14.1", "topic": "bayes", "kind": "multi_step", "prompt": "P(A given B)?", "options": { "A": "5/24", "B": "13/60", "C": "1/6", "D": "1/4" }, "expected": [ "B" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0165, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.38812073416459414 }, { "exam": "final_sp25", "sample": "follow_up", "id": "14.2", "topic": "bayes", "kind": "multi_step", "prompt": "P(C given A)?", "options": { "A": "2/5", "B": "1/4", "C": "1/2", "D": "3/5" }, "expected": [ "D" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0067, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.28850952576306876 }, { "exam": "final_sp25", "sample": "follow_up", "id": "15.1", "topic": "probability", "kind": "calculation", "prompt": "Independent Bernoulli X,Y each have mean 1/2. P(X=Y)?", "options": { "A": "1", "B": "1/4", "C": "1/2", "D": "3/4" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1765, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4294135902937379 }, { "exam": "final_sp25", "sample": "follow_up", "id": "15.2a", "topic": "moments", "kind": "multi_step", "prompt": "E[XY]?", "options": { "A": "1/4", "B": "3/8", "C": "1/2", "D": "1/8" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0097, "default_truncated": false, "context": "Bernoulli X,Y each have mean 1/2 and correlation 1/2.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8313516547495263 }, { "exam": "final_sp25", "sample": "follow_up", "id": "15.2b", "topic": "probability", "kind": "multi_step", "prompt": "P(X=Y)?", "options": { "A": "1/2", "B": "3/4", "C": "3/8", "D": "1/4" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1562, "default_truncated": false, "context": "Bernoulli X,Y each have mean 1/2 and correlation 1/2.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6739467757277202 }, { "exam": "final_sp25", "sample": "follow_up", "id": "16.1", "topic": "expectation", "kind": "recognition", "prompt": "E[Xi]?", "options": { "A": "p(1−p)", "B": "1−p", "C": "p", "D": "np" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0327, "default_truncated": false, "context": "Sample n people independently from a population with positive-test probability p. Xi indicates a positive test.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9683031646393272 }, { "exam": "final_sp25", "sample": "follow_up", "id": "16.2", "topic": "bounds", "kind": "recognition", "prompt": "Sharp p-independent upper bound on Var(Xi)?", "options": { "A": "1", "B": "p", "C": "1/4", "D": "1/2" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0109, "default_truncated": false, "context": "Sample n people independently from a population with positive-test probability p. Xi indicates a positive test.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4258861893623922 }, { "exam": "final_sp25", "sample": "follow_up", "id": "16.3", "topic": "bounds", "kind": "multi_step", "prompt": "50 of 100 independently sampled people test positive. Using Chebyshev and the worst-case Bernoulli variance, give a 95% confidence interval for p.", "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)]" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0346, "default_truncated": false, "context": "Sample n people independently from a population with positive-test probability p. Xi indicates a positive test.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.41795982077644706 }, { "exam": "final_sp25", "sample": "follow_up", "id": "17.1", "topic": "proof", "kind": "multi_step", "prompt": "Prove every tournament (one directed edge per unordered vertex pair) has a Hamiltonian path.", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.3951, "default_truncated": true, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9700906790035261 }, { "exam": "final_sp25", "sample": "follow_up", "id": "17.2a", "topic": "counting", "kind": "calculation", "prompt": "Orient every edge of a complete n-vertex graph independently with a fair coin. Number of possible tournaments?", "options": { "A": "C(n,2)", "B": "2^C(n,2)", "C": "2^(n²)", "D": "n!" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0744, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9940676430684721 }, { "exam": "final_sp25", "sample": "follow_up", "id": "17.2b", "topic": "expectation", "kind": "multi_step", "prompt": "Orient every edge of a complete n-vertex graph independently with a fair coin. Expected Hamiltonian paths in the resulting tournament?", "options": { "A": "n!/2^(n−1)", "B": "n!/2^C(n,2)", "C": "2^(n−1)", "D": "n/2" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.048, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5279255085818403 }, { "exam": "final_sp25", "sample": "follow_up", "id": "18.1", "topic": "regression", "kind": "recognition", "prompt": "The best affine least-squares estimator of Y from X passes through the origin iff E[Y] equals what?", "options": { "A": "Cov(X,Y)/E[X]", "B": "0", "C": "Cov(X,Y)*E[X]/Var(X)", "D": "E[X]" }, "expected": [ "C" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0535, "default_truncated": false, "context": "Random variables have finite second moments; Var(X)>0.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.36253010780209793 }, { "exam": "final_sp25", "sample": "follow_up", "id": "18.2", "topic": "moments", "kind": "recognition", "prompt": "E[X²] in terms of variance and mean?", "options": { "A": "Var(X)−E[X]²", "B": "E[X]²", "C": "Var(X)+E[X]²", "D": "Var(X)²+E[X]" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0067, "default_truncated": false, "context": "Random variables have finite second moments; Var(X)>0.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9960661868778539 }, { "exam": "final_sp25", "sample": "follow_up", "id": "18.3", "topic": "moments", "kind": "recognition", "prompt": "If E[X]=E[Y]=t, what t makes Cov(X,Y)=E[XY]?", "options": { "A": "0", "B": "1/2", "C": "−1", "D": "1" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0969, "default_truncated": false, "context": "Random variables have finite second moments; Var(X)>0.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6395831334705606 }, { "exam": "final_sp25", "sample": "follow_up", "id": "18.4", "topic": "regression", "kind": "recognition", "prompt": "If X,Y are independent, best affine least-squares estimator of Y given X?", "options": { "A": "E[XY]", "B": "E[X]", "C": "E[Y]", "D": "X" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1172, "default_truncated": false, "context": "Random variables have finite second moments; Var(X)>0.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.739652598526696 }, { "exam": "final_sp25", "sample": "follow_up", "id": "19.1", "topic": "distributions", "kind": "recognition", "prompt": "Independent X,Y~Bin(n,p). Distribution of X+Y?", "options": { "A": "Poisson(2np)", "B": "Bin(n,2p)", "C": "Bin(2n,p)", "D": "Bin(2n,p²)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1017, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9515716343688497 }, { "exam": "final_sp25", "sample": "follow_up", "id": "19.2", "topic": "distributions", "kind": "recognition", "prompt": "Independent X,Y~Geom(p), counting trials from 1. Distribution of min(X,Y)?", "options": { "A": "Geom(2p−p²)", "B": "Geom(p²)", "C": "Geom(p)", "D": "Geom(2p)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0209, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9228212367499056 }, { "exam": "final_sp25", "sample": "follow_up", "id": "19.3", "topic": "distributions", "kind": "recognition", "prompt": "Fixed lambda>0 and nonnegative integer i. Limit as n->infinity of C(n,i)(lambda/n)^i(1−lambda/n)^(n−i)?", "options": { "A": "exp(−lambda)", "B": "exp(−i)*i^lambda/lambda!", "C": "lambda^i/i!", "D": "exp(−lambda)*lambda^i/i!" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0099, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9659418143921974 }, { "exam": "final_sp25", "sample": "follow_up", "id": "19.4a", "topic": "distributions", "kind": "recognition", "prompt": "Marginal density of X?", "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" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0147, "default_truncated": false, "context": "First X~Uniform(0,1]; conditional on X=x, draw Y~Uniform[0,x].", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.730384050321913 }, { "exam": "final_sp25", "sample": "follow_up", "id": "19.4b", "topic": "distributions", "kind": "recognition", "prompt": "Describe the joint support region, replacing the requested shading.", "options": { "A": "0<=x,y<=1, the full square", "B": "0=0", "D": "0=a)?", "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)?", "options": { "A": "(t−s)/(b−s)", "B": "(t−s)/(t−a)", "C": "(t−s)/(b−a)", "D": "(t−a)/(b−a)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0201, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9415362489505729 }, { "exam": "final_sp25", "sample": "follow_up", "id": "20.2b", "topic": "distributions", "kind": "recognition", "prompt": "A nondegenerate bounded uniform distribution is memoryless.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0388, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.991422514586288 }, { "exam": "final_sp25", "sample": "follow_up", "id": "20.3a", "topic": "distributions", "kind": "multi_step", "prompt": "X is the minimum of m independent Uniform[0,1] draws. Its density?", "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" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0334, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.725084453649814 }, { "exam": "final_sp25", "sample": "follow_up", "id": "20.3b", "topic": "expectation", "kind": "calculation", "prompt": "Expected minimum of m independent Uniform[0,1] draws?", "options": { "A": "1/(m+1)", "B": "m/(m+1)", "C": "1/m", "D": "1/2" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0248, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5643139187624726 }, { "exam": "final_sp25", "sample": "follow_up", "id": "21.1a", "topic": "markov", "kind": "recognition", "prompt": "Is this chain irreducible?", "options": { "A": "Yes", "B": "No" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1969, "default_truncated": false, "context": "States 0..29. From i, move to i+3 or i+5 modulo 30, each with probability 1/2.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5926665999540697 }, { "exam": "final_sp25", "sample": "follow_up", "id": "21.1b", "topic": "markov", "kind": "multi_step", "prompt": "Period of state 0?", "options": { "A": "10", "B": "1", "C": "2", "D": "6" }, "expected": [ "C" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.004, "default_truncated": false, "context": "States 0..29. From i, move to i+3 or i+5 modulo 30, each with probability 1/2.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3925874504024267 }, { "exam": "final_sp25", "sample": "follow_up", "id": "21.1c", "topic": "markov", "kind": "recognition", "prompt": "The uniform distribution on these states is invariant.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1439, "default_truncated": false, "context": "States 0..29. From i, move to i+3 or i+5 modulo 30, each with probability 1/2.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7981867777396212 }, { "exam": "final_sp25", "sample": "follow_up", "id": "21.2a", "topic": "markov", "kind": "multi_step", "prompt": "Expected minutes until first reaching Andy?", "options": { "A": "4", "B": "8", "C": "5", "D": "2" }, "expected": [ "C" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0585, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.33424000363035195 }, { "exam": "final_sp25", "sample": "follow_up", "id": "21.2b", "topic": "markov", "kind": "multi_step", "prompt": "Probability of reaching Jay before Gavin?", "options": { "A": "2/3", "B": "1/4", "C": "3/4", "D": "1/2" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.099, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.48383755173009263 }, { "exam": "final_su25", "sample": "follow_up", "id": "3.1", "topic": "logic", "kind": "recognition", "prompt": "(NOT P implies P) is equivalent to P.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.07, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9669140216112958 }, { "exam": "final_su25", "sample": "follow_up", "id": "3.2", "topic": "logic", "kind": "recognition", "prompt": "[(P OR Q) AND (P implies R) AND (Q implies R)] is equivalent to R.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.5133, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7310585786300049 }, { "exam": "final_su25", "sample": "follow_up", "id": "3.3", "topic": "logic", "kind": "recognition", "prompt": "exists x [P(x) AND NOT Q(x)] is equivalent to forall x [NOT P(x) implies Q(x)].", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0386, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9796676466573412 }, { "exam": "final_su25", "sample": "follow_up", "id": "3.4", "topic": "logic", "kind": "recognition", "prompt": "exists x [P(x) AND NOT Q(x)] is equivalent to NOT forall x [P(x) implies Q(x)].", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1488, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9706877692486436 }, { "exam": "final_su25", "sample": "follow_up", "id": "3.5", "topic": "logic", "kind": "recognition", "prompt": "Simplify (P AND Q) OR (P AND NOT Q).", "options": { "A": "Q", "B": "P OR Q", "C": "False", "D": "P" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.037, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8870143638353266 }, { "exam": "final_su25", "sample": "follow_up", "id": "3.6", "topic": "logic", "kind": "recognition", "prompt": "Simplify P AND (P implies Q).", "options": { "A": "P AND Q", "B": "Q", "C": "P OR Q", "D": "P" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2356, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.40148861260281904 }, { "exam": "final_su25", "sample": "follow_up", "id": "4.1", "topic": "proof", "kind": "recognition", "prompt": "Which proof technique uses [P(1) AND ... AND P(k)] implies P(k+1)?", "options": { "A": "Contraposition", "B": "Strong induction", "C": "Proof by cases", "D": "Diagonalization" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0064, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.672735715405311 }, { "exam": "final_su25", "sample": "follow_up", "id": "4.2", "topic": "proof", "kind": "multi_step", "prompt": "Real x,y have (x+y)/2=a. Prove x<=a or y<=a.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0885, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9983075783681356 }, { "exam": "final_su25", "sample": "follow_up", "id": "5.1", "topic": "graphs", "kind": "recognition", "prompt": "Every graph with maximum degree at most two is planar.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0394, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9669140216112958 }, { "exam": "final_su25", "sample": "follow_up", "id": "5.2", "topic": "graphs", "kind": "recognition", "prompt": "A graph containing a vertex of degree six cannot be planar.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2503, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8807970779778823 }, { "exam": "final_su25", "sample": "follow_up", "id": "5.3", "topic": "graphs", "kind": "recognition", "prompt": "A simple cycle on n vertices is bipartite iff n is odd.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.186, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9579122720843811 }, { "exam": "final_su25", "sample": "follow_up", "id": "5.4a", "topic": "graphs", "kind": "recognition", "prompt": "Minimum vertex colors for a d-dimensional hypercube, d>=1?", "options": { "A": "d+1", "B": "2", "C": "d", "D": "2^d" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0024, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9386401911259745 }, { "exam": "final_su25", "sample": "follow_up", "id": "5.4b", "topic": "proof", "kind": "multi_step", "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.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0043, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9847937609959219 }, { "exam": "final_su25", "sample": "follow_up", "id": "5.4c", "topic": "proof", "kind": "multi_step", "prompt": "Prove the four-dimensional hypercube is nonplanar.", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0235, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9605810487966544 }, { "exam": "final_su25", "sample": "follow_up", "id": "6.1", "topic": "matching", "kind": "recognition", "prompt": "If all candidates rank the same job last, job-proposing deferred acceptance cannot finish in one day.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1887, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5312093733737563 }, { "exam": "final_su25", "sample": "follow_up", "id": "6.2", "topic": "matching", "kind": "recognition", "prompt": "If all jobs have the same strict ranking of candidates, the stable matching is unique.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1063, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7549149868676283 }, { "exam": "final_su25", "sample": "follow_up", "id": "6.3", "topic": "matching", "kind": "recognition", "prompt": "It is impossible for every candidate to get its favorite job in a job-optimal stable matching.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1356, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7310585786300049 }, { "exam": "final_su25", "sample": "follow_up", "id": "6.4", "topic": "matching", "kind": "recognition", "prompt": "A job and candidate who rank each other first are paired in every stable matching.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0035, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.679178699175393 }, { "exam": "final_su25", "sample": "follow_up", "id": "7.1a", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "There exists integer x with x=1 mod 10 and x=2 mod 15.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0434, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9399133498259924 }, { "exam": "final_su25", "sample": "follow_up", "id": "7.1b", "topic": "proof", "kind": "multi_step", "prompt": "Why are x=1 mod 10 and x=2 mod 15 inconsistent?", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0246, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9273723380979241 }, { "exam": "final_su25", "sample": "follow_up", "id": "7.2a", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "There exists integer x with x=6 mod 10 and x=1 mod 15.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0689, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8175744761936437 }, { "exam": "final_su25", "sample": "follow_up", "id": "7.2b", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Give a witness satisfying x=6 mod 10 and x=1 mod 15.", "options": { "A": "31", "B": "16", "C": "46+1", "D": "6" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0378, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6928657060230099 }, { "exam": "final_su25", "sample": "follow_up", "id": "7.3a", "topic": "proof", "kind": "multi_step", "prompt": "Prove x²=(m−x)² modulo positive m.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0515, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9557787183094788 }, { "exam": "final_su25", "sample": "follow_up", "id": "7.3b", "topic": "proof", "kind": "multi_step", "prompt": "For positive even n, prove sum(k=1..n) (−1)^k k² is divisible by n+1.", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.025, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8977247823367789 }, { "exam": "final_su25", "sample": "follow_up", "id": "8.1", "topic": "polynomials", "kind": "calculation", "prompt": "Over GF(5), how many formal polynomials of degree at most 3 pass through one specified point?", "options": { "A": "25", "B": "625", "C": "125", "D": "100" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0085, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.45971651129010377 }, { "exam": "final_su25", "sample": "follow_up", "id": "8.2a", "topic": "secret_sharing", "kind": "multi_step", "prompt": "Find the secret.", "options": { "A": "1", "B": "4", "C": "2", "D": "3" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0783, "default_truncated": false, "context": "A Shamir polynomial of degree at most two over GF(5) gives shares (1,3),(2,4),(4,2). Secret is P(0).", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.49708930713115246 }, { "exam": "final_su25", "sample": "follow_up", "id": "8.2b", "topic": "secret_sharing", "kind": "multi_step", "prompt": "What value belongs in a share at x=3?", "options": { "A": "1", "B": "2", "C": "4", "D": "3" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1319, "default_truncated": false, "context": "A Shamir polynomial of degree at most two over GF(5) gives shares (1,3),(2,4),(4,2). Secret is P(0).", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.29539205153207015 }, { "exam": "final_su25", "sample": "follow_up", "id": "8.3a", "topic": "coding", "kind": "calculation", "prompt": "Eight Reed–Solomon evaluations are transmitted over GF(11), correcting up to two corrupted values. Maximum message length in field symbols?", "options": { "A": "6", "B": "2", "C": "4", "D": "8" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0126, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3785916633064982 }, { "exam": "final_su25", "sample": "follow_up", "id": "8.3b", "topic": "coding", "kind": "calculation", "prompt": "Minimal error locator is E(x)=x²−6x+8 over GF(11). Which evaluation coordinates are corrupted?", "options": { "A": "6 and 8", "B": "1 and 8", "C": "2 and 4", "D": "3 and 5" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0456, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.40322669517609055 }, { "exam": "final_su25", "sample": "follow_up", "id": "9.1", "topic": "counting", "kind": "multi_step", "prompt": "Ten rounds give +30 for a win, −30 for a loss, starting at zero. Number of ordered outcomes ending at +60?", "options": { "A": "C(10,4)", "B": "2^10", "C": "C(10,2)", "D": "C(10,5)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0768, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3939044996684532 }, { "exam": "final_su25", "sample": "follow_up", "id": "9.2", "topic": "counting", "kind": "calculation", "prompt": "Ten-digit IDs allow 0..9 in every position, with no equal adjacent digits. Count?", "options": { "A": "9*10^9", "B": "10*9^9", "C": "10^10", "D": "10!" