id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-22d4cb0472599aa01fff02052d70da60186dfbc66ac42aa7f081089ad8bd319a | What is the probability of three customers arriving in a five-minute time interval at an ATM where customers arrive at a rate of 30 customers per hour? | [
"0.21375"
] | [
"What is the probability of three customers arriving in a five-minute time interval at an ATM where customers arrive at a rate of 30 customers per hour?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
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} | 44 | 0 | 0 |
omr-ws-2517fbb162a15881cdd69d1d8ee1212fe7285b081f6a51124c64c946e19b414b | What is the probability of throwing a six-faced die five times and getting five different results? | [
"\\dfrac{5}{54}"
] | [
"What is the probability of throwing a six-faced die five times and getting five different results?"
] | [
"aops_c7_college_math"
] | [
"cot"
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"cot": 16
} | 16 | 0 | 0 |
omr-ws-d36fa3f07f1e780d2fdfc9a2438ffaeb2b1f6de5b74a3a0720a496dbe776c338 | What is the probability of two consecutive sixes appearing for the first time after the nth roll of a standard fair die? | [
"\\frac{3 - \\sqrt{5}}{30} \\left( \\frac{5 + 3\\sqrt{5}}{12} \\right)^n + \\frac{3 + \\sqrt{5}}{30} \\left( \\frac{5 - 3\\sqrt{5}}{12} \\right)^n"
] | [
"What is the probability of two consecutive sixes appearing for the first time after the nth roll of a standard fair die?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-fda1723089b2f68a07c5d55ec7897d08fda80160dafc86d2414f65753c5dfd88 | What is the probability of winning a 2-digit lottery game where you pick a number ranging from 1 to 31? | [
"\\(\\frac{1}{961}\\)"
] | [
"What is the probability of winning a 2-digit lottery game where you pick a number ranging from 1 to 31?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-29f7ff2a8c45b23fc9486dae24ff5fcb808ebfdfd3bbeced3cf155b93d1970b0 | What is the probability of winning a 3 attacker vs. 2 defender play in Risk if you are the attacker? In such a play, the attacker rolls three dice and the defender rolls two. The highest rolls of the attacker are matched to the highest rolls of the defender respectively. The higher roll wins, and the defender wins in t... | [
"0.5787"
] | [
"What is the probability of winning a 3 attacker vs. 2 defender play in Risk if you are the attacker? In such a play, the attacker rolls three dice and the defender rolls two. The highest rolls of the attacker are matched to the highest rolls of the defender respectively. The higher roll wins, and the defender wins... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef75074ed39c47fc332c1d6375f5d2d016e35724cf034d39e8a8c8fafe8d0203 | What is the probability of winning a game by flipping exactly four heads in a row, where the game requires flipping \( n + 1 \) heads in a row after flipping \( n \) tails? Express your answer as a common fraction. | [
"\\(\\frac{21}{1024}\\)"
] | [
"What is the probability of winning a game by flipping exactly four heads in a row, where the game requires flipping \\( n + 1 \\) heads in a row after flipping \\( n \\) tails? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot",
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] | {
"cot": 8,
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} | 24 | 0 | 0 |
omr-ws-a884913f32d2e165f04de16032b8bd785e6b1a4728d44d86dc77c88030b63de4 | What is the probability of winning all three games, given that a team has three games to play and each game can result in either a win (W) or a loss (L)? | [
"\\dfrac{1}{8}"
] | [
"What is the probability of winning all three games, given that a team has three games to play and each game can result in either a win (W) or a loss (L)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-00ecac814c248b8baaa3a34915be4773c03ba064e4c543d8bc179f552cc81fde | What is the probability of winning at least 5 out of 8 races, given that the probability of winning a single race is 90%? | [
"0.99497565"
] | [
"What is the probability of winning at least 5 out of 8 races, given that the probability of winning a single race is 90%?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-308bb1560c1dbd45de4040b0d8589ee46ec71b3679fc3b120885c4db11535dc3 | What is the probability of winning the dice game "craps"? | [
"\\(\\frac{244}{495}\\)"
] | [
"What is the probability of winning the dice game \"craps\"?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-0e859210803393a98a73303fb34bad06e47d93c80333375e759586ebbd508e4f | What is the probability of winning the main prize in a lottery in which six numbers must be guessed out of 48? What is the probability of guessing live, four or three numbers? | [
"$\\frac{1}{C(48,6)};\\ \\frac{C(6,5)C(42,1)}{C(48,6)};\\ \\frac{C(6,4)C(42,2)}{C(48,6)};\\ \\frac{C(6,3)C(42,3)}{C(48,6)}$"
] | [
"What is the probability of winning the main prize in a lottery in which six numbers must be guessed out of 48? What is the probability of guessing live, four or three numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b923256154a8ff9b3fb6bbb92024beaab96ac88752c133489f1e0d6fd036b2e3 | What is the probability of winning when $X=6$ in a game where you roll a fair six-sided die repeatedly and win if the sum of the rolls hits exactly $X$ at some point? | [
"\\dfrac{16807}{46656}"
] | [
"What is the probability of winning when $X=6$ in a game where you roll a fair six-sided die repeatedly and win if the sum of the rolls hits exactly $X$ at some point?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4d9b98d85f63f8866a47773b14fe57f2e713ac1fb7f9b9b8e7ce376be2ddf9c1 | What is the probability that $A \cap B \cap C = \emptyset$ for subsets $A, B,$ and $C$ of $S = \{1, 2, 3, 4\}$? | [
"\\(\\frac{2401}{4096}\\)"
] | [
"What is the probability that $A \\cap B \\cap C = \\emptyset$ for subsets $A, B,$ and $C$ of $S = \\{1, 2, 3, 4\\}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-cb4328776eed05e408fccd7c3f00df5b10716ff861d5af80c40440cb6e81604e | What is the probability that $A \cup B = S$ and $n(A \cap B) = 3$ for subsets $A$ and $B$ of $S = \{1, 2, 3, 4\}$? | [
"\\(\\frac{1}{32}\\)"
] | [
"What is the probability that $A \\cup B = S$ and $n(A \\cap B) = 3$ for subsets $A$ and $B$ of $S = \\{1, 2, 3, 4\\}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-266b99e078d907e54da2144601898b51fad6e2c7a46ccf38199218c0018b733f | What is the probability that $\sin^2 x$, $\cos^2 x$, and $\sin x \cos x$ do not form a triangle when $x$ is randomly chosen from $(0^{\circ},90^{\circ})$? | [
"\\(\\frac{\\arctan{2}}{90}\\)"
] | [
"What is the probability that $\\sin^2 x$, $\\cos^2 x$, and $\\sin x \\cos x$ do not form a triangle when $x$ is randomly chosen from $(0^{\\circ},90^{\\circ})$?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-0fc8583d05c64782fba690a5e26380cf66d36f7e753b78270133b628fff91d46 | What is the probability that $n$ integers, each ranging from 0 to $m$, have a greatest common divisor (GCD) of 1? | [
"$\\frac{1}{\\zeta(n)}$"
] | [
"What is the probability that $n$ integers, each ranging from 0 to $m$, have a greatest common divisor (GCD) of 1?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-942e03238aebc7fcdec259b9439ed00fecd32b97d232b906acf5592a96ff5a80 | What is the probability that $n$ points, selected independently and uniformly at random within the unit square, form a convex $n$-gon? | [
"\\frac{n!}{4^{n-1}}"
] | [
"What is the probability that $n$ points, selected independently and uniformly at random within the unit square, form a convex $n$-gon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-0cb79e4bcd875eba868ac9b6bf1017f41d79a1ca5db87e3b18fab4273cb7ab9d | What is the probability that $n$ random points on a circle are contained in a semicircle? | [
"\\( \\frac{4}{2^n} \\)"
] | [
"What is the probability that $n$ random points on a circle are contained in a semicircle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cad1a22542499f4fec25c55fa25569f0497883b9c5096bc92d74eef2324aa98c | What is the probability that $n(A) > n(B) > n(C) \geq 1$ for subsets $A, B,$ and $C$ of $S = \{1, 2, 3, 4\}$? | [
"\\(\\frac{3}{512}\\)"
] | [
"What is the probability that $n(A) > n(B) > n(C) \\geq 1$ for subsets $A, B,$ and $C$ of $S = \\{1, 2, 3, 4\\}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b36b67fd547377531ec9d4a3e812bcf0491785b57a16e5f8112a4dbf7cb566d | What is the probability that $n^{100}$ has a remainder of 16 when divided by 70, given that $n$ is a positive integer such that $1 \leq n \leq 70$? | [
"\\(\\frac{4}{35}\\)"
] | [
"What is the probability that $n^{100}$ has a remainder of 16 when divided by 70, given that $n$ is a positive integer such that $1 \\leq n \\leq 70$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 30
} | 52 | 0 | 0 |
omr-ws-ce7bb5e5bb61f1fd4183263360d70d87eeae1871dbc6643e25980ff42a858b4e | What is the probability that 10 coins selected randomly from 100 coins (which consist of pennies, nickels, dimes, and quarters) add up to $1? Each coin has an equal probability of being selected. | [
"\\(\\frac{\\binom{25}{3}\\binom{25}{2}\\binom{25}{5} + \\binom{25}{2}\\binom{25}{2}\\binom{25}{6} + \\binom{25}{1}\\binom{25}{6}\\binom{25}{3} + \\binom{25}{10}}{\\binom{100}{10}}\\)"
] | [
"What is the probability that 10 coins selected randomly from 100 coins (which consist of pennies, nickels, dimes, and quarters) add up to $1? Each coin has an equal probability of being selected."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d99c564f39e4750477da9c7d3a59a729feba6a48db74ed018e8204cde40fe26a | What is the probability that 11 points on a sphere lie in the same hemisphere? | [
"\\dfrac{7}{128}"
] | [
"What is the probability that 11 points on a sphere lie in the same hemisphere?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-7160436d1dc0b2e084c879fdc879c45f881fef00d928c85f17806f3e6b34281b | What is the probability that 17 points on the surface of a sphere lie on the same hemisphere? | [
"\\dfrac{137}{65536}"
] | [
"What is the probability that 17 points on the surface of a sphere lie on the same hemisphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-b13e80d5394c02a259d5ff20df7b0233ddc3c7000a65a7c9d6e9c3250e5a17ac | What is the probability that 3 cards chosen from a standard 52-card deck form a pair (i.e., 2 cards match in rank and 1 is different)? | [
"\\(\\frac{72}{425}\\)"
] | [
"What is the probability that 3 cards chosen from a standard 52-card deck form a pair (i.e., 2 cards match in rank and 1 is different)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-a8ecac8791014669903382a1ed449f777936a366384e3fc07bc108cd2763dfbb | What is the probability that 3 consecutive heads appear before 2 consecutive tails when a fair coin is tossed repeatedly? | [
"\\dfrac{3}{10}"
] | [
"What is the probability that 3 consecutive heads appear before 2 consecutive tails when a fair coin is tossed repeatedly?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f9748e0c58e8307fcc3bd3dc63eac3be346989a3858ecdf607107a03e15a702c | What is the probability that 3 out of 5 balls drawn from a box containing 5 white, 3 black, and 4 red balls are white? | [
"\\(\\frac{35}{132}\\)"
] | [
"What is the probability that 3 out of 5 balls drawn from a box containing 5 white, 3 black, and 4 red balls are white?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-183c1332f10137e3b59764e14deced9fd184a5e082d87430ea9ac3dd80415a96 | What is the probability that 3 out of 5 people will get back their own gifts when the boxes are given back randomly? | [
"\\( \\frac{1}{6} \\)"
] | [
"What is the probability that 3 out of 5 people will get back their own gifts when the boxes are given back randomly?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eb7470e51ccc434ea3be4b8db009ef3104bc2ec7af4cec0438b7f5d55b6e5568 | What is the probability that 3 randomly chosen people from a group of 10 seated at a round table are seated consecutively? | [
"\\(\\frac{1}{18}\\)"
] | [
"What is the probability that 3 randomly chosen people from a group of 10 seated at a round table are seated consecutively?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f7e522d934aa0cf5554092a16a676804bd109a9fd1b3c5b13cce9d5e4dd1e22 | What is the probability that 3 randomly chosen points from 2010 evenly spaced points on a circle form the vertices of a right triangle? | [