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0081, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9773192129390049 }, { "exam": "final_su25", "sample": "follow_up", "id": "9.3", "topic": "counting", "kind": "calculation", "prompt": "Distribute eight integer points into three named stats; first nonnegative, others strictly positive. Count?", "options": { "A": "3^8", "B": "C(8,2)", "C": "C(7,2)", "D": "C(10,2)" }, "expected": [ "B" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0118, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.39855634852953997 }, { "exam": "final_su25", "sample": "follow_up", "id": "9.4", "topic": "counting", "kind": "calculation", "prompt": "Choose distinct captain and lieutenant from eleven people. Count?", "options": { "A": "121", "B": "110", "C": "55", "D": "22" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0082, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7941527303887939 }, { "exam": "final_su25", "sample": "follow_up", "id": "9.5", "topic": "counting", "kind": "calculation", "prompt": "Seven distinct collectible items; choose a subset with at least two. Count?", "options": { "A": "120", "B": "21", "C": "127", "D": "128" }, "expected": [ "A" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0101, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6522291226369561 }, { "exam": "final_su25", "sample": "follow_up", "id": "9.6", "topic": "counting", "kind": "multi_step", "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.", "options": { "A": "n!(n−1)!/2", "B": "n!(n−1)!", "C": "(2n−1)!", "D": "(n!)²" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0178, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.43589456091525647 }, { "exam": "final_su25", "sample": "follow_up", "id": "9.7", "topic": "counting", "kind": "calculation", "prompt": "Number of functions from an n-element set to itself that are not bijections?", "options": { "A": "n^n−n!", "B": "n^n−n", "C": "n!−n", "D": "2^n−n!" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.3712, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9310451107511357 }, { "exam": "final_su25", "sample": "follow_up", "id": "10.1", "topic": "countability", "kind": "recognition", "prompt": "Functions N->{True,False} form a countable set.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1531, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5621765008857981 }, { "exam": "final_su25", "sample": "follow_up", "id": "10.2", "topic": "countability", "kind": "recognition", "prompt": "Finite binary strings form a countable set.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.3059, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9902915235185259 }, { "exam": "final_su25", "sample": "follow_up", "id": "10.3", "topic": "computability", "kind": "recognition", "prompt": "It is impossible to write a halting analyzer that correctly handles even some programs.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0323, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9796676466573412 }, { "exam": "final_su25", "sample": "follow_up", "id": "10.4", "topic": "computability", "kind": "recognition", "prompt": "No always-terminating analyzer can correctly decide for all programs whether they ever print a particular fixed string.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.069, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9626731126558706 }, { "exam": "final_su25", "sample": "follow_up", "id": "10.5", "topic": "computability", "kind": "recognition", "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.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0855, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7981867777396212 }, { "exam": "final_su25", "sample": "follow_up", "id": "10.6", "topic": "proof", "kind": "multi_step", "prompt": "Why can no total analyzer decide whether an arbitrary Python program ever assigns a string to a given variable? Assume unbounded theoretical computation.", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0591, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9914970910137124 }, { "exam": "final_su25", "sample": "follow_up", "id": "11.1", "topic": "probability", "kind": "calculation", "prompt": "P(A)?", "options": { "A": "1/2", "B": "3/5", "C": "5/8", "D": "3/4" }, "expected": [ "D" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.028, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4839056816586273 }, { "exam": "final_su25", "sample": "follow_up", "id": "11.2", "topic": "probability", "kind": "calculation", "prompt": "P(NOT B)?", "options": { "A": "2/5", "B": "1/2", "C": "5/8", "D": "3/8" }, "expected": [ "D" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0113, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.33544561004293627 }, { "exam": "final_su25", "sample": "follow_up", "id": "11.3", "topic": "probability", "kind": "calculation", "prompt": "P(A union B)?", "options": { "A": "3/4", "B": "1", "C": "5/8", "D": "7/8" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0335, "default_truncated": false, "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.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.31283638571410965 }, { "exam": "final_su25", "sample": "follow_up", "id": "11.4", "topic": "probability", "kind": "multi_step", "prompt": "P(A given B)?", "options": { "A": "4/5", "B": "3/4", "C": "1/2", "D": "5/8" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0857, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4743223188344241 }, { "exam": "final_su25", "sample": "follow_up", "id": "11.5", "topic": "probability", "kind": "multi_step", "prompt": "P(A intersection B given A union B)?", "options": { "A": "4/7", "B": "4/5", "C": "1/2", "D": "7/8" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0519, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5171959964471434 }, { "exam": "final_su25", "sample": "follow_up", "id": "11.6", "topic": "probability", "kind": "recognition", "prompt": "P(A intersection NOT B given B)?", "options": { "A": "0", "B": "3/8", "C": "1", "D": "4/5" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0205, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.39672043498534576 }, { "exam": "final_su25", "sample": "follow_up", "id": "11.7", "topic": "probability", "kind": "multi_step", "prompt": "Now vary P(1)=p<1 and keep the other four outcomes equally likely. Which p makes A and B independent?", "options": { "A": "1/5", "B": "1/2", "C": "1/3", "D": "0" }, "expected": [ "C" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0036, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3499320087587727 }, { "exam": "final_su25", "sample": "follow_up", "id": "12.1", "topic": "probability", "kind": "recognition", "prompt": "Must p1+p2 equal one?", "options": { "A": "No, it cannot.", "B": "Yes, always.", "C": "Depends; it can but need not." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.039, "default_truncated": false, "context": "A detector flags plagiarized submissions with probability p1 and original submissions with probability p2. Class CS1 has 10% plagiarized submissions; CS100 has 2%.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6671044428044404 }, { "exam": "final_su25", "sample": "follow_up", "id": "12.2a", "topic": "bayes", "kind": "multi_step", "prompt": "p1=0.8,p2=0.2. Given a CS1 submission was flagged, probability it was plagiarized?", "options": { "A": "4/5", "B": "4/13", "C": "1/4", "D": "1/10" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0053, "default_truncated": false, "context": "A detector flags plagiarized submissions with probability p1 and original submissions with probability p2. Class CS1 has 10% plagiarized submissions; CS100 has 2%.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7696754653020688 }, { "exam": "final_su25", "sample": "follow_up", "id": "12.2b", "topic": "bayes", "kind": "multi_step", "prompt": "p1=0.8,p2=0.2. Given a CS100 submission was flagged, probability it was plagiarized?", "options": { "A": "4/13", "B": "4/5", "C": "1/50", "D": "4/53" }, "expected": [ "D" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0834, "default_truncated": false, "context": "A detector flags plagiarized submissions with probability p1 and original submissions with probability p2. Class CS1 has 10% plagiarized submissions; CS100 has 2%.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4146252635361017 }, { "exam": "final_su25", "sample": "follow_up", "id": "12.3", "topic": "bayes", "kind": "multi_step", "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?", "options": { "A": "1/10", "B": "1/2", "C": "5/6", "D": "4/13" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0431, "default_truncated": false, "context": "A detector flags plagiarized submissions with probability p1 and original submissions with probability p2. Class CS1 has 10% plagiarized submissions; CS100 has 2%.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.35909976350004963 }, { "exam": "final_su25", "sample": "follow_up", "id": "13.1", "topic": "probability", "kind": "calculation", "prompt": "Probability the two California senators sit adjacent?", "options": { "A": "2/(2n)", "B": "2/(2n−1)", "C": "1/(2n−1)", "D": "1/n" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0957, "default_truncated": false, "context": "There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.45971651129010377 }, { "exam": "final_su25", "sample": "follow_up", "id": "13.2", "topic": "probability", "kind": "multi_step", "prompt": "Probability the California pair is adjacent given the New Mexico pair is adjacent?", "options": { "A": "2/(2n−1)", "B": "2/(2n−2)", "C": "1/(2n−2)", "D": "4/((2n−1)(2n−2))" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0809, "default_truncated": false, "context": "There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4855401664512152 }, { "exam": "final_su25", "sample": "follow_up", "id": "13.3", "topic": "probability", "kind": "multi_step", "prompt": "One California senator is happy if adjacent to either of two specified other senators. Probability of happiness?", "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)" }, "expected": [ "A" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0385, "default_truncated": false, "context": "There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.31911523504877376 }, { "exam": "final_su25", "sample": "follow_up", "id": "13.4a", "topic": "probability", "kind": "calculation", "prompt": "Ni means senator i is adjacent to their state partner. P(Ni)?", "options": { "A": "1/(2n−1)", "B": "2/(2n−2)", "C": "1/n", "D": "2/(2n−1)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0613, "default_truncated": false, "context": "There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3431547327443468 }, { "exam": "final_su25", "sample": "follow_up", "id": "13.4b", "topic": "probability", "kind": "recognition", "prompt": "Are the events Ni pairwise independent over all senators?", "options": { "A": "Yes", "B": "No" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1818, "default_truncated": false, "context": "There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7310585786300049 }, { "exam": "final_su25", "sample": "follow_up", "id": "13.4c", "topic": "proof", "kind": "multi_step", "prompt": "Why are the Ni not pairwise independent?", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0556, "default_truncated": false, "context": "There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9002022849296801 }, { "exam": "final_su25", "sample": "follow_up", "id": "13.4d", "topic": "expectation", "kind": "calculation", "prompt": "Expected number of senators adjacent to their state partner?", "options": { "A": "4n/(2n−2)", "B": "4n/(2n−1)", "C": "2", "D": "2n/(2n−1)" }, "expected": [ "B" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0475, "default_truncated": false, "context": "There are n>=2 states, each with two distinct senators; all 2n senators are uniformly assigned seats around a round table.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.3689108554330873 }, { "exam": "final_su25", "sample": "follow_up", "id": "14.1", "topic": "probability", "kind": "recognition", "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?", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0128, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9998233321583648 }, { "exam": "final_su25", "sample": "follow_up", "id": "14.2", "topic": "proof", "kind": "multi_step", "prompt": "Prove variance of a sum of pairwise independent finite-variance variables equals the sum of variances.", "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)).", "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)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0521, "default_truncated": false, "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).", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5948722009020091 }, { "exam": "final_su25", "sample": "follow_up", "id": "15.1e", "topic": "moments", "kind": "multi_step", "prompt": "Instead choose uniformly from the four points in S with x,y>=0. Give (E[X],E[Y],Var(Y),Cov(X,Y)).", "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)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0052, "default_truncated": false, "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).", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6184496414007788 }, { "exam": "final_su25", "sample": "follow_up", "id": "16.1a", "topic": "distributions", "kind": "calculation", "prompt": "P(X+Y=k)?", "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)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1853, "default_truncated": false, "context": "Independent X~Bin(n,p),Y~Bin(m,p).", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7956467427827978 }, { "exam": "final_su25", "sample": "follow_up", "id": "16.1b", "topic": "distributions", "kind": "recognition", "prompt": "Distribution of X+Y?", "options": { "A": "Bin(n+m,p)", "B": "Poisson((n+m)p)", "C": "Bin(nm,p)", "D": "Bin(n+m,2p)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1673, "default_truncated": false, "context": "Independent X~Bin(n,p),Y~Bin(m,p).", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9983415193609757 }, { "exam": "final_su25", "sample": "follow_up", "id": "16.1c", "topic": "proof", "kind": "multi_step", "prompt": "Explain the distribution of X+Y without a probability formula.", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1309, "default_truncated": false, "context": "Independent X~Bin(n,p),Y~Bin(m,p).", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9918915390754022 }, { "exam": "final_su25", "sample": "follow_up", "id": "16.2a", "topic": "bounds", "kind": "multi_step", "prompt": "X counts heads in 100 independent fair flips. Chebyshev bound for P(X<=10 or X>=90)?", "options": { "A": "1/16", "B": "1/8", "C": "1/64", "D": "1/40" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0244, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.37134179371136994 }, { "exam": "final_su25", "sample": "follow_up", "id": "16.2b", "topic": "probability", "kind": "recognition", "prompt": "For X counting heads in 100 independent fair flips, P(X<=10)=P(X>=90).", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2703, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7981867777396212 }, { "exam": "final_su25", "sample": "follow_up", "id": "16.2c", "topic": "bounds", "kind": "multi_step", "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).", "options": { "A": "2P(Z>=4)", "B": "2P(Z>=8)", "C": "2P(Z>=40)", "D": "P(Z>=8)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0316, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.722215489287567 }, { "exam": "final_su25", "sample": "follow_up", "id": "17.1", "topic": "expectation", "kind": "calculation", "prompt": "Expected net winnings with no rerolls?", "options": { "A": "7", "B": "−1", "C": "−2", "D": "1" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0203, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4641850402782907 }, { "exam": "final_su25", "sample": "follow_up", "id": "17.2", "topic": "expectation", "kind": "multi_step", "prompt": "Reroll each initial 1 for free. Expected net winnings?", "options": { "A": "−1/6", "B": "−1", "C": "1/6", "D": "5/6" }, "expected": [ "A" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0546, "default_truncated": false, "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.", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.33065623127838456 }, { "exam": "final_su25", "sample": "follow_up", "id": "17.3", "topic": "expectation", "kind": "multi_step", "prompt": "Choose integer threshold c in advance and reroll each die <=c for free. Give optimal c and expected net winnings.", "options": { "A": "(2,1/3)", "B": "(4,2/3)", "C": "(3,3/2)", "D": "(3,1/2)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0125, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.35347162922091135 }, { "exam": "final_su25", "sample": "follow_up", "id": "17.4", "topic": "expectation", "kind": "multi_step", "prompt": "Each reroll costs $1. Give optimal integer threshold c and expected net winnings.", "options": { "A": "(1,−1/6)", "B": "(3,−1/2)", "C": "(2,−1/3)", "D": "(2,1/3)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0156, "default_truncated": false, "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.", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3423962339378881 }, { "exam": "final_su25", "sample": "follow_up", "id": "17.5", "topic": "expectation", "kind": "multi_step", "prompt": "Smallest integer reroll fee d such that any reroll strictly decreases conditional expected net winnings?", "options": { "A": "4", "B": "2", "C": "3", "D": "6" }, "expected": [ "C" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0084, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3819020577651715 }, { "exam": "final_su25", "sample": "follow_up", "id": "18.1a", "topic": "distributions", "kind": "recognition", "prompt": "PDF fR(x) for x>=0?", "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)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0635, "default_truncated": false, "context": "Independent T1~Exp(rate lambda1),T2~Exp(rate lambda2), both rates positive. R=min(T1,T2).", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9719506331820598 }, { "exam": "final_su25", "sample": "follow_up", "id": "18.1b", "topic": "expectation", "kind": "recognition", "prompt": "E[R]?", "options": { "A": "1/(lambda1+lambda2)", "B": "1/lambda1+1/lambda2", "C": "min(1/lambda1,1/lambda2)", "D": "1/(lambda1*lambda2)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0511, "default_truncated": false, "context": "Independent T1~Exp(rate lambda1),T2~Exp(rate lambda2), both rates positive. R=min(T1,T2).", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9975060135303174 }, { "exam": "final_su25", "sample": "follow_up", "id": "18.1c", "topic": "probability", "kind": "recognition", "prompt": "Choose an integral for P(T1<=T2).", "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" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0016, "default_truncated": true, "context": "Independent T1~Exp(rate lambda1),T2~Exp(rate lambda2), both rates positive. R=min(T1,T2).", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4887027005548193 }, { "exam": "final_su25", "sample": "follow_up", "id": "18.1d", "topic": "probability", "kind": "multi_step", "prompt": "Evaluate P(T1<=T2).", "options": { "A": "1/2", "B": "lambda1*lambda2/(lambda1+lambda2)", "C": "lambda2/(lambda1+lambda2)", "D": "lambda1/(lambda1+lambda2)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0307, "default_truncated": false, "context": "Independent T1~Exp(rate lambda1),T2~Exp(rate lambda2), both rates positive. R=min(T1,T2).", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9824534268610069 }, { "exam": "final_su25", "sample": "follow_up", "id": "18.2a", "topic": "distributions", "kind": "multi_step", "prompt": "PDF of S for 0<=x<=10?", "options": { "A": "(10−x)/50", "B": "(10−x)²/100", "C": "x/50", "D": "1/10" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0122, "default_truncated": false, "context": "Independent T1,T2~Uniform[0,10], S=min(T1,T2).", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5906171549085213 }, { "exam": "final_su25", "sample": "follow_up", "id": "18.2b", "topic": "expectation", "kind": "multi_step", "prompt": "E[S]?", "options": { "A": "20/3", "B": "5", "C": "5/2", "D": "10/3" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0343, "default_truncated": false, "context": "Independent T1,T2~Uniform[0,10], S=min(T1,T2).", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.31483005318115603 }, { "exam": "final_su25", "sample": "follow_up", "id": "18.2c", "topic": "probability", "kind": "multi_step", "prompt": "P(|T1−T2|<=1)?", "options": { "A": "1/10", "B": "9/100", "C": "1/5", "D": "19/100" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2253, "default_truncated": false, "context": "Independent T1,T2~Uniform[0,10], S=min(T1,T2).", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.30601362565976303 }, { "exam": "final_su25", "sample": "follow_up", "id": "19.1", "topic": "expectation", "kind": "multi_step", "prompt": "X~Exp(rate lambda>0), Y=min(X,2/lambda). Choose a derivation of E[Y].", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0239, "default_truncated": true, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9737072740547735 }, { "exam": "final_su25", "sample": "follow_up", "id": "20.1", "topic": "markov", "kind": "recognition", "prompt": "This chain is irreducible.