"\\(\\frac{3}{2009}\\)"
] | [
"What is the probability that 3 randomly chosen points from 2010 evenly spaced points on a circle form the vertices of a right triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 59
} | 90 | 0 | 0 |
omr-ws-599aee89e42b3a2821d2169026cdf6de7407f2abe3a4c481de18ec8feae01f39 | What is the probability that 3 randomly chosen points on a circle form an acute triangle? | [
"\\dfrac{1}{4}"
] | [
"What is the probability that 3 randomly chosen points on a circle form an acute triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a8e221c5611a68148953851288a2f82527a700d3aaa68a880ad698da98998f3d | What is the probability that 3 randomly chosen points out of a 4×3 grid lie on a straight line? | [
"\\(\\frac{1}{11}\\)"
] | [
"What is the probability that 3 randomly chosen points out of a 4×3 grid lie on a straight line?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 30
} | 44 | 0 | 0 |
omr-ws-9c17defe34dd4e4b1a3e1b32b291e25b06e02bb217f4270ad5ced3d126a8e7bd | What is the probability that 3 randomly drawn cards from a box containing 3 green, 3 red, 3 blue, and 3 yellow cards (each numbered 1, 2, or 3) are of the same color? | [
"\\(\\frac{3}{220}\\)"
] | [
"What is the probability that 3 randomly drawn cards from a box containing 3 green, 3 red, 3 blue, and 3 yellow cards (each numbered 1, 2, or 3) are of the same color?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-571b0e201e1416b4a6595115a52f99d7efaa7fd6ce152e37ef17c3f22870cfcb | What is the probability that 3 randomly drawn cards from a box containing 3 green, 3 red, 3 blue, and 3 yellow cards (each numbered 1, 2, or 3) are of three different colors? | [
"\\(\\frac{108}{220}\\)"
] | [
"What is the probability that 3 randomly drawn cards from a box containing 3 green, 3 red, 3 blue, and 3 yellow cards (each numbered 1, 2, or 3) are of three different colors?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-21420a7fc85ac1acad5b2deabd4f299edba5dcb54d51ef83cdd6872f29100df6 | What is the probability that 3 randomly drawn cards from a box containing 3 green, 3 red, 3 blue, and 3 yellow cards (each numbered 1, 2, or 3) have numbers 1, 2, and 3 regardless of their colors? | [
"$\\frac{64}{220}$"
] | [
"What is the probability that 3 randomly drawn cards from a box containing 3 green, 3 red, 3 blue, and 3 yellow cards (each numbered 1, 2, or 3) have numbers 1, 2, and 3 regardless of their colors?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-64c1a02e055f6b381778e6b336cde7509588e6d5e1f39c11a2aee687af5c06d9 | What is the probability that 3 randomly drawn cards from a box containing 3 green, 3 red, 3 blue, and 3 yellow cards (each numbered 1, 2, or 3) have the same number on them? | [
"\\(\\frac{12}{220}\\)"
] | [
"What is the probability that 3 randomly drawn cards from a box containing 3 green, 3 red, 3 blue, and 3 yellow cards (each numbered 1, 2, or 3) have the same number on them?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-b679d88171bd83dcafa1b11f82b92f991fb5f13a057e096472648cad1f4ba0ae | What is the probability that 4 knights and 4 jokers seated around a round table will be seated alternately? | [
"\\(\\frac{1}{35}\\)"
] | [
"What is the probability that 4 knights and 4 jokers seated around a round table will be seated alternately?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 11
} | 28 | 0 | 0 |
omr-ws-046f50edf57f3825518e17ef71ec361f606833cd6d820e781147faa38ca7f53f | What is the probability that 4 or more people will have to be tested before 2 people with a certain gene are found, given that the probability of carrying the gene is 0.1? | [
"0.972"
] | [
"What is the probability that 4 or more people will have to be tested before 2 people with a certain gene are found, given that the probability of carrying the gene is 0.1?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-0d493ef25219de75ce03aaee382be62a969a0984abc719f5edd5bdc1561707bb | What is the probability that 4 points, selected independently and uniformly at random within the unit square, form a convex quadrilateral? | [
"\\( 1 - \\frac{35}{12\\pi^2} \\)"
] | [
"What is the probability that 4 points, selected independently and uniformly at random within the unit square, form a convex quadrilateral?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-00ea1a8300ed1d92c0e80791593af5696f9f506505a66e6f5796b6903dc29a18 | What is the probability that 4 real numbers chosen in the interval (0,1] can be the side lengths of a quadrilateral with positive area? | [
"\\(\\frac{1}{8}\\)"
] | [
"What is the probability that 4 real numbers chosen in the interval (0,1] can be the side lengths of a quadrilateral with positive area?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f64584cd47894428c5f4ef4334282255ef2adabe27fca8900caf165f229d947 | What is the probability that 5 children will get the flavors of their choice if they are allowed to choose their own flavor, given that there are 16 candies in total, with 4 each of grape, strawberry, chocolate, and coffee flavors? | [
"\\dfrac{255}{256}"
] | [
"What is the probability that 5 children will get the flavors of their choice if they are allowed to choose their own flavor, given that there are 16 candies in total, with 4 each of grape, strawberry, chocolate, and coffee flavors?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7baee46eb9f871eb4c364f4bfd7b729d0e816104cdad709d8f62a7ce9770cf0f | What is the probability that 5 dice rolled together all show different numbers? | [
"\\(\\frac{5}{54}\\)"
] | [
"What is the probability that 5 dice rolled together all show different numbers?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-c0d2e9f3a48fdcfdd9ad1796bd22c4db7bf27013c3016ddc55a799cac0795dd7 | What is the probability that 5 friends stay in separate hotels, given there are 8 hotels in a town? | [
"\\(\\frac{105}{512}\\)"
] | [
"What is the probability that 5 friends stay in separate hotels, given there are 8 hotels in a town?"
] | [
"aops_c4_high_school_math"
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"cot",
"tir"
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"cot": 16,
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} | 44 | 0 | 0 |
omr-ws-2c7c7b60e0d31e79c610ae2b6c6055f4d24dc16dd685daf79804217656346827 | What is the probability that 5 people in a group of 20 are left-handed, given that 10% of the population are left-handed? | [
"0.3182"
] | [
"What is the probability that 5 people in a group of 20 are left-handed, given that 10% of the population are left-handed?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-05fcb1c0251517445e6a04ea15a6d708c1c0316ad20b6fbe4c6f04adf6a55d8e | What is the probability that 50 passengers will overload an aircraft with a load limit of 11,000 lbs, given that passengers (with luggage) weigh an average of 200 lbs with a standard deviation of 50 lbs? | [
"0.0023"
] | [
"What is the probability that 50 passengers will overload an aircraft with a load limit of 11,000 lbs, given that passengers (with luggage) weigh an average of 200 lbs with a standard deviation of 50 lbs?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
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"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-ff950962ca853974c2a8309f924d0d6a8824d7efb07520c6308050b6d4d5dec6 | What is the probability that 9 specific friends will all be on the same team as you and a specific person will not be on your team, given there are 5 possible teams to be on? | [
"\\(\\frac{4}{5^{10}}\\)"
] | [
"What is the probability that 9 specific friends will all be on the same team as you and a specific person will not be on your team, given there are 5 possible teams to be on?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
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"cot": 15,
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} | 41 | 0 | 0 |
omr-ws-788940f6d151f8dc489a23d1a64450074acda9ad56d911bc26997d44ca359e42 | What is the probability that 9 specific friends will not be on the same team as you and a specific person will be on your team, given there are 5 possible teams to be on? | [
"\\(\\frac{4^9}{5^{10}}\\)"
] | [
"What is the probability that 9 specific friends will not be on the same team as you and a specific person will be on your team, given there are 5 possible teams to be on?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
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"cot": 15,
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omr-ws-5bfb3f7774e916f3ce98c1151b178bcd629184eb95f16f198d34ab25a2809e77 | What is the probability that A and B will get the same number of heads when A tosses (n+1) coins and B tosses n coins? | [
"\\(\\frac{1}{n+2}\\)"
] | [
"What is the probability that A and B will get the same number of heads when A tosses (n+1) coins and B tosses n coins?"
] | [
"aops_c7_college_math"
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"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-175da9bd0cc88e2e13590c582f2bf59bdbde61ec21054562747429af74040d84 | What is the probability that Al and Bob will not be on a randomly selected 4-person subcommittee from a group of Al, Bob, Chris, Dave, Ed, and Frank? | [
"\\(\\frac{1}{15}\\)"
] | [
"What is the probability that Al and Bob will not be on a randomly selected 4-person subcommittee from a group of Al, Bob, Chris, Dave, Ed, and Frank?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
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"cot": 13,
"tir": 16
} | 29 | 0 | 0 |
omr-ws-edfcabb1fd88f74829795a517510c66112b2cec86ed72beeff002d264818f970 | What is the probability that Alan and Betty both require the same number of throws to roll a $6$ on their six-sided dice? | [
"\\(\\frac{1}{11}\\)"
] | [
"What is the probability that Alan and Betty both require the same number of throws to roll a $6$ on their six-sided dice?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
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"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-1e8053e277375e1864d7469a7123e2efed157b5437af64b9e55c4a2d74e72c48 | What is the probability that Alex finishes his shave before Bernie finishes his haircut, given that the time to get a shave is exponentially distributed with parameter $\lambda=4$ per hour and the time to get a haircut is exponentially distributed with parameter $\mu=3$ per hour? | [
"\\(\\frac{4}{7}\\)"
] | [
"What is the probability that Alex finishes his shave before Bernie finishes his haircut, given that the time to get a shave is exponentially distributed with parameter $\\lambda=4$ per hour and the time to get a haircut is exponentially distributed with parameter $\\mu=3$ per hour?"
] | [
"aops_c7_college_math"
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"cot",
"tir"
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"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-efa0c02af99eaa87b14ab372a3b3ea883ff79fcbc9a783a21daa70e4c0cf4315 | What is the probability that Alexander wins, given that he bets on a total sum of 12 occurring first and Eliza bets on two consecutive sums of 7 appearing first when rolling a standard pair of 6-sided dice? | [
"\\(\\frac{6}{13}\\)\n\nExplanation:\nThe probability that Alexander wins can be derived from the probability that Eliza wins, which is given as \\(\\frac{7}{13}\\). Since there are only two outcomes (either Alexander wins or Eliza wins), the probability that Alexander wins is \\(1 - \\frac{7}{13} = \\frac{6}{13}\\... | [
"What is the probability that Alexander wins, given that he bets on a total sum of 12 occurring first and Eliza bets on two consecutive sums of 7 appearing first when rolling a standard pair of 6-sided dice?"