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2995, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8175744761936437 }, { "exam": "final_su25", "sample": "follow_up", "id": "20.2", "topic": "markov", "kind": "recognition", "prompt": "Period of the chain?", "options": { "A": "3", "B": "2", "C": "4", "D": "1" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0212, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.49171754910554016 }, { "exam": "final_su25", "sample": "follow_up", "id": "20.3", "topic": "proof", "kind": "multi_step", "prompt": "Why does the distribution converge to a unique stationary distribution from every initial state?", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0945, "default_truncated": true, "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.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8843512198853015 }, { "exam": "final_su25", "sample": "follow_up", "id": "20.4", "topic": "markov", "kind": "recognition", "prompt": "Choose stationary equations, plus pi0+pi1+pi2+pi3=1.", "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" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0473, "default_truncated": true, "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.", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5959823694129226 }, { "exam": "final_su25", "sample": "follow_up", "id": "20.5", "topic": "markov", "kind": "multi_step", "prompt": "Expected steps to first reach state 3, starting at 0?", "options": { "A": "4", "B": "8", "C": "16", "D": "12" }, "expected": [ "C" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0183, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.49651941445493847 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "3.1a", "topic": "logic", "kind": "recognition", "prompt": "P implies Q is equivalent to NOT P implies NOT Q.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.195, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9046505351008906 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "3.1b", "topic": "logic", "kind": "recognition", "prompt": "Complete the contrapositive: (P AND Q) implies R is equivalent to NOT R implies __.", "options": { "A": "NOT P OR NOT Q", "B": "NOT P AND NOT Q", "C": "P OR Q", "D": "P AND Q" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0114, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9623903401328848 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "3.2", "topic": "logic", "kind": "recognition", "prompt": "NOT exists integer x [P(x) OR Q(x)] is equivalent to forall integer x [NOT P(x) AND NOT Q(x)].", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0383, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9953904278206259 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "3.3", "topic": "logic", "kind": "recognition", "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)].", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.6321, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5926665999540697 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "3.4", "topic": "number_theory", "kind": "recognition", "prompt": "For all natural a and positive integer b, a/b is not sqrt(2).", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.3131, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7981867777396212 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "3.5", "topic": "number_theory", "kind": "recognition", "prompt": "Every natural n is either a perfect square or has irrational square root.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0521, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7057850278370112 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "4.1", "topic": "proof", "kind": "multi_step", "prompt": "For integers a,b,c, does a|b and a|c imply a|(b+5c)? Justify.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2081, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9728455875349389 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "4.2", "topic": "proof", "kind": "multi_step", "prompt": "Does b²=n imply n divides b for all positive integers b,n?", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.169, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5037210203436986 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "4.3", "topic": "proof", "kind": "multi_step", "prompt": "Why must a positive nonsquare integer have an odd exponent in its prime factorization?", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0097, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.908480748920783 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "5.1", "topic": "proof", "kind": "multi_step", "prompt": "x1=1; xn=sqrt(1+x(n−1)) for n>=2. Prove xn<2.", "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.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0949, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.542748037123821 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "6.1", "topic": "matching", "kind": "recognition", "prompt": "A pair present in both proposer-optimal matchings is present in every stable matching.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0916, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "6.2", "topic": "matching", "kind": "recognition", "prompt": "If some pair appears in both proposer-optimal matchings, then the entire stable matching is unique.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0336, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8175744761936437 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "6.3", "topic": "matching", "kind": "recognition", "prompt": "With two jobs and two candidates, every stable matching is optimal for one proposer side or both.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0864, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6224593312018546 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "6.4", "topic": "matching", "kind": "recognition", "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.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1664, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9626731126558706 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "7.1", "topic": "graphs", "kind": "calculation", "prompt": "Smallest N such that every n-dimensional hypercube with n>=N has an even edge count?", "options": { "A": "1", "B": "3", "C": "2", "D": "4" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0661, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.3392357547465047 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "7.2", "topic": "graphs", "kind": "recognition", "prompt": "Kn has an even number of edges for every n>3.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1227, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6513548646660542 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "7.3a", "topic": "graphs", "kind": "calculation", "prompt": "In Kn, how many edges cross a cut with k vertices on one side?", "options": { "A": "n−1", "B": "C(k,2)", "C": "k(n−k)", "D": "k*n" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0457, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9814287373037599 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "7.3b", "topic": "graphs", "kind": "recognition", "prompt": "Minimum edges crossing a nonempty proper cut of an n-vertex tree, n>=2?", "options": { "A": "n−1", "B": "0", "C": "2", "D": "1" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0148, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6938735876930563 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "7.3c", "topic": "graphs", "kind": "recognition", "prompt": "Maximum edges crossing a cut of an n-vertex tree?", "options": { "A": "n", "B": "1", "C": "n−1", "D": "floor(n/2)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0356, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7419039172011334 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "7.4", "topic": "graphs", "kind": "recognition", "prompt": "Adding a new edge either reduces the component count or creates a cycle containing that edge.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0286, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9399133498259924 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "7.5", "topic": "graphs", "kind": "recognition", "prompt": "Minimum possible component count with n>=1 vertices and e edges?", "options": { "A": "n−1", "B": "e−n+1", "C": "n−e", "D": "max(n−e,1)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.089, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6077367252546441 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "7.6", "topic": "graphs", "kind": "recognition", "prompt": "Smallest edge-count threshold guaranteeing a cycle in a simple n-vertex graph?", "options": { "A": "n", "B": "2n", "C": "n+1", "D": "n−1" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0274, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9152003859409163 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "8.1a", "topic": "graphs", "kind": "recognition", "prompt": "The n-dimensional hypercube is edge-colorable with n colors.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.019, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8807970779778823 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "8.1b", "topic": "graphs", "kind": "recognition", "prompt": "Every graph of maximum degree d is edge-colorable with d colors.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0428, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5621765008857981 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "8.1c", "topic": "graphs", "kind": "recognition", "prompt": "Every bipartite graph is edge-colorable with two colors.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1991, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9579122720843811 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "8.1di", "topic": "graphs", "kind": "calculation", "prompt": "Number of edges?", "options": { "A": "nd", "B": "d/2", "C": "nd/2", "D": "n+d" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0927, "default_truncated": false, "context": "A bipartite graph on n>2 vertices is d-regular, and its edges partition into Hamiltonian cycles.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8380222804600492 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "8.1dii", "topic": "graphs", "kind": "calculation", "prompt": "Number of cycles in this edge partition?", "options": { "A": "nd/2", "B": "d/2", "C": "d", "D": "n/2" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0166, "default_truncated": false, "context": "A bipartite graph on n>2 vertices is d-regular, and its edges partition into Hamiltonian cycles.", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.40544191343510666 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "8.1diii", "topic": "proof", "kind": "multi_step", "prompt": "Why can the graph be edge-colored with d colors?", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1609, "default_truncated": false, "context": "A bipartite graph on n>2 vertices is d-regular, and its edges partition into Hamiltonian cycles.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5407528490639676 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "8.2", "topic": "proof", "kind": "multi_step", "prompt": "In a d-regular bipartite graph, d>0, prove every S on the left satisfies |N(S)|>=|S|.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0954, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9994031310847863 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "9.1", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "Compute 7^50 modulo 35.", "options": { "A": "14", "B": "1", "C": "7", "D": "0" }, "expected": [ "A" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0323, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.36779178791187 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "9.2a", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Prime q, N=q². How many residues modulo N are coprime to N?", "options": { "A": "q", "B": "q²−q", "C": "q²−1", "D": "q−1" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0786, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6116506886253091 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "9.2b", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "If q is prime and gcd(a,q)=1, then a has a multiplicative inverse modulo q².", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0007, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9669140216112958 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "9.2c", "topic": "modular_arithmetic", "kind": "recognition", "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.", "options": { "A": "q", "B": "q−1", "C": "q(q−1)+1", "D": "q²" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0017, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5016048947027452 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "9.3", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "For all moduli n, ab=0 mod n implies a=0 mod n or b=0 mod n.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0023, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9046505351008906 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "9.4a", "topic": "number_theory", "kind": "recognition", "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.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0149, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9820137900379085 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "9.4b", "topic": "number_theory", "kind": "recognition", "prompt": "For positive x,y, the least positive integer linear combination is min(x,y).", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1466, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5312093733737563 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "9.4c", "topic": "number_theory", "kind": "recognition", "prompt": "For positive x,y, the least positive integer linear combination is gcd(x,y).", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2806, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9875683491468141 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "9.5", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "If x=a mod m and x=a mod n, with coprime positive m,n, then x=a mod mn.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0022, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9840936082881853 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "9.6a", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Positive m,n have gcd q. Smallest positive k with a+kn=a mod m?", "options": { "A": "n/q", "B": "q", "C": "m/q", "D": "mn/q" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0121, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5101839830120496 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "9.6b", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Positive m,n have gcd q. How many x in 0..mn−1 satisfy x=a mod m and x=a mod n?", "options": { "A": "q", "B": "1", "C": "mn/q", "D": "m+n" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0052, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.44178089647797625 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "10.1", "topic": "proof", "kind": "multi_step", "prompt": "For 1<=n2 has p−1=q1*...*qk, a product of distinct primes.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9855857132800594 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "10.2", "topic": "proof", "kind": "multi_step", "prompt": "For each fixed i, prove there exists a≠0,1 modulo p with a^qi=1.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0153, "default_truncated": true, "context": "Prime p>2 has p−1=q1*...*qk, a product of distinct primes.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9295945523420504 }, { "exam": "midterm_fa23", "sample": "follow_up", "id": "10.3", "topic": "proof", "kind": "multi_step", "prompt": "If x has order d modulo prime p and x^q=1 mod p, prove d divides q.", "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>=1. Euclidean division gives P=DQ+R, with R zero or deg R4. Hint: separate square and nonsquare factorizations.", "options": { "A": "Write n=ab with 12 and a,2a=1.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0445, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7085382251365335 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "6.1a", "topic": "matching", "kind": "multi_step", "prompt": "Job-optimal matching?", "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" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0061, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3468666740605408 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "6.1b", "topic": "matching", "kind": "multi_step", "prompt": "Candidate-optimal matching?", "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" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0669, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.3949756198399488 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "6.1c", "topic": "matching", "kind": "multi_step", "prompt": "Another stable matching beyond the two proposer-optimal ones?", "options": { "A": "A–2, B–1, C–3", "B": "A–3, B–2, C–1", "C": "A–1, B–2, C–3", "D": "None" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0211, "default_truncated": false, "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.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5102876641018756 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "6.2", "topic": "matching", "kind": "recognition", "prompt": "A job receiving its least-preferred candidate in the job-optimal matching must be every candidate’s least-preferred job.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0009, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5926665999540697 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "6.3", "topic": "matching", "kind": "recognition", "prompt": "If a job has the same candidate in every stable matching, that candidate must be its first choice.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2317, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5312093733737563 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "6.4", "topic": "matching", "kind": "recognition", "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.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2391, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5621765008857981 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "6.5", "topic": "matching", "kind": "recognition", "prompt": "A job–candidate pair present in both proposer-optimal matchings is present in every stable matching.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0196, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5621765008857981 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "7.1a", "topic": "graphs", "kind": "recognition", "prompt": "A graph with n vertices and m>=n−1 edges must be connected.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1045, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9465966702001757 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "7.1b", "topic": "graphs", "kind": "recognition", "prompt": "A graph with n vertices and m<=n−1 edges must be acyclic.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0001, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8807970779778823 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "7.1c", "topic": "graphs", "kind": "calculation", "prompt": "Number of edges in Kn?", "options": { "A": "C(n,2)", "B": "n²", "C": "2n", "D": "n−1" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0908, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9851409072922443 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "7.1d", "topic": "graphs", "kind": "calculation", "prompt": "A hypercube has n vertices. Number of edges?", "options": { "A": "n log2(n)", "B": "n log2(n)/2", "C": "n/2", "D": "2^n" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1491, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.873904337275191 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "7.1e", "topic": "graphs", "kind": "recognition", "prompt": "A graph with an Eulerian tour must have an even number of edges.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0551, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8519528019683106 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "7.1f", "topic": "graphs", "kind": "recognition", "prompt": "Minimum possible connected components for a simple graph with n>=1 vertices and feasible edge count m?", "options": { "A": "max(1,n−m)", "B": "n−m", "C": "n−1", "D": "max(1,m−n)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0387, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8066651435528208 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "7.1g", "topic": "graphs", "kind": "recognition", "prompt": "A connected acyclic graph with at least two vertices has at least how many degree-one vertices? Give the sharp universal bound.", "options": { "A": "3", "B": "1", "C": "2", "D": "n−1" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0188, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8750436574932761 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "7.1h", "topic": "graphs", "kind": "recognition", "prompt": "A forest has n vertices and c components. Number of edges?", "options": { "A": "c−1", "B": "n−1", "C": "n+c", "D": "n−c" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1214, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9828911028022888 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "7.2", "topic": "graphs", "kind": "recognition", "prompt": "Every graph has an even number of odd-degree vertices.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0268, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9971990730328789 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "7.3", "topic": "graphs", "kind": "recognition", "prompt": "Minimum edge colors for a d-dimensional hypercube?", "options": { "A": "d+1", "B": "d", "C": "2^d", "D": "2d" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0059, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9904645952410364 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "7.4", "topic": "graphs", "kind": "recognition", "prompt": "Minimum edge colors for an odd cycle?", "options": { "A": "4", "B": "2", "C": "1", "D": "3" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0061, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8956868588341913 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "7.5", "topic": "graphs", "kind": "multi_step", "prompt": "A simple graph has n vertices and n+1 edges. Largest constant degree guaranteed for at least one vertex?", "options": { "A": "3", "B": "2", "C": "4", "D": "n−1" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0109, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.45360944044158075 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "8.1", "topic": "proof", "kind": "multi_step", "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.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0167, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9939222489945714 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "8.2", "topic": "proof", "kind": "multi_step", "prompt": "Partition the edges of a bipartite Eulerian graph into two spanning subgraphs with half of each vertex’s degree.", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0319, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9781909273895838 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "9.1a", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "For integers a,x,y, gcd(x,y)=1 and a divides xy imply a divides x or a divides y.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2457, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6224593312018546 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "9.1b", "topic": "proof", "kind": "multi_step", "prompt": "Choose a counterexample to: gcd(x,y)=1 and a|xy imply a|x or a|y.", "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" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0232, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6605489699221654 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "9.2a", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Number of residues x mod N satisfying qx=0 mod N?", "options": { "A": "1", "B": "pqr", "C": "q", "D": "pr" }, "expected": [ "C" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0697, "default_truncated": false, "context": "N=pqr for three distinct primes p,q,r.", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3615085256056106 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "9.2b", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "If gcd(x,N)=1, what is x^((p−1)(q−1)(r−1)) mod N?", "options": { "A": "N−1", "B": "x", "C": "0", "D": "1" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0905, "default_truncated": false, "context": "N=pqr for three distinct primes p,q,r.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8253558393433151 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "9.2c", "topic": "proof", "kind": "multi_step", "prompt": "Why is x^((p−1)(q−1)(r−1))=1 mod N when gcd(x,N)=1?", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.031, "default_truncated": false, "context": "N=pqr for three distinct primes p,q,r.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9926248488929696 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "10.1", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Multiplicative order of 3 modulo 5?", "options": { "A": "3", "B": "5", "C": "2", "D": "4" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0339, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4847303831708126 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "10.2", "topic": "proof", "kind": "multi_step", "prompt": "Let t be the order of a modulo n, gcd(a,n)=1. Prove a^m=1 mod n iff t divides m.", "options": { "A": "Multiples of t give powers of a^t=1. Conversely write m=qt+r, 0<=r1 with n dividing 2^n−1.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1414, "default_truncated": true, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9882016709059792 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "11.1", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "7^26 modulo 10?", "options": { "A": "1", "B": "9", "C": "3", "D": "7" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0222, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.35581602402486456 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "11.2", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "Inverse of 7 modulo 68 in 0..67?", "options": { "A": "29", "B": "39", "C": "10", "D": "61" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0616, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.43185985038551544 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "11.3", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "Distinct integers x,y are congruent modulo positive m and n, gcd(m,n)=d. Sharp lower bound on |x−y|?", "options": { "A": "mn", "B": "m+n", "C": "mn/d", "D": "d" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0317, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9163418856826213 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "11.4", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Number of x in 0..504 solving 10x=5 mod 505?", "options": { "A": "1", "B": "5", "C": "10", "D": "0" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.019, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3460058095032025 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "11.5a", "topic": "rsa", "kind": "recognition", "prompt": "For valid textbook RSA encryption E and decryption D, D(E(x))=E(D(x)) modulo N for every message x.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0447, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6224593312018546 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "11.5b", "topic": "rsa", "kind": "recognition", "prompt": "Given y=E(E(x)) in valid RSA, which expression recovers x?", "options": { "A": "D(D(y))", "B": "E(E(y))", "C": "E(D(y))", "D": "D(y)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0248, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.42363828119916935 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "11.5c", "topic": "rsa", "kind": "recognition", "prompt": "In valid textbook RSA, a=E(x), b=E(y). What is D(ab) modulo N?", "options": { "A": "xy", "B": "x^y", "C": "x+y", "D": "x/y" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0394, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9885505512388347 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "12.1", "topic": "polynomials", "kind": "multi_step", "prompt": "Over GF(5), give a degree-at-most-2 polynomial through (0,1),(1,0),(3,0).", "options": { "A": "2(x−1)(x−3)", "B": "3(x−1)(x−3)", "C": "(x−1)(x−3)", "D": "2(x+1)(x+3)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.3764, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.3804775568859664 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "12.2", "topic": "polynomials", "kind": "multi_step", "prompt": "Over GF(5), ax²+bx+c passes through (0,0),(1,1),(2,4). Give (a,b,c).", "options": { "A": "(1,1,0)", "B": "(2,4,0)", "C": "(1,0,0)", "D": "(0,1,0)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0077, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.47453432981901295 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "12.3", "topic": "polynomials", "kind": "calculation", "prompt": "Over GF(p), p>=5, four points have distinct x-coordinates. Number of degree-at-most-4 polynomials through them?", "options": { "A": "p²", "B": "1", "C": "p", "D": "p−1" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0647, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4482780905009404 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "12.4a", "topic": "polynomials", "kind": "recognition", "prompt": "Exact degree of Q?", "options": { "A": "d−d′", "B": "d′−1", "C": "d+d′", "D": "d/d′" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0038, "default_truncated": false, "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)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.005, "default_truncated": false, "context": "Five TAs each hold a nonzero-coordinate share of P, three professors a share of Q; P(0)=Q(0)=s. A strict majority of either group should reconstruct, and smaller coalitions within that group should not.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8039229105439095 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "13.2a", "topic": "coding", "kind": "calculation", "prompt": "What is b0 in 0..4?", "options": { "A": "1", "B": "0", "C": "4", "D": "2" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0098, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3690058809048272 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "13.2bi", "topic": "coding", "kind": "recognition", "prompt": "Degree d of Q?", "options": { "A": "3", "B": "4", "C": "2", "D": "1" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0355, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3967757573269966 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "13.2bii", "topic": "coding", "kind": "recognition", "prompt": "Leading coefficient qd of Q?", "options": { "A": "1", "B": "4", "C": "3", "D": "2" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0201, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4797225835875461 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "13.2biii", "topic": "coding", "kind": "recognition", "prompt": "Constant coefficient q0 in terms of b0?", "options": { "A": "2b0", "B": "3b0", "C": "b0", "D": "3+b0" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0486, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3583275739447147 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "14.1", "topic": "counting", "kind": "calculation", "prompt": "Number of formal polynomials of degree exactly d over GF(p)?", "options": { "A": "(p−1)p^d", "B": "(p−1)^(d+1)", "C": "p^d", "D": "p^(d+1)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2256, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.48383755173009263 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "14.2", "topic": "polynomials", "kind": "recognition", "prompt": "Every degree-exactly-one polynomial over a field has exactly one root.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0683, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9149009549929797 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "14.3", "topic": "polynomials", "kind": "recognition", "prompt": "Choose a quadratic over GF(p) having x=1 as its only distinct root, for every prime p.", "options": { "A": "x²−1", "B": "(x−1)²", "C": "x²+1", "D": "x(x−1)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1214, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.975714550220758 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "14.4", "topic": "counting", "kind": "calculation", "prompt": "Number of distinct formal polynomials a(x−r)² over GF(p), a nonzero?", "options": { "A": "(p−1)C(p,2)", "B": "(p−1)p", "C": "p−1", "D": "p²" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0085, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5970437583423148 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "14.5", "topic": "counting", "kind": "calculation", "prompt": "Number of a(x−r)(x−s), a nonzero and r≠s, over GF(p)?", "options": { "A": "C(p,2)", "B": "p²", "C": "(p−1)p(p−1)", "D": "(p−1)C(p,2)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0724, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.43871174808085056 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "14.6", "topic": "counting", "kind": "multi_step", "prompt": "Number of degree-exactly-two polynomials over GF(p) with no field roots?", "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" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.141, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4429666554483415 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "14.7", "topic": "counting", "kind": "calculation", "prompt": "For fixed distinct field elements r1,...,rk, count polynomials product (x−ri)^bi with each bi>=1 and sum bi=d>=k.", "options": { "A": "C(d,k)", "B": "k^d", "C": "C(d+k−1,k−1)", "D": "C(d−1,k−1)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0961, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6495404303421376 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "15.1a", "topic": "counting", "kind": "calculation", "prompt": "Number of unordered 13-card hands from 52 cards?", "options": { "A": "C(64,13)", "B": "C(52,13)", "C": "52!/39!", "D": "52^13" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0198, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9883874412011823 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "15.1b", "topic": "counting", "kind": "calculation", "prompt": "Deal all 52 distinct cards to four named players, 13 each, unordered within each hand. Number of deals?", "options": { "A": "C(52,13)", "B": "52!/(4!*(13!)^4)", "C": "52!/(13!)^4", "D": "4^52" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0174, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9920739449556016 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "15.2", "topic": "counting", "kind": "calculation", "prompt": "Count integer k-tuples with xi>=−1 and sum xi=n, n>=0, k>=1.", "options": { "A": "C(n+2k−1,k−1)", "B": "C(n+k−1,k−1)", "C": "k^n", "D": "C(n−1,k−1)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1786, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.572689141194751 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "15.3a", "topic": "counting", "kind": "calculation", "prompt": "Number of anagrams of CANTSTANFURD?", "options": { "A": "12!/2!", "B": "11!/(2!2!2!)", "C": "12!/(3!3!)", "D": "12!/(2!2!2!)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.3204, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.3963287975238086 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "15.3b", "topic": "counting", "kind": "calculation", "prompt": "Let x be the number of anagrams of CANTSTANFURD. How many put C before S before D, without an adjacency requirement?", "options": { "A": "x/2", "B": "x/12", "C": "x/6", "D": "x/3" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1011, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5074099707555233 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "15.3c", "topic": "counting", "kind": "calculation", "prompt": "Let x be the number of anagrams of CANTSTANFURD. How many put both As before S, without an adjacency requirement?", "options": { "A": "x/2", "B": "2x/3", "C": "x/6", "D": "x/3" }, "expected": [ "D" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0237, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.2821833983601388 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "15.4", "topic": "counting", "kind": "calculation", "prompt": "Number of length-n bitstrings with k ones?", "options": { "A": "C(n,k)", "B": "2^k", "C": "n!/k!", "D": "n^k" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0442, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9968000301524156 }, { "exam": "midterm_fa24", "sample": "follow_up", "id": "15.5", "topic": "counting", "kind": "calculation", "prompt": "Distribute five identical dollar bills among four named people, allowing zero. Number of allocations?", "options": { "A": "C(5,4)", "B": "C(8,3)", "C": "5^4", "D": "4^5" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0451, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8343114106971716 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "1.1", "topic": "logic", "kind": "recognition", "prompt": "If 6 is prime, then trees can walk.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0347, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "1.2", "topic": "logic", "kind": "recognition", "prompt": "Is NOT(P implies Q) equivalent to (NOT P) AND (NOT Q)?", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1065, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9740426428022031 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "1.3", "topic": "proof", "kind": "multi_step", "prompt": "For integers a,m,n, prove: if a does not divide mn, then a divides neither m nor n. Which argument is valid?", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1399, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7404320837930707 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "2.1", "topic": "logic", "kind": "recognition", "prompt": "Express: Bloof is not healing unless it is not in motion.", "options": { "A": "NOT M implies NOT H", "B": "M implies NOT H", "C": "NOT H implies M", "D": "H implies M" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.029, "default_truncated": false, "context": "J means jumping, G on ground, M in motion, H healing, W touching a wall.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6568557825067092 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "2.2", "topic": "logic", "kind": "recognition", "prompt": "Give the contrapositive of J implies (NOT G AND M).", "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)" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0998, "default_truncated": false, "context": "J means jumping, G on ground, M in motion, H healing, W touching a wall.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9956411441935573 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "2.3", "topic": "logic", "kind": "multi_step", "prompt": "H is true. What can be concluded about W?", "options": { "A": "True", "B": "Could be either", "C": "False" }, "expected": [ "C" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0956, "default_truncated": false, "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).", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.43214049950399813 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "3.1", "topic": "proof", "kind": "multi_step", "prompt": "Which proves that an integer n is odd iff 5n+3 is even?", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0167, "default_truncated": true, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8954410577188929 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "3.2", "topic": "proof", "kind": "multi_step", "prompt": "Prove 7 divides A_n=2^(n+2)+3^(2n+1), for integers n>=0. Which induction works?", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0517, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9524100293798948 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "4.1a", "topic": "matching", "kind": "multi_step", "prompt": "Is matching A–1, B–2, C–3 stable? Include a valid reason.", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0486, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9066200907893139 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "4.1b", "topic": "matching", "kind": "multi_step", "prompt": "Which matching is stable?", "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" }, "expected": [ "D" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0314, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3690058809048272 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "4.2", "topic": "matching", "kind": "recognition", "prompt": "In a stable matching, it is impossible for two partners to be each other’s least preferred option.