] | [
"aops_c4_high_school_math"
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"cot"
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"cot": 15
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omr-ws-8cf6284bbe736844faae61eb075f8e2c8e448f0f500268cb6ce0e62a6ab9b099 | What is the probability that Angus is dealt two pairs from a standard 52-card deck? | [
"\\(\\frac{198}{4165}\\)"
] | [
"What is the probability that Angus is dealt two pairs from a standard 52-card deck?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-ab34c5d186c96cece058a2f4ddd39a0e0edba6e0e58642bbef75f7d2579282a7 | What is the probability that Ann wins the game, where Ann wins if she rolls a "six" on a fair die and Bob wins if he tosses a "head" on a fair coin, with the game continuing in an alternating pattern until one of them wins? | [
"\\(\\frac{2}{7}\\)"
] | [
"What is the probability that Ann wins the game, where Ann wins if she rolls a \"six\" on a fair die and Bob wins if he tosses a \"head\" on a fair coin, with the game continuing in an alternating pattern until one of them wins?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-04648a740a3cd2e7e1450bacf442c8981967ede90aa7f6a2f7c89147076032b9 | What is the probability that Arnold wins after taking exactly 7 shots, given that he wins once he has made 3 shots through the same hoop, and each shot has an equal probability of falling into one of three hoops? | [
"\\(\\frac{10}{81}\\)"
] | [
"What is the probability that Arnold wins after taking exactly 7 shots, given that he wins once he has made 3 shots through the same hoop, and each shot has an equal probability of falling into one of three hoops?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-3793bf13458b03a50f33ece7ff96aaad737c407a5dc4f11dd6d0f5ccd508f52a | What is the probability that Becky sees a shooting star in the first 16 minutes, given that there's an 80% chance she'll see a shooting star in any given hour? | [
"0.35"
] | [
"What is the probability that Becky sees a shooting star in the first 16 minutes, given that there's an 80% chance she'll see a shooting star in any given hour?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-46b06f5230fcd81e2e8caf23103312d832c709bc577e7c393be146442a29884e | What is the probability that Becky will see a shooting star in the first 15 minutes, given that there is an 80% chance of seeing a shooting star in any given hour? | [
"\\(1 - \\sqrt[4]{0.2}\\)"
] | [
"What is the probability that Becky will see a shooting star in the first 15 minutes, given that there is an 80% chance of seeing a shooting star in any given hour?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
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] | {
"cot": 14,
"genselect": 5,
"tir": 29
} | 48 | 0 | 0 |
omr-ws-4528e66489176836ce925b636ef3b6f0193f9f7961456af9c0fd622915371a9d | What is the probability that Becky would see a shooting star in the first 15 minutes, given that there is an 80% chance she will see a shooting star in any given hour? | [
"\\( 1 - \\sqrt[4]{0.2} \\)"
] | [
"What is the probability that Becky would see a shooting star in the first 15 minutes, given that there is an 80% chance she will see a shooting star in any given hour?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-01ff7c313fd557949198630df184903ed2bf5eb0f7eebc4a2004aff327b9bf2d | What is the probability that Bob gets 3 heads in a row before his 11th flip when flipping a two-sided coin until he gets 3 heads? | [
"\\dfrac{65}{128}"
] | [
"What is the probability that Bob gets 3 heads in a row before his 11th flip when flipping a two-sided coin until he gets 3 heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-4bc7dd7323fff309879a116dd6a9e4c2c5b396be3a7f8c33218e3450d91acfca | What is the probability that Bob gets at least 3 questions correct on a test with 5 questions, given the probabilities of getting each question correct are as follows: 100% for the first, 80% for the second, 60% for the third, 40% for the fourth, and 20% for the fifth? | [
"\\(\\frac{447}{625}\\)"
] | [
"What is the probability that Bob gets at least 3 questions correct on a test with 5 questions, given the probabilities of getting each question correct are as follows: 100% for the first, 80% for the second, 60% for the third, 40% for the fourth, and 20% for the fifth?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-79e9bfa96c6e04a4c1bd0887b4b85e6ed12441d01910b34db8499f96f68c667d | What is the probability that Bob picks one club and two red cards when he chooses 5 cards from a deck of 52 cards? | [
"\\(\\frac{323297}{2399040}\\)"
] | [
"What is the probability that Bob picks one club and two red cards when he chooses 5 cards from a deck of 52 cards?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-fdc218c2e548a7daa40ec633aa0a8d8833cecb77b03d978a9c28993a76d8ab5f | What is the probability that Cecilia, Annie, and Kimi are the first 3 gymnasts to perform, in any order, out of a total of seven people? | [
"\\(\\frac{1}{35}\\)"
] | [
"What is the probability that Cecilia, Annie, and Kimi are the first 3 gymnasts to perform, in any order, out of a total of seven people?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-b6d2ee57b3f106aba344d954e3c3ce364445ebb7df6a09afd9050d6d5eaedd13 | What is the probability that Charles flips more heads than both Albert and Brian, given that Albert flips 5 fair coins, Brian flips 6 fair coins, and Charles flips 7 fair coins? | [
"\\(\\frac{104696}{262144}\\)"
] | [
"What is the probability that Charles flips more heads than both Albert and Brian, given that Albert flips 5 fair coins, Brian flips 6 fair coins, and Charles flips 7 fair coins?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 21
} | 42 | 0 | 0 |
omr-ws-2c6958d41670517b0e489a7c031d58206d3807aa1a1bddc794b758d40347930d | What is the probability that Cyrus the frog lands less than 1 unit away from his starting position after jumping 2 units in one direction and then 2 more units in another direction? | [