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1215, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.679178699175393 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "4.3", "topic": "matching", "kind": "recognition", "prompt": "If a job has the same partner in every stable matching, that partner must be its first choice.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1667, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "4.4", "topic": "matching", "kind": "recognition", "prompt": "Job-proposing and candidate-proposing deferred acceptance produce the same matching. How many stable matchings exist?", "options": { "A": "Exactly one", "B": "None", "C": "Cannot be determined", "D": "More than one" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.5533, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.35628748125978893 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "5.1", "topic": "graphs", "kind": "recognition", "prompt": "K4 has an Eulerian tour.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0363, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9399133498259924 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "5.2", "topic": "graphs", "kind": "recognition", "prompt": "K6 is planar.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0955, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5621765008857981 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "5.3", "topic": "graphs", "kind": "calculation", "prompt": "A graph has vertex degrees 4,3,3,2,1,1. How many edges?", "options": { "A": "8", "B": "14", "C": "6", "D": "7" }, "expected": [ "D" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0399, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5746624415494149 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "5.4", "topic": "graphs", "kind": "calculation", "prompt": "In the 5-dimensional hypercube, what is the distance between 10010 and 01000?", "options": { "A": "4", "B": "3", "C": "5", "D": "2" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0738, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7068861512998341 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "6.1", "topic": "graphs", "kind": "recognition", "prompt": "Is this graph a tree?", "options": { "A": "No", "B": "Yes" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0159, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9465966702001757 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "6.2", "topic": "graphs", "kind": "recognition", "prompt": "Is this graph bipartite?", "options": { "A": "No", "B": "Yes" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1432, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5926665999540697 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "6.3", "topic": "graphs", "kind": "calculation", "prompt": "What is the minimum number of colors for a proper edge coloring?", "options": { "A": "3", "B": "5", "C": "4", "D": "2" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0093, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3607860911977634 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "6.4", "topic": "graphs", "kind": "multi_step", "prompt": "Does the graph have an Eulerian walk? If not, choose an edge to add that creates one.", "options": { "A": "No; add Berkeley–Coliseum.", "B": "Yes; no added edge needed.", "C": "No; add Oakland–Daly City.", "D": "No; add Walnut Creek–Fremont." }, "expected": [ "D" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0829, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.45227865708619125 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "6.5", "topic": "graphs", "kind": "multi_step", "prompt": "Does the graph have an Eulerian tour? If not, choose two edges that create one.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0115, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4349598892232299 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "6.6", "topic": "graphs", "kind": "multi_step", "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?", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0201, "default_truncated": true, "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.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6795199456895651 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "7.1", "topic": "proof", "kind": "multi_step", "prompt": "Which induction proves the sum of vertex degrees is twice the edge count?", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0072, "default_truncated": true, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9511381833271031 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "8.1", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "For every positive integer n, 5 divides 6^n−1.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1325, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7981867777396212 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "8.2", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "If xy=0 modulo 6, then x=0 or y=0 modulo 6.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0622, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9626731126558706 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "8.3", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "For every positive odd m, 9 divides 4^m+5^m.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0756, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8175744761936437 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "8.4", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "If p>3 and both p and p+2 are prime, what is p modulo 3?", "options": { "A": "0", "B": "2", "C": "1", "D": "Not determined" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1477, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5552289140656056 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "8.5", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "What is the last decimal digit of 9^68?", "options": { "A": "1", "B": "8", "C": "6", "D": "9" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0276, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.30702389681474196 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "8.6", "topic": "rsa", "kind": "multi_step", "prompt": "RSA has p=5, q=7 and public exponent e=5. Which is a valid decryption exponent d?", "options": { "A": "24", "B": "7", "C": "12", "D": "5" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0602, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.554735809640914 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "8.7", "topic": "rsa", "kind": "multi_step", "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.", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1813, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8974610864323499 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "9.1", "topic": "proof", "kind": "multi_step", "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?", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0275, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9887086870931818 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "10.1a", "topic": "polynomials", "kind": "recognition", "prompt": "Polynomials f,g over a field have degrees k>j. What is deg(f+g)?", "options": { "A": "Could be any value below k", "B": "k+j", "C": "j", "D": "k" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0342, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9571253274620543 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "10.1b", "topic": "polynomials", "kind": "recognition", "prompt": "Nonzero polynomials f,g over a field have degrees k,j. What is deg(fg)?", "options": { "A": "k−j", "B": "k*j", "C": "k+j", "D": "max(k,j)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1063, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9864586549634923 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "10.2", "topic": "polynomials", "kind": "multi_step", "prompt": "Over GF(5), which degree-at-most-4 polynomial is equal as a function to 4x^84+x^23+2x^5+3?", "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" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0293, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6158307782580915 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "10.3", "topic": "polynomials", "kind": "multi_step", "prompt": "A degree-2 polynomial over GF(7) passes through (0,1),(3,0),(4,0). Which is it?", "options": { "A": "4(x−3)(x−4)", "B": "3(x−3)(x−4)", "C": "3(x−2)(x−4)", "D": "(x−3)(x−4)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0756, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.502873446678164 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "10.4", "topic": "polynomials", "kind": "calculation", "prompt": "How many formal polynomials of degree at most 4 over GF(5) pass through two given points with distinct x-coordinates?", "options": { "A": "5", "B": "625", "C": "125", "D": "25" }, "expected": [ "C" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0071, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4703895888525571 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "11.1", "topic": "secret_sharing", "kind": "multi_step", "prompt": "Choose a valid Shamir configuration (degree; shares per professor, head TA, TA; sufficient prime bound). Secret s is a nonnegative integer.", "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))" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0349, "default_truncated": true, "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.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5916888672429292 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "11.2", "topic": "coding", "kind": "calculation", "prompt": "A polynomial of degree 9 is transmitted by evaluations. At most two values are corrupted. How many evaluations suffice for unique recovery?", "options": { "A": "12", "B": "13", "C": "18", "D": "14" }, "expected": [ "D" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0171, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3035988665653362 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "12.1", "topic": "counting", "kind": "calculation", "prompt": "How many length-30 bit strings contain exactly 12 ones?", "options": { "A": "C(30,12)", "B": "30^12", "C": "2^12", "D": "30!/12!" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0028, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9981398039509597 }, { "exam": "midterm_fa25", "sample": "follow_up", "id": "12.2", "topic": "counting", "kind": "calculation", "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.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.06, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9974813097555685 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "5.1", "topic": "proof", "kind": "multi_step", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1452, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.997694189960457 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "5.2", "topic": "proof", "kind": "multi_step", "prompt": "How does this finish the induction?", "options": { "A": "The inequality shows m−Fk=0, so every natural is itself Fibonacci.", "B": "Represent m−Fk and append F(k−1), whose value equals Fk.", "C": "Allow a repeated Fk; repeated indices are permitted by the theorem.", "D": "Represent m−Fk using the hypothesis. Since its value is less than F(k−1), its nonnegative summands have indices at most k−2; adjoining Fk preserves distinct, nonconsecutive indices." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0356, "default_truncated": true, "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?", "options": { "A": "1", "B": "n−1", "C": "2", "D": "0" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0489, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7708528350077796 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "8.5a", "topic": "graphs", "kind": "calculation", "prompt": "Maximum edges in a bipartite graph with fixed parts A,B?", "options": { "A": "C(|A|+|B|,2)", "B": "|A|*|B|", "C": "min(|A|,|B|)", "D": "|A|+|B|−1" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0341, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9986770489214637 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "8.5b", "topic": "graphs", "kind": "recognition", "prompt": "Every graph of maximum degree at most two is bipartite.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1039, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.679178699175393 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "8.5c", "topic": "graphs", "kind": "recognition", "prompt": "Smallest constant D such that every nonempty bipartite planar graph has a vertex of degree at most D?", "options": { "A": "4", "B": "3", "C": "5", "D": "2" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.044, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6107874071036075 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "8.5d", "topic": "graphs", "kind": "recognition", "prompt": "For the three-dimensional hypercube, choose one side A of a bipartition.", "options": { "A": "{001,011,101,111}", "B": "{000,011,110,101}", "C": "{000,001,010,011}", "D": "{000,111}" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0108, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5110391347635618 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "8.6", "topic": "graphs", "kind": "recognition", "prompt": "Largest possible vertex degree in a simple planar graph on n vertices?", "options": { "A": "6", "B": "5", "C": "3n−6", "D": "n−1" }, "expected": [ "D" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0052, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8854192530131136 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "8.7", "topic": "graphs", "kind": "multi_step", "prompt": "In a simple n-vertex graph, n>=3, what sharp threshold on du+dv guarantees distinct vertices u,v share a neighbor?", "options": { "A": "n−1", "B": "n", "C": "2n−2", "D": "n+1" }, "expected": [ "D" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0499, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.28108825044289903 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "8.8", "topic": "proof", "kind": "multi_step", "prompt": "Must a connected planar graph with more than 100 vertices have at least three vertices of degree below six?", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0436, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9830362757582279 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "8.9", "topic": "graphs", "kind": "recognition", "prompt": "Remove a degree-d vertex v from G. The rest has a given k-coloring. What bound follows when restoring v?", "options": { "A": "k+d", "B": "max(k,1+min(d,k))", "C": "min(k,d)", "D": "k−1" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0133, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8696704536468565 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "9.1a", "topic": "graphs", "kind": "recognition", "prompt": "Every 3-regular graph has an even number of vertices.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1024, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5621765008857981 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "9.1b", "topic": "proof", "kind": "multi_step", "prompt": "Justify the parity constraint for a 3-regular graph on n vertices.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.057, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9784833340454555 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "9.2a", "topic": "graphs", "kind": "recognition", "prompt": "Every 4-regular graph has an odd number of vertices.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1088, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9046505351008906 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "9.2b", "topic": "proof", "kind": "multi_step", "prompt": "Give a counterexample to the claim that every 4-regular graph has an odd number of vertices.", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0236, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7288422088351846 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "9.3", "topic": "proof", "kind": "multi_step", "prompt": "Prove any simple n-vertex graph, n>=2, has two equal degrees.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0217, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9973647356891014 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "10.1", "topic": "functions", "kind": "recognition", "prompt": "A={0,1,2,3}, B={0,1,2}; f(0)=f(1)=0 and f(2)=f(3)=1. Find k.", "options": { "A": "Unknown", "B": "1", "C": "2", "D": "3" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.064, "default_truncated": false, "context": "A function is k-uniform when each output has either zero or exactly k distinct preimages.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8212328360370813 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "10.2", "topic": "functions", "kind": "recognition", "prompt": "g is k1-uniform and h is k2-uniform. What k is guaranteed for g composed with h?", "options": { "A": "k1+k2", "B": "k1*k2", "C": "Unknown; the composition need not be uniform.", "D": "max(k1,k2)" }, "expected": [ "C" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0209, "default_truncated": false, "context": "A function is k-uniform when each output has either zero or exactly k distinct preimages.", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8370536304452664 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "10.3a", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "f(x)=ax mod prime m, a nonzero, domain and codomain all residues. Find k.", "options": { "A": "1", "B": "a", "C": "m", "D": "Unknown" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0646, "default_truncated": false, "context": "A function is k-uniform when each output has either zero or exactly k distinct preimages.", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.31483005318115603 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "10.3b", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "f(x)=ax mod m, gcd(a,m)=d, domain and codomain all residues. Find k.", "options": { "A": "Unknown", "B": "1", "C": "d", "D": "m/d" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.024, "default_truncated": false, "context": "A function is k-uniform when each output has either zero or exactly k distinct preimages.", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5482601385401136 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "10.4", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "m=pq for distinct primes; f(x)=ax mod m on the unit residues, gcd(a,m)=1. Find k.", "options": { "A": "Unknown", "B": "(p−1)(q−1)", "C": "1", "D": "p−1" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0333, "default_truncated": false, "context": "A function is k-uniform when each output has either zero or exactly k distinct preimages.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5327251004904839 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "10.5", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "m=pq for distinct primes; f(x)=ax mod m on unit residues, gcd(a,m)=p. Codomain all residues. Find k.", "options": { "A": "q−1", "B": "p", "C": "p−1", "D": "1" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0126, "default_truncated": false, "context": "A function is k-uniform when each output has either zero or exactly k distinct preimages.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4129089721781154 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "11.1", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "x=3 mod 5 and x=2 mod 11. Find x mod 55 in 0..54.", "options": { "A": "13", "B": "23", "C": "2", "D": "3" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0186, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6245228453112519 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "11.2a", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "x=a mod 10 and x=b mod 15, where b−a is divisible by 5. Number of solutions modulo 150?", "options": { "A": "0", "B": "5", "C": "1", "D": "10" }, "expected": [ "B" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0682, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.48783852322368093 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "11.2b", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "x=a mod 10 and x=b mod 15, where b−a is not divisible by 5. Number of solutions modulo 150?", "options": { "A": "0", "B": "5", "C": "1", "D": "15" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0546, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.408483547916507 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "11.3", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "For every integer a and prime p, p divides a^p−a.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0348, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9902915235185259 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "11.4", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "For distinct primes p,q and every integer a, choose a guaranteed divisor of a*(a^((p−1)(q−1))−1).", "options": { "A": "pq", "B": "p²q", "C": "p+q", "D": "p²q²" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0557, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8053041565700648 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "11.5a", "topic": "rsa", "kind": "recognition", "prompt": "RSA ciphertexts are y1=x1^e and y2=x2^e modulo N. Ciphertext of x1*x2?", "options": { "A": "(y1*y2)^d mod N", "B": "y1*y2 mod N", "C": "y1+y2 mod N", "D": "(y1+y2)^e mod N" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0057, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8463189387099178 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "11.5b", "topic": "rsa", "kind": "recognition", "prompt": "In valid RSA, y1=x1^e and y2=x2^e modulo N, with decryption exponent d. Express x1*x2 modulo N.", "options": { "A": "(y1*y2)^e mod N", "B": "(y1*y2)^d mod N", "C": "y1*y2 mod N", "D": "(y1+y2)^d mod N" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0064, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9336103040573762 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "12.1", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Least nonnegative residue of 2^n modulo p?", "options": { "A": "2", "B": "0", "C": "1", "D": "p−1" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0784, "default_truncated": false, "context": "p=2^n−1 is a Mersenne prime, n>=2.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5262386562523383 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "12.2", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "For positive integer a, least nonnegative residue of (2^a)^n modulo p?", "options": { "A": "0", "B": "1", "C": "a", "D": "2^a" }, "expected": [ "B" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0149, "default_truncated": false, "context": "p=2^n−1 is a Mersenne prime, n>=2.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8261425789867147 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "12.3", "topic": "proof", "kind": "multi_step", "prompt": "Show there are at least n distinct residues x with x^n=1 mod p.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0204, "default_truncated": false, "context": "p=2^n−1 is a Mersenne prime, n>=2.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9928208841288604 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "13.1", "topic": "polynomials", "kind": "calculation", "prompt": "Choose a polynomial over GF(7) through (1,0),(2,5).", "options": { "A": "5x+1", "B": "5x+2", "C": "x−1", "D": "2x+5" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0671, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.48997015892458795 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "13.2", "topic": "polynomials", "kind": "calculation", "prompt": "Over GF(p), p>d, how many formal polynomials of degree at most d pass through d−1 points with distinct x-coordinates?", "options": { "A": "p²", "B": "p^(d−1)", "C": "p", "D": "1" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1028, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3142819961202565 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "13.3a", "topic": "polynomials", "kind": "recognition", "prompt": "How many distinct roots does Delta1 have?", "options": { "A": "d", "B": "d+1", "C": "d−1", "D": "Unknown" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1255, "default_truncated": false, "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.", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.37134179371136994 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "13.3bi", "topic": "polynomials", "kind": "recognition", "prompt": "Formal degree of Delta1*Delta2?", "options": { "A": "d+1", "B": "2d", "C": "d", "D": "Unknown" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1089, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.33482349867891054 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "13.3bii", "topic": "polynomials", "kind": "recognition", "prompt": "Number of distinct roots of Delta1*Delta2?", "options": { "A": "d+1", "B": "2d", "C": "d", "D": "Unknown" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2338, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.35347162922091135 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "13.3c", "topic": "polynomials", "kind": "recognition", "prompt": "Every real polynomial of degree d has exactly d distinct real roots.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1241, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9988304897349445 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "14.1", "topic": "coding", "kind": "recognition", "prompt": "Packets sufficient to recover n message symbols with at most e erased packets and k corruptions?", "options": { "A": "n+2e+k", "B": "n+e+2k", "C": "n+e+k", "D": "n+2(e+k)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0954, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7354006848459316 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "14.2", "topic": "coding", "kind": "calculation", "prompt": "Minimal error locator is x²−1 over GF(13). Give the error coordinates in 0..12.", "options": { "A": "{1,2}", "B": "{0,1}", "C": "{1,12}", "D": "{1,6}" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1103, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6319849081601967 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "14.3", "topic": "secret_sharing", "kind": "calculation", "prompt": "Shamir shares secret 5 with ten participants, any three reconstruct. Smallest prime modulus, with nonzero distinct share coordinates?", "options": { "A": "7", "B": "5", "C": "11", "D": "13" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0028, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4159628150951531 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "14.4", "topic": "secret_sharing", "kind": "multi_step", "prompt": "Design a scheme requiring a strict majority of both a five-person group and a seven-person group.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1566, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.57595191084186 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "15.1", "topic": "countability", "kind": "recognition", "prompt": "The set of finite subsets of a countably infinite set is uncountable.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.3303, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6224593312018546 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "15.2", "topic": "countability", "kind": "recognition", "prompt": "The set of all subsets of a countably infinite set is uncountable.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0154, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.995929862284104 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "15.3a", "topic": "computability", "kind": "recognition", "prompt": "Every real number has a finite program that halts on input n and returns its nth digit.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0006, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9626731126558706 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "15.3b", "topic": "proof", "kind": "multi_step", "prompt": "Why are some real numbers not computable digit-by-digit?", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0989, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9985174059046341 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "16.1", "topic": "counting", "kind": "calculation", "prompt": "Distinct anagrams of ABRACADABRA?", "options": { "A": "11!/5!", "B": "11!/(5!2!2!)", "C": "11!/(4!2!2!)", "D": "10!/(5!2!2!)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1767, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9130209025277846 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "16.2", "topic": "counting", "kind": "calculation", "prompt": "Strings with exactly four As and three Bs?", "options": { "A": "4!*3!", "B": "C(7,3)", "C": "2^7", "D": "7!" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0166, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9139234691319885 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "16.3", "topic": "counting", "kind": "calculation", "prompt": "Distribute n1 identical apples and n2 identical bananas among k named people, allowing zero. Count?", "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)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1284, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9012590725683363 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "16.4", "topic": "counting", "kind": "multi_step", "prompt": "Walk starts at 0 and ends at n>=0 after n+2k unit steps, each ±1, unrestricted positions. Number of step sequences?", "options": { "A": "2^(n+2k)", "B": "C(n+2k,n)", "C": "C(n+k,k)", "D": "C(n+2k,k)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0962, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6413163990929579 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "16.5a", "topic": "counting", "kind": "multi_step", "prompt": "Sn counts ordered sums of ones and twos equaling n. Give a summation.", "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)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.4904, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9502492090918537 }, { "exam": "midterm_sp23", "sample": "follow_up", "id": "16.5b", "topic": "recurrences", "kind": "recognition", "prompt": "With S0=S1=1, recurrence for ordered sums of ones and twos equaling n, n>=2?", "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)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0105, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9024846001889492 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "1.1", "topic": "logic", "kind": "recognition", "prompt": "A judge says: if Bob stole the diamonds, he did not act alone. A truthful attorney denies that statement. What must be true?", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "E" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1851, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.398345768222268 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "2.1", "topic": "computability", "kind": "recognition", "prompt": "If the halting problem is computable, then pigs can fly.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2478, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7772998611746911 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "2.2", "topic": "logic", "kind": "recognition", "prompt": "forall x [P(x) implies Q(x)] is equivalent to NOT exists x [NOT P(x) AND Q(x)].", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0018, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9947798743064417 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "2.3", "topic": "logic", "kind": "recognition", "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).", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0356, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8175744761936437 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "2.4", "topic": "logic", "kind": "recognition", "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).", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.5115, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.679178699175393 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "2.5", "topic": "computability", "kind": "recognition", "prompt": "There is an algorithm deciding whether an input finite decimal string is the corresponding prefix of pi’s decimal expansion.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0763, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6224593312018546 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "2.6", "topic": "computability", "kind": "recognition", "prompt": "There is an algorithm deciding, for arbitrary program P and strings x,y, whether P on x outputs y.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1196, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9669140216112958 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "3.1", "topic": "matching", "kind": "recognition", "prompt": "A pair present in both the job-proposing and candidate-proposing stable matchings is present in every stable matching.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1081, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7310585786300049 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "3.2", "topic": "matching", "kind": "recognition", "prompt": "For every n, there can be a job-optimal stable matching containing a pair who rank each other last.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1243, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6513548646660542 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "3.3", "topic": "matching", "kind": "recognition", "prompt": "Two participants who rank each other first must be paired in every stable matching.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0021, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5312093733737563 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "3.4", "topic": "matching", "kind": "recognition", "prompt": "Two disjoint pairs, each pair ranking one another last, cannot both occur in a stable matching.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1124, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8933094060543487 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "3.5", "topic": "matching", "kind": "recognition", "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.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0059, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8807970779778823 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "4.a", "topic": "logic", "kind": "recognition", "prompt": "Express: the robot is not in motion unless powered on.", "options": { "A": "P AND M", "B": "NOT M implies NOT P", "C": "P implies M", "D": "M implies P" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0644, "default_truncated": false, "context": "P: powered on. M: in motion. A: alarm activated. R: rerouting. S: senses an obstacle.", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8140724970502541 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "4.b", "topic": "logic", "kind": "recognition", "prompt": "Contrapositive of: if powered on, it is in motion or its alarm is activated.", "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" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.3726, "default_truncated": false, "context": "P: powered on. M: in motion. A: alarm activated. R: rerouting. S: senses an obstacle.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8136299637574461 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "4.c", "topic": "logic", "kind": "multi_step", "prompt": "Given M true, what follows about R,S?", "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" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0577, "default_truncated": false, "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).", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.335666620665722 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "4.d", "topic": "logic", "kind": "recognition", "prompt": "L(t) means low battery at t, C(t) means charging. Express: whenever battery is low, it charges at some strictly later time.", "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)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0724, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9963364525355027 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "5.a", "topic": "proof", "kind": "multi_step", "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.", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0108, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8494959364785251 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "5.b", "topic": "proof", "kind": "multi_step", "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.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0354, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9903635927640659 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "6.a", "topic": "proof", "kind": "multi_step", "prompt": "F0=0,F1=1,Fn=F(n−1)+F(n−2). Which induction proves sum(Fi²,i=0..n)=Fn*F(n+1)?", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0008, "default_truncated": true, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9923133932184344 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "6.b", "topic": "proof", "kind": "multi_step", "prompt": "Prove sum(i³,i=1..n)=[sum(i,i=1..n)]². You may use sum(i)=n(n+1)/2.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.028, "default_truncated": true, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9856234670372559 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "7.a", "topic": "graphs", "kind": "multi_step", "prompt": "Which one new edge creates an Eulerian tour?", "options": { "A": "b–d", "B": "a–e", "C": "a–d", "D": "b–e" }, "expected": [ "D" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.005, "default_truncated": false, "context": "Vertices a,b,c,d,e,f. Edges ab,af,bf,bc,fc,fe,ce,cd,de. There are no other edges.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.35540643906208824 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "7.b", "topic": "graphs", "kind": "multi_step", "prompt": "Which one vertex must be removed so the remaining graph has a Hamiltonian cycle?", "options": { "A": "g", "B": "b", "C": "c", "D": "a" }, "expected": [ "C" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0223, "default_truncated": false, "context": "Vertices a,b,c,d,e,f,g. Edges ab,ad,bc,bf,ce,de,ef,fg,eg. There are no other edges.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4871416572550292 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "7.c", "topic": "proof", "kind": "multi_step", "prompt": "Prove a tree with a degree-k vertex has at least k leaves.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0267, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9822156828964075 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "7.d", "topic": "graphs", "kind": "multi_step", "prompt": "A connected graph has 80 vertices, 135 edges and girth 5. Is it planar? Justify.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.082, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9780406283719822 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "8.ai", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "Compute 1!+3!+5!+...+99! modulo 24.", "options": { "A": "1", "B": "6", "C": "7", "D": "0" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0706, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.626001444845002 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "8.aii", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Solve 5x=7 mod 11, x in 0..10.", "options": { "A": "2", "B": "5", "C": "9", "D": "8" }, "expected": [ "D" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0059, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3528925139185056 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "8.aiii", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Units digit of 7^96?", "options": { "A": "1", "B": "3", "C": "9", "D": "7" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0519, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3568632977686014 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "8.aiv", "topic": "rsa", "kind": "calculation", "prompt": "Give the public key (N,e).", "options": { "A": "(77,7)", "B": "(77,11)", "C": "(60,11)", "D": "(18,11)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0637, "default_truncated": false, "context": "RSA uses primes p=7,q=11 and public exponent e=11.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9963040755415415 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "8.av", "topic": "rsa", "kind": "multi_step", "prompt": "Encrypt message 3.", "options": { "A": "27", "B": "11", "C": "33", "D": "47" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0117, "default_truncated": false, "context": "RSA uses primes p=7,q=11 and public exponent e=11.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.423442608492013 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "8.avi", "topic": "rsa", "kind": "multi_step", "prompt": "Decrypt ciphertext 47.", "options": { "A": "3", "B": "7", "C": "47", "D": "11" }, "expected": [ "A" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0026, "default_truncated": false, "context": "RSA uses primes p=7,q=11 and public exponent e=11.