"\\(\\frac{2\\arcsin\\left(\\frac{1}{4}\\right)}{\\pi}\\)"
] | [
"What is the probability that Cyrus the frog lands less than 1 unit away from his starting position after jumping 2 units in one direction and then 2 more units in another direction?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 2,
"tir": 4
} | 21 | 0 | 0 |
omr-ws-6f5b4af6cde680fb1727a714792b369a041853e3a1248e28af512e080fb7d870 | What is the probability that Fatima gets fewer heads than tails if she flips 10 coins? | [
"\\dfrac{193}{512}"
] | [
"What is the probability that Fatima gets fewer heads than tails if she flips 10 coins?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a077f46f0612466609461860a1f6ec6c63d24f7a06dd919a7090e838200d06d3 | What is the probability that Fisher wins the match, given that each game is won by Fisher with probability 0.4, by Spassky with probability 0.3, and is a draw with probability 0.3, independent of previous games? | [
"0.5714"
] | [
"What is the probability that Fisher wins the match, given that each game is won by Fisher with probability 0.4, by Spassky with probability 0.3, and is a draw with probability 0.3, independent of previous games?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 20
} | 34 | 0 | 0 |
omr-ws-00cb894415b5534c165c230f747b63647361ee689aa991bc73e76b0f802f25e7 | What is the probability that I stop flipping a coin once more than 3/4 of my flips up to that point are heads? | [
"\\(\\frac{8}{15}\\)"
] | [
"What is the probability that I stop flipping a coin once more than 3/4 of my flips up to that point are heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-bde042e48c1ca52208d4a2e295a094e31a8a4183400866c86dbe36ea35d76994 | What is the probability that Jack will get exactly 5 questions right if he guesses on all 25 multiple choice questions, where each question has 5 choices? | [
"\\(\\binom{25}{5} \\left(\\frac{1}{5}\\right)^5 \\left(\\frac{4}{5}\\right)^{20}\\) or approximately \\(\\frac{1}{5}\\)"
] | [
"What is the probability that Jack will get exactly 5 questions right if he guesses on all 25 multiple choice questions, where each question has 5 choices?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 3,
"tir": 1
} | 12 | 0 | 0 |
omr-ws-9b145d10505b6afdf206d8f9c762105091a2562f170aa4815627e21fa94872a7 | What is the probability that Janet's running total will equal 3 at some point when she rolls a standard 6-sided die 4 times? Note: The answer choices are provided but not included in the problem statement. | [
"\\(\\frac{49}{216}\\)"
] | [
"What is the probability that Janet's running total will equal 3 at some point when she rolls a standard 6-sided die 4 times?\n\nNote: The answer choices are provided but not included in the problem statement."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-c25b204319e5326a829dcd8e41e343bfab63e1299ea942394f9d572f4daeb8e0 | What is the probability that Jason will stop on two consecutive tosses of 3 when he tosses a die repeatedly until he gets a 6 or two consecutive tosses of 3? | [
"\\(\\frac{1}{8}\\)"
] | [
"What is the probability that Jason will stop on two consecutive tosses of 3 when he tosses a die repeatedly until he gets a 6 or two consecutive tosses of 3?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-e3ef18a79c812cd10f4600fb21c22fd020fc507cd3385f7b6b0dfb703f4819fa | What is the probability that Jelena wins a set of tennis 6 games to 4, given that Martina beats Jelena in 2 games out of 3? | [
"\\(\\frac{224}{6561}\\)"
] | [
"What is the probability that Jelena wins a set of tennis 6 games to 4, given that Martina beats Jelena in 2 games out of 3?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-c600eb238b8384b7c90933efd0ea16a4d59ee4ee02a7d32d010412997269bc98 | What is the probability that Jenny and Alex each flipping four fair coins get the same number of tails? | [
"\\(\\frac{35}{128}\\)"
] | [
"What is the probability that Jenny and Alex each flipping four fair coins get the same number of tails?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-047e5519750aaf08488e4d4e9f43d9b0f4b86145b84c891e5f3a13f2f3c58188 | What is the probability that Judy would win the prize, given that she returned home at some time after 2:20 PM, if the radio station calls a random phone number at a random time between 2:00 PM and 3:00 PM? Give your answer as a reduced fraction. | [
"\\(\\frac{1}{3}\\)"
] | [
"What is the probability that Judy would win the prize, given that she returned home at some time after 2:20 PM, if the radio station calls a random phone number at a random time between 2:00 PM and 3:00 PM? Give your answer as a reduced fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-0847f20f36a3a0f9274cc893dbfed088c3bcef62f4b000cb707930bf40f5677c | What is the probability that LA wins the game given that Kobe has an 82% chance of making each free throw and a 50% chance of missing the second shot if he misses the first? | [
"0.6724"
] | [
"What is the probability that LA wins the game given that Kobe has an 82% chance of making each free throw and a 50% chance of missing the second shot if he misses the first?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-996f280b8f7ec8912beda7f3fdcd812cf9db9fe84abf5aeaded697b79079c579 | What is the probability that Lisa, Frank, and Maria will be in the same group when Mrs. Johnson divides her 15 kids into groups of 3? Express your answer as a common fraction. | [
"\\(\\frac{1}{91}\\)"
] | [
"What is the probability that Lisa, Frank, and Maria will be in the same group when Mrs. Johnson divides her 15 kids into groups of 3? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-716f1060507eb3836b67b321eeb2661954adfa84db3df492f20d016402ce4ae2 | What is the probability that Louie's name is drawn to compete in at least two of the three diving events, given that only two out of three divers (Huey, Dewey, and Louie) are allowed to compete in each event and the divers' names are drawn randomly for each event? | [