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5935802900633251 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "8.b", "topic": "modular_arithmetic", "kind": "multi_step", "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.", "options": { "A": "97", "B": "42", "C": "165", "D": "152" }, "expected": [ "D" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0265, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3331832353540621 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "9.a", "topic": "polynomials", "kind": "recognition", "prompt": "Hints needed to determine the treasure location?", "options": { "A": "3", "B": "4", "C": "1", "D": "2" }, "expected": [ "A" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0103, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.34031470637689565 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "9.b", "topic": "polynomials", "kind": "calculation", "prompt": "After one hint, number of still-possible treasure locations?", "options": { "A": "p−1", "B": "p²", "C": "p", "D": "1" }, "expected": [ "B" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0115, "default_truncated": false, "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.", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5202594698638454 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "9.c", "topic": "polynomials", "kind": "calculation", "prompt": "After two hints, number of still-possible treasure locations?", "options": { "A": "p−1", "B": "1", "C": "p²", "D": "p" }, "expected": [ "D" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0129, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.43940420719120843 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "9.d", "topic": "polynomials", "kind": "multi_step", "prompt": "Hints P(0)=5,P(1)=5,P(3)=11. Find P. Hint: solve for coefficients rather than use Lagrange.", "options": { "A": "x²−x+5", "B": "x²+x+5", "C": "x+5", "D": "2x²−2x+5" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0473, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5250523182647024 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "9.e", "topic": "polynomials", "kind": "multi_step", "prompt": "For p=89 and hints P(0)=5,P(1)=5,P(3)=11, find the treasure coordinates.", "options": { "A": "(16,69)", "B": "(6,29)", "C": "(5,11)", "D": "(95,385)" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0317, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6590610711989869 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "9.f", "topic": "coding", "kind": "recognition", "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?", "options": { "A": "5", "B": "2", "C": "3", "D": "4" }, "expected": [ "D" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0018, "default_truncated": false, "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.", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.37320173824469816 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "9.g", "topic": "coding", "kind": "recognition", "prompt": "One hint is corrupted at an unknown location. Minimum hints guaranteeing recovery of the correct polynomial?", "options": { "A": "6", "B": "4", "C": "3", "D": "5" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0078, "default_truncated": false, "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.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.33482349867891054 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "10.a", "topic": "countability", "kind": "multi_step", "prompt": "Is the set of integer grid points (x,y) with gcd(x,y)=1 countable? Justify.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0719, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.993568806019467 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "10.b", "topic": "countability", "kind": "multi_step", "prompt": "A vehicle starts at the origin facing up and chooses left, straight or right at each grid intersection. Are its finite routes countable?", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1889, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9908478197133418 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "10.c", "topic": "countability", "kind": "multi_step", "prompt": "A vehicle starts at the origin facing up and chooses left, straight or right at each grid intersection. Are all infinite routes countable?", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0155, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8723623266666565 }, { "exam": "midterm_sp24", "sample": "follow_up", "id": "10.d", "topic": "countability", "kind": "multi_step", "prompt": "At each (x,y) in Z² there are |x|+|y| parking spaces. Is the set of all spaces countable?", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.359, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8621541670874971 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "2.1", "topic": "course_context", "kind": "multi_step", "prompt": "How many TAs are there this semester?", "options": { "A": "13", "B": "14", "C": "12", "D": "11" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0867, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5682517316796698 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "3.1a", "topic": "logic", "kind": "recognition", "prompt": "(NOT P AND R) OR (NOT P AND Q) is equivalent to NOT P AND (R OR Q).", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2218, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9982992775885648 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "3.1b", "topic": "logic", "kind": "recognition", "prompt": "(P AND Q) AND (NOT P OR NOT Q) is equivalent to P AND NOT P.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2283, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8670357598021706 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "3.1c", "topic": "logic", "kind": "recognition", "prompt": "(P implies Q) AND (P implies NOT Q) is equivalent to NOT P.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2782, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8519528019683106 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "3.1d", "topic": "logic", "kind": "recognition", "prompt": "(P implies R) AND (P implies NOT R) is equivalent to NOT R.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7057850278370112 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "3.2a", "topic": "logic", "kind": "recognition", "prompt": "Negate forall n in N, P(n), without using forall.", "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)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0183, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9575681107642394 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "3.2b", "topic": "logic", "kind": "recognition", "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)].", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1899, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7549149868676283 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "3.2c", "topic": "logic", "kind": "recognition", "prompt": "forall n [Q(n) AND P(n)] is equivalent to (forall n Q(n)) AND (forall n P(n)).", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.7472, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5621765008857981 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "3.2d", "topic": "logic", "kind": "recognition", "prompt": "forall n [Q(n) implies P(n)] is equivalent to (forall n NOT Q(n)) OR (forall n P(n)).", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0042, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9399133498259924 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "4.1", "topic": "proof", "kind": "multi_step", "prompt": "Three people each like exactly one other person. Must someone be liked by nobody? Give a proof or counterexample.", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0472, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8459376412747526 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "4.2", "topic": "proof", "kind": "multi_step", "prompt": "Positive integers a,b,c satisfy a²+b²=c² and gcd(a,b,c)=1. Why must c be odd?", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0381, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9813176798934244 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "5.1", "topic": "proof", "kind": "multi_step", "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.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0188, "default_truncated": true, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9939019111912427 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "6.1", "topic": "matching", "kind": "calculation", "prompt": "Among all preference instances with n jobs and n candidates, what is the maximum possible number of blocking pairs in a complete pairing?", "options": { "A": "n(n−1)", "B": "n−1", "C": "n²", "D": "n(n−1)/2" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0369, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4339080564556254 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "6.2", "topic": "matching", "kind": "recognition", "prompt": "If a stable matching gives one particular job its best attainable stable partner, must that candidate have its worst attainable stable partner?", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2082, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7057850278370112 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "6.3", "topic": "matching", "kind": "recognition", "prompt": "In job-proposing deferred acceptance, if a candidate rejects n−1 distinct jobs, that candidate ends with its favorite job.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.07, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7057850278370112 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "6.4", "topic": "matching", "kind": "recognition", "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.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2721, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5312093733737563 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "6.5a", "topic": "matching", "kind": "recognition", "prompt": "If a job and candidate rank each other first, they are paired in every stable matching.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1432, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5621765008857981 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "6.5b", "topic": "proof", "kind": "multi_step", "prompt": "Construct, for each n>=2, an instance with exactly one stable matching.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0498, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8996923695100589 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "6.5c", "topic": "proof", "kind": "multi_step", "prompt": "Construct an instance with exactly two stable matchings for every n>=2.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0059, "default_truncated": true, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7720326085723336 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "7.1a", "topic": "graphs", "kind": "calculation", "prompt": "A connected simple planar graph has v>=3 vertices. Maximum number of faces?", "options": { "A": "3v−6", "B": "2v−4", "C": "2v−2", "D": "v−2" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0093, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3076566816396497 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "7.1b", "topic": "graphs", "kind": "multi_step", "prompt": "A connected simple planar graph contains a cycle and has girth at least g>=3. Complete the bound e <= [g/(g−2)]v − __.", "options": { "A": "g/(g−2)", "B": "g−2", "C": "2", "D": "2g/(g−2)" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0799, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7834068092280078 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "7.1c", "topic": "proof", "kind": "multi_step", "prompt": "Why is every planar graph with no cycle shorter than six edges 3-colorable?", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.127, "default_truncated": true, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.990073480520138 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "7.2a", "topic": "graphs", "kind": "calculation", "prompt": "How many edges are in the complement of a tree on n vertices?", "options": { "A": "C(n,2)−n+1", "B": "C(n−1,2)−1", "C": "C(n,2)−n", "D": "n−1" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.062, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8884408570938974 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "7.2b", "topic": "graphs", "kind": "calculation", "prompt": "How many edges are in the complement of a d-dimensional hypercube?", "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" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0176, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.49713963877346956 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "7.3a", "topic": "graphs", "kind": "calculation", "prompt": "If G has n vertices and m edges, how many edges does its double have?", "options": { "A": "m+2n", "B": "2m+n", "C": "2m+2n", "D": "2m" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0538, "default_truncated": false, "context": "Doubling a graph means taking two disjoint copies and adding an edge between each original vertex and its corresponding copy.", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7284590384590237 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "7.3b", "topic": "graphs", "kind": "recognition", "prompt": "If G is bipartite, its double is bipartite.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.7275, "default_truncated": false, "context": "Doubling a graph means taking two disjoint copies and adding an edge between each original vertex and its corresponding copy.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8519528019683106 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "7.3c", "topic": "graphs", "kind": "calculation", "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?", "options": { "A": "2c", "B": "c", "C": "c+1", "D": "c−1" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0557, "default_truncated": false, "context": "Doubling a graph means taking two disjoint copies and adding an edge between each original vertex and its corresponding copy.", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5001925230688294 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "7.3f", "topic": "graphs", "kind": "recognition", "prompt": "If G contains a Hamiltonian cycle, its double also contains a Hamiltonian cycle.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.6201, "default_truncated": false, "context": "Doubling a graph means taking two disjoint copies and adding an edge between each original vertex and its corresponding copy.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6513548646660542 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "7.3g", "topic": "graphs", "kind": "recognition", "prompt": "If a graph with at least two vertices contains an Eulerian tour, its double contains an Eulerian tour.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0179, "default_truncated": false, "context": "Doubling a graph means taking two disjoint copies and adding an edge between each original vertex and its corresponding copy.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5621765008857981 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "8.1", "topic": "proof", "kind": "multi_step", "prompt": "Prove the complement of a disconnected graph is connected.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0849, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9976298843697197 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "9.1", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Inverse of 5 modulo 24, in 0..23?", "options": { "A": "19", "B": "1", "C": "7", "D": "5" }, "expected": [ "D" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0368, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5937452201987418 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "9.2", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "Compute 2^75 modulo 35. Hint: consider RSA with p=7,q=5,e=5.", "options": { "A": "2", "B": "1", "C": "8", "D": "32" }, "expected": [ "C" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0404, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4042843624714489 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "9.3", "topic": "rsa", "kind": "multi_step", "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.", "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." }, "expected": [ "C" ], "jev_selected": [ "D" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.4993, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5120516638183038 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "9.4", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "For positive integers y=0, prove a^k=a^(k mod (p−1)) mod p.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0204, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.997950122255985 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "9.6c", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "If gcd(k,p−1)=1, multiplication by k permutes the residue classes modulo p−1.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2315, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8175744761936437 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "9.6d", "topic": "proof", "kind": "multi_step", "prompt": "Prime p, positive k, and a^k=1 mod p. Why must gcd(k,p−1)>1 or a=1 mod p?", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0045, "default_truncated": true, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.999160494334246 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "9.6e", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "If a^k=1 mod prime p, k>0, and a is not 1 mod p, then gcd(k,p−1)>1.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0016, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.8670357598021706 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "10.1", "topic": "polynomials", "kind": "multi_step", "prompt": "Over GF(5), ax²+bx+c passes through (0,1),(1,2),(2,0). Give (a,b,c).", "options": { "A": "(1,0,1)", "B": "(1,1,0)", "C": "(2,4,1)", "D": "(0,1,1)" }, "expected": [ "A" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0811, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.36694471738720585 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "10.2a", "topic": "coding", "kind": "multi_step", "prompt": "Recover the polynomial.", "options": { "A": "3x+1", "B": "2x+1", "C": "x+1", "D": "2x+3" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0045, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.46793683914590584 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "10.2b", "topic": "coding", "kind": "multi_step", "prompt": "Find the monic minimal error locator.", "options": { "A": "x", "B": "x−3", "C": "x−2", "D": "x−1" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0131, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.43589456091525647 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "10.3", "topic": "polynomials", "kind": "recognition", "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4429666554483415 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "10.4b", "topic": "polynomials", "kind": "recognition", "prompt": "Minimum distinct roots of PQ?", "options": { "A": "r+s", "B": "max(r,s)", "C": "min(r,s)", "D": "0" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.034, "default_truncated": false, "context": "Over GF(p) with p>2d and d>=1, P and Q have degree exactly d, with r and s distinct roots respectively.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5200545237355652 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "10.4c", "topic": "polynomials", "kind": "recognition", "prompt": "Degree of PQ?", "options": { "A": "d²", "B": "d", "C": "At most d−1", "D": "2d" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0058, "default_truncated": false, "context": "Over GF(p) with p>2d and d>=1, P and Q have degree exactly d, with r and s distinct roots respectively.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5623660377662696 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "10.4d", "topic": "polynomials", "kind": "recognition", "prompt": "Maximum degree of P+Q?", "options": { "A": "0", "B": "2d", "C": "d−1", "D": "d" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.029, "default_truncated": false, "context": "Over GF(p) with p>2d and d>=1, P and Q have degree exactly d, with r and s distinct roots respectively.", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9723563867702055 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "10.5", "topic": "proof", "kind": "multi_step", "prompt": "Why does a degree-one polynomial over a field have exactly one root?", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0389, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9493321312021362 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "10.6a", "topic": "counting", "kind": "calculation", "prompt": "Over GF(p), count degree-exactly-two polynomials that split into linear factors, including repeated roots.", "options": { "A": "p²", "B": "(p−1)C(p,2)", "C": "(p−1)(C(p,2)+p)", "D": "(p−1)p²" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0937, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.3469506810908233 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "10.6b", "topic": "counting", "kind": "calculation", "prompt": "Over GF(p), count all formal polynomials of degree exactly two.", "options": { "A": "(p−1)(C(p,2)+p)", "B": "(p−1)³", "C": "p³", "D": "(p−1)p²" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0057, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.48591457614046446 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "11.1", "topic": "coding", "kind": "multi_step", "prompt": "What is the largest number of adversarial delegates guaranteed tolerable? Give the threshold argument.