"$\\frac{20}{27}$"
] | [
"What is the probability that Louie's name is drawn to compete in at least two of the three diving events, given that only two out of three divers (Huey, Dewey, and Louie) are allowed to compete in each event and the divers' names are drawn randomly for each event?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-f0f4d8c94e949f3f462afd70471537d105458d48b04bd5da65bbf8be35ec3461 | What is the probability that Marcia will be a defence attorney on the third day and have other roles for the remaining days, given that she rolls a die each day where rolling a 1 means she is "in" as a defence attorney and rolling a 2, 3, 4, 5, or 6 means she is "out"? | [
"\\(\\frac{125}{1296}\\)"
] | [
"What is the probability that Marcia will be a defence attorney on the third day and have other roles for the remaining days, given that she rolls a die each day where rolling a 1 means she is \"in\" as a defence attorney and rolling a 2, 3, 4, 5, or 6 means she is \"out\"?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-78796e10a76c9c61c9302abdf2f7bef2d1c62b27afe46a873eb91e2e6b00d709 | What is the probability that Mr. T will be able to take out five chess pieces, with exactly four of them being white, from a box containing eight white and eight black chess pieces? | [
"\\(\\frac{35}{273}\\)"
] | [
"What is the probability that Mr. T will be able to take out five chess pieces, with exactly four of them being white, from a box containing eight white and eight black chess pieces?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 30
} | 39 | 0 | 0 |
omr-ws-a1520aa0b50d5682ab2352b8fa9181fb516afe61c41c9dc4016ac26974c570ef | What is the probability that Player A will win if Player A bets on rolling a 12 first and Player B bets on rolling a 7 first, with the dice being rolled repeatedly until one of these outcomes occurs? | [
"\\(\\frac{1}{7}\\)"
] | [
"What is the probability that Player A will win if Player A bets on rolling a 12 first and Player B bets on rolling a 7 first, with the dice being rolled repeatedly until one of these outcomes occurs?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-a8ddde043c4c1fe5b5aa13edb0bfe513677c94bd2141023acb76dfb92543c419 | What is the probability that Rocky's roll of one six-sided die is greater than or equal to Bullwinkle's larger number when Bullwinkle rolls two six-sided dice? | [
"\\(\\frac{161}{216}\\)"
] | [
"What is the probability that Rocky's roll of one six-sided die is greater than or equal to Bullwinkle's larger number when Bullwinkle rolls two six-sided dice?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3f6b0f56634105253730ced4e95d29934ce48dd2ac58269be1a2161fc97e502e | What is the probability that Steven gets at least half of the questions correct on a 15 question multiple choice test, where each question has two choices (one correct and one incorrect), if he guesses randomly on each question? Express your answer as a common fraction. | [
"\\(\\frac{309}{2048}\\)"
] | [
"What is the probability that Steven gets at least half of the questions correct on a 15 question multiple choice test, where each question has two choices (one correct and one incorrect), if he guesses randomly on each question? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b4217e4fc2429fb66be75a03ba2f2d355ff52276ce395bff25b1de679139d0f | What is the probability that Team A wins the series and is always leading the series against Team B, given that the first team to win 4 out of 7 games wins the series and each team is equally likely to win each game? | [
"\\(\\frac{15}{128}\\)"
] | [
"What is the probability that Team A wins the series and is always leading the series against Team B, given that the first team to win 4 out of 7 games wins the series and each team is equally likely to win each game?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d771b4188870bd206cc5c704f18e06aed17ab13d0d1674643b44c17a63297d49 | What is the probability that \( X_n \) is a negative number after throwing a dice with sides labeled \( 0, 2, 3, \sqrt{2}, \sqrt{3}, i \) \( n \) times and defining \( X_n \) as the product of the outcomes? | [
" \\left( \\frac{5}{6} \\right)^n \\cdot \\frac{1}{4} \\left( 1 + \\left( \\frac{3}{5} \\right)^n - 2 \\left( \\frac{\\sqrt{17}}{5} \\right)^n \\cos\\left( n \\arctan\\left( \\frac{1}{4} \\right) \\right) \\right) "
] | [
"What is the probability that \\( X_n \\) is a negative number after throwing a dice with sides labeled \\( 0, 2, 3, \\sqrt{2}, \\sqrt{3}, i \\) \\( n \\) times and defining \\( X_n \\) as the product of the outcomes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-824528e760bec5a3cc106f1200ab9e49e73a6bed733aab7103ae0ea12233039c | What is the probability that \( X_n \) is a real number after throwing a dice with sides labeled \( 0, 2, 3, \sqrt{2}, \sqrt{3}, i \) \( n \) times and defining \( X_n \) as the product of the outcomes? | [
"1"
] | [
"What is the probability that \\( X_n \\) is a real number after throwing a dice with sides labeled \\( 0, 2, 3, \\sqrt{2}, \\sqrt{3}, i \\) \\( n \\) times and defining \\( X_n \\) as the product of the outcomes?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bf0800ad935a5ede737191c311e2a61e3a2f1cb122701b9bee41dc9840198fd2 | What is the probability that \( n \) points chosen randomly on the surface of a sphere all lie on some hemisphere of the sphere? (Note: "Randomly" means that the probability of a point being chosen in a region is proportional to the area of that region relative to the total surface area of the sphere.) | [
"\\(\\frac{n^2 - n + 2}{2^n}\\)"
] | [
"What is the probability that \\( n \\) points chosen randomly on the surface of a sphere all lie on some hemisphere of the sphere? \n\n(Note: \"Randomly\" means that the probability of a point being chosen in a region is proportional to the area of that region relative to the total surface area of the sphere.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 5,
"tir": 2
} | 32 | 0 | 0 |
omr-ws-350e2235945de3a1edcf710a59e3da7ca573a55ce6fda5cd91edfc1f17f2fdca | What is the probability that \( y^2 > \sqrt{x(1-x)} \) given that \( 0 < x, y < 1 \)? | [
"0.382"
] | [
"What is the probability that \\( y^2 > \\sqrt{x(1-x)} \\) given that \\( 0 < x, y < 1 \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 15
} | 17 | 0 | 0 |
omr-ws-d0a5faa119f0359e2530abb1dcbe65d8304628f650c4bc9be938f35acc363d18 | What is the probability that \( y^2 > \sqrt{x} \cdot \sqrt{1-x} \) given \( 0 < x, y < 1 \)? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the probability that \\( y^2 > \\sqrt{x} \\cdot \\sqrt{1-x} \\) given \\( 0 < x, y < 1 \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe2a81e6ca4d28f5622470ee3dd8e41fcd309269acfa13cf29c306f73a9e69d2 | What is the probability that a $10$-inch-long stick, broken in a random place, is split into two pieces whose lengths differ by at most $2$ inches? | [
"\\(\\frac{2}{5}\\)"
] | [
"What is the probability that a $10$-inch-long stick, broken in a random place, is split into two pieces whose lengths differ by at most $2$ inches?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-136cfa9ece85cf81c655f2f3026a7fb53727ee60b21c925e77ba7f52ae24b922 | What is the probability that a $\sqrt{17}$ cm rod can be broken down into 3 smaller rods that can form a triangle? | [
"\\(\\frac{1}{4}\\)"
] | [
"What is the probability that a $\\sqrt{17}$ cm rod can be broken down into 3 smaller rods that can form a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-a42a5c0ecdf8dc304de88028c7c51a376256f1688d4d5c77b26f1c348dcdf861 | What is the probability that a 1 appears consecutively when a die is rolled 4 times? | [
"\\(\\frac{2}{27}\\)"
] | [
"What is the probability that a 1 appears consecutively when a die is rolled 4 times?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-47fb37ac3aeb6c1ae0d56c42b2e716e0f0d7d401021067bfc6b5173a5bf71981 | What is the probability that a 13-card bridge hand will have 2 voids (a void is no cards in the same suit)? | [
"\\( 9.827126847 \\cdot 10^{-5} \\)"
] | [
"What is the probability that a 13-card bridge hand will have 2 voids (a void is no cards in the same suit)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-3c7b7dae48b3dcb773ddb44905aeafdc6b5b215938d5bbd97d909420bc32be3d | What is the probability that a 4x4 matrix filled with 0s and 1s will have an odd determinant? | [
"\\(\\frac{20160}{65536}\\)"
] | [
"What is the probability that a 4x4 matrix filled with 0s and 1s will have an odd determinant?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-a3a3f705488ff7d9008552db437cee435fa4d3b142eafc59b3886208d1a08c60 | What is the probability that a 5-card poker hand has exactly 2 cards of the same value, but no other cards duplicate? | [
"\\(\\frac{{13\\choose 1}{4\\choose 2}{12\\choose 3}{4\\choose 1}^3}{{52 \\choose 5}}\\)"
] | [
"What is the probability that a 5-card poker hand has exactly 2 cards of the same value, but no other cards duplicate?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 17
} | 29 | 0 | 0 |
omr-ws-898177c4916b1d7cb4ecbc29f183590607e48643d1b7a4193c0bb04eb1a52cb1 | What is the probability that a 5-in-a-row bingo (including main diagonals) will contain the free space, assuming all 24 squares have an equal likelihood of being picked? Consider both scenarios: with replacement and without replacement. | [
"\\dfrac{1}{3}"
] | [
"What is the probability that a 5-in-a-row bingo (including main diagonals) will contain the free space, assuming all 24 squares have an equal likelihood of being picked? Consider both scenarios: with replacement and without replacement."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-c4d3c699a42d79ed27b51968967441ba3573628ea628fec189f0cbfe2680cc31 | What is the probability that a 7-card poker hand has a value of less than one pair? | [
"\\(\\frac{23294460}{133784560}\\) or approximately \\(17.4\\%\\)."
] | [
"What is the probability that a 7-card poker hand has a value of less than one pair?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 9
} | 23 | 0 | 0 |
omr-ws-86189977cc4b8e27c86d7a1d1c3195b147049fbd6ce0460e2589a0c3a6ca846a | What is the probability that a 9-digit integer formed by arranging the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 in random order is divisible by 11? | [
"\\(\\frac{11}{126}\\)"
] | [
"What is the probability that a 9-digit integer formed by arranging the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 in random order is divisible by 11?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 24
} | 38 | 0 | 0 |
omr-ws-bfd7f856b34758a414116c633129caf9ba471d2f264111c90276835d208b8754 | What is the probability that a best-of-7 series between two evenly matched teams (each with a 0.5 probability of winning any game) will require all seven games to determine a winner? | [
"\\(\\frac{5}{16}\\)"
] | [
"What is the probability that a best-of-7 series between two evenly matched teams (each with a 0.5 probability of winning any game) will require all seven games to determine a winner?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-0e1226a78718cafaa88591da8978e2f56c3b811498beb5d20eda9745e820f8bf | What is the probability that a bishop and a rook randomly placed on an $8 \times 8$ chessboard can attack each other? | [
"\\dfrac{13}{36}"
] | [
"What is the probability that a bishop and a rook randomly placed on an $8 \\times 8$ chessboard can attack each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed289201deddbf216585ac3bcba8ea06614869b3afe639fa1f4c1cec173c0734 | What is the probability that a bridge hand is void in at least one suit? | [
"\