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0081, "default_truncated": true, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6923076609241594 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "12.1", "topic": "counting", "kind": "calculation", "prompt": "Distinct arrangements of BAA?", "options": { "A": "3", "B": "1", "C": "6", "D": "2" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0638, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5469058326002219 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "12.2", "topic": "counting", "kind": "calculation", "prompt": "Distinct arrangements of 126?", "options": { "A": "3", "B": "6", "C": "1", "D": "9" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0162, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.31292412927280244 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "12.3", "topic": "counting", "kind": "calculation", "prompt": "n distinct bears, k distinct children, n>k. Assign exactly one distinct bear to each child. How many assignments?", "options": { "A": "k^n", "B": "C(n,k)", "C": "n^k", "D": "n!/(n−k)!" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0828, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.502533789699026 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "12.4", "topic": "counting", "kind": "calculation", "prompt": "Number of positive integer n-tuples summing to k, k>=n>=1?", "options": { "A": "C(k+n−1,n−1)", "B": "C(k−1,n−1)", "C": "n^k", "D": "C(k,n)" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0428, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.9908781880280934 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "12.5", "topic": "counting", "kind": "calculation", "prompt": "Number of n-bit strings with exactly k ones, 0<=k<=n?", "options": { "A": "n!/k!", "B": "C(n,k)", "C": "2^k", "D": "n^k" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0434, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.988920267449846 }, { "exam": "midterm_sp25", "sample": "follow_up", "id": "12.6", "topic": "counting", "kind": "calculation", "prompt": "Circular arrangements of n distinct beads, identifying rotations but not reflections?", "options": { "A": "(n−1)!", "B": "(n−1)!/2", "C": "2^n", "D": "n!" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1293, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9054826800168108 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "2.1a", "topic": "logic", "kind": "recognition", "prompt": "What is the truth status of (P AND NOT P) implies Q?", "options": { "A": "Could be either", "B": "True", "C": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0347, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9354499482139648 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "2.1b", "topic": "logic", "kind": "recognition", "prompt": "What is the truth status of P implies (Q OR NOT Q)?", "options": { "A": "Could be either", "B": "False", "C": "True" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0197, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9980695521035058 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "2.1c", "topic": "logic", "kind": "recognition", "prompt": "What is the truth status of (P OR NOT P) implies (Q AND NOT Q)?", "options": { "A": "Could be either", "B": "False", "C": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0291, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5922008117092141 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "2.1d", "topic": "logic", "kind": "recognition", "prompt": "What is the truth status of (P OR Q) implies (P AND Q)?", "options": { "A": "False", "B": "Could be either", "C": "True" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0643, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5401783173692418 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "2.2a", "topic": "logic", "kind": "recognition", "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?", "options": { "A": "Equivalent", "B": "Not equivalent" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1726, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7310585786300049 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "2.2b", "topic": "logic", "kind": "recognition", "prompt": "Are exists x [P(x) implies forall y R(x,y)] and forall x [P(x) OR exists y R(x,y)] equivalent?", "options": { "A": "Not equivalent", "B": "Equivalent" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0016, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7057850278370112 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "2.3", "topic": "logic", "kind": "multi_step", "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.", "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))" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1459, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7166787101418413 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "3.1", "topic": "proof", "kind": "multi_step", "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.", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0994, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4387718460501942 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "3.2a", "topic": "inequalities", "kind": "recognition", "prompt": "State AM–GM for two nonnegative numbers x,y.", "options": { "A": "sqrt(x+y) >= x*y", "B": "x+y >= xy", "C": "(x+y)/2 >= sqrt(xy)", "D": "(x+y)/2 <= sqrt(xy)" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0373, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9828038905382875 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "3.2b", "topic": "proof", "kind": "multi_step", "prompt": "Which proves AM–GM for nonnegative x,y?", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0105, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9984311637772874 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "4.1a", "topic": "recurrences", "kind": "multi_step", "prompt": "Smallest integer c with Sn<=c*2^n for all n>=0?", "options": { "A": "3", "B": "2", "C": "4", "D": "1" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0169, "default_truncated": false, "context": "S0=3, S1=2, S2=1; for n>=3, Sn=S(n−1)+S(n−2)+2S(n−3).", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.35823796793174434 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "4.1b", "topic": "proof", "kind": "multi_step", "prompt": "Which is a valid induction proving Sn<=3*2^n?", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.007, "default_truncated": true, "context": "S0=3, S1=2, S2=1; for n>=3, Sn=S(n−1)+S(n−2)+2S(n−3).", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9972806573482469 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "4.1c", "topic": "recurrences", "kind": "multi_step", "prompt": "Smallest integer c with Sn<=c*2^n for every n>=4?", "options": { "A": "1", "B": "3", "C": "2", "D": "0" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0046, "default_truncated": false, "context": "S0=3, S1=2, S2=1; for n>=3, Sn=S(n−1)+S(n−2)+2S(n−3).", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4844571435978946 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "4.2", "topic": "proof", "kind": "multi_step", "prompt": "Which induction proves 7 divides 11^n−4^n for n>=0?", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0253, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9623012493648266 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "5.1a", "topic": "matching", "kind": "multi_step", "prompt": "Find the job-optimal stable matching.", "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" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0384, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.3468666740605408 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "5.1b", "topic": "matching", "kind": "multi_step", "prompt": "Find the candidate-optimal stable matching.", "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" }, "expected": [ "D" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0267, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.29539205153207015 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "5.1c", "topic": "matching", "kind": "multi_step", "prompt": "Is there any other stable matching beyond the outputs of the two proposer directions?", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1486, "default_truncated": false, "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.", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.35909976350004963 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "5.2", "topic": "matching", "kind": "recognition", "prompt": "If every job has a different first-choice candidate, there must be exactly one stable matching.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0232, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.7310585786300049 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "5.3", "topic": "matching", "kind": "recognition", "prompt": "In the job-optimal stable matching, at least one job must get its first-choice candidate.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2025, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5312093733737563 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "5.4", "topic": "matching", "kind": "recognition", "prompt": "A job can receive its least-preferred candidate in the job-optimal stable matching.", "options": { "A": "False", "B": "True" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0168, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7310585786300049 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "5.5", "topic": "matching", "kind": "recognition", "prompt": "If a job is matched to its favorite candidate, that candidate cannot participate in a blocking pair.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0839, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.679178699175393 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "5.6", "topic": "matching", "kind": "recognition", "prompt": "For every preference instance, modifying only candidate preferences can change the job-proposing algorithm’s output.", "options": { "A": "False", "B": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0153, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8519528019683106 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "5.7", "topic": "proof", "kind": "multi_step", "prompt": "For every n>=2, construct preferences where job-proposing matches every job to its favorite and every candidate to its least favorite.", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.053, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5769654320158285 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "6.1a", "topic": "graphs", "kind": "recognition", "prompt": "A six-vertex graph with four edges is connected.", "options": { "A": "False", "B": "Could be either", "C": "True" }, "expected": [ "A" ], "jev_selected": [ "B" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0348, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.36166446309228156 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "6.1b", "topic": "graphs", "kind": "recognition", "prompt": "A six-vertex graph with six edges is connected.", "options": { "A": "Could be either", "B": "True", "C": "False" }, "expected": [ "A" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0048, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.37212220057302725 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "6.1c", "topic": "graphs", "kind": "recognition", "prompt": "A six-vertex graph with six edges is acyclic.", "options": { "A": "Could be either", "B": "True", "C": "False" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0059, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7174510311561372 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "6.1d", "topic": "graphs", "kind": "recognition", "prompt": "A six-vertex graph with thirteen edges is connected.", "options": { "A": "True", "B": "False", "C": "Could be either" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.2153, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.6460855563489776 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "6.1e", "topic": "graphs", "kind": "recognition", "prompt": "A six-vertex graph with thirteen edges is planar.", "options": { "A": "False", "B": "Could be either", "C": "True" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0632, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.7571610884557699 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "6.1f", "topic": "graphs", "kind": "calculation", "prompt": "Maximum edges of a simple graph on six vertices?", "options": { "A": "15", "B": "12", "C": "18", "D": "30" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1724, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9716710989485109 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "6.2", "topic": "graphs", "kind": "recognition", "prompt": "Minimum edges of an undirected graph with V vertices and C connected components?", "options": { "A": "C−1", "B": "V+C", "C": "V−1", "D": "V−C" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0228, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9650123086739246 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "6.3a", "topic": "graphs", "kind": "recognition", "prompt": "K7 has an Eulerian tour.", "options": { "A": "True", "B": "False" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0042, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6224593312018546 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "6.3b", "topic": "proof", "kind": "multi_step", "prompt": "Why does K7 have an Eulerian tour?", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0461, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9866576352829299 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "6.4a", "topic": "graphs", "kind": "recognition", "prompt": "K8 has an Eulerian tour.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0128, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.5312093733737563 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "6.4b", "topic": "proof", "kind": "multi_step", "prompt": "Why does K8 lack an Eulerian tour?", "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." }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "B" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0445, "default_truncated": false, "context": "", "semif_selected": [ "D" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9492261638636077 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "6.5", "topic": "graphs", "kind": "calculation", "prompt": "A connected planar embedding has 15 vertices and 4 faces including the outer face. How many edges?", "options": { "A": "13", "B": "17", "C": "19", "D": "11" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0236, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.4886653048003729 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "7.1", "topic": "proof", "kind": "multi_step", "prompt": "Prove that a graph all of whose vertices have odd degree has an even number of vertices.", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0395, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9944031200105473 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "7.2", "topic": "proof", "kind": "multi_step", "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?", "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." }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0591, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9838700411930208 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "8.1", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Compute 5^42 modulo 41.", "options": { "A": "1", "B": "0", "C": "5", "D": "25" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0447, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.4722904878490655 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "8.2", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Last digit of 519^487?", "options": { "A": "5", "B": "9", "C": "7", "D": "1" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0093, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.513313078784462 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "8.3", "topic": "modular_arithmetic", "kind": "multi_step", "prompt": "Compute 43^21 modulo 55.", "options": { "A": "43", "B": "1", "C": "23", "D": "12" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0067, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.35888231301687173 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "8.4", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "Choose nonzero a modulo m for which ax=0 modulo m has a nonzero solution.", "options": { "A": "a=4,m=9", "B": "a=2,m=5", "C": "a=3,m=7", "D": "a=2,m=4" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "D" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0185, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6042524401334828 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "8.5", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "For positive integers k1?", "options": { "A": "3", "B": "2", "C": "7", "D": "5" }, "expected": [ "C" ], "jev_selected": [ "A" ], "jev_correct": false, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0115, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.59394751363645 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "8.8b", "topic": "rsa", "kind": "multi_step", "prompt": "RSA has p=11,q=7,e=29. What is the least positive decryption exponent?", "options": { "A": "31", "B": "29", "C": "59", "D": "7" }, "expected": [ "B" ], "jev_selected": [ "C" ], "jev_correct": false, "laya_selected": [ "B" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0473, "default_truncated": false, "context": "", "semif_selected": [ "C" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.49909316711463353 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "9.1", "topic": "proof", "kind": "multi_step", "prompt": "Without induction, why is 11^n−4^n divisible by 7 for all nonnegative n?", "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." }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0094, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9676162947017731 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "9.2", "topic": "proof", "kind": "multi_step", "prompt": "If a has multiplicative order p−1 modulo prime p, why do a^0,...,a^(p−2) cover all nonzero residues?", "options": { "A": "They are nonzero. A repeated pair i1.", "options": { "A": "2", "B": "0", "C": "m−1", "D": "1" }, "expected": [ "D" ], "jev_selected": [ "D" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0493, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.8878604924576408 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "9.3b", "topic": "modular_arithmetic", "kind": "recognition", "prompt": "The residue m−1 is a nontrivial square root of 1 modulo m, where trivial roots mean ±1.", "options": { "A": "True", "B": "False" }, "expected": [ "B" ], "jev_selected": [ "B" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.2649, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.5926665999540697 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "9.3c", "topic": "proof", "kind": "multi_step", "prompt": "Why can a prime modulus p have no square root of 1 other than ±1?", "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." }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.1215, "default_truncated": false, "context": "", "semif_selected": [ "A" ], "semif_correct": true, "semif_valid": true, "semif_top_option_probability": 0.9995107883134526 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "9.3d", "topic": "modular_arithmetic", "kind": "calculation", "prompt": "Which is a nontrivial square root of 1 modulo 15?", "options": { "A": "4", "B": "14", "C": "0", "D": "1" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.0837, "default_truncated": false, "context": "", "semif_selected": [ "B" ], "semif_correct": false, "semif_valid": true, "semif_top_option_probability": 0.6866053578926755 }, { "exam": "midterm_su25", "sample": "follow_up", "id": "10.1a", "topic": "polynomials", "kind": "recognition", "prompt": "Largest possible degree of P−Q?", "options": { "A": "max(d1,d2)", "B": "d1*d2", "C": "min(d1,d2)", "D": "d1+d2" }, "expected": [ "A" ], "jev_selected": [ "A" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.015, "default_truncated": false, "context": "P,Q are distinct formal polynomials over GF(p), with degrees d1,d2=2. Smallest possible degree of P−Q?", "options": { "A": "d1", "B": "Undefined because it must be zero", "C": "0", "D": "d1−1" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "A" ], "laya_correct": false, "laya_valid": true, "laya_confidence": 0.1221, "default_truncated": false, "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)?", "options": { "A": "1", "B": "d1", "C": "0", "D": "p" }, "expected": [ "C" ], "jev_selected": [ "C" ], "jev_correct": true, "laya_selected": [ "C" ], "laya_correct": true, "laya_valid": true, "laya_confidence": 0.0825, "default_truncated": false, "context": "P,Q are distinct formal polynomials over GF(p), with degrees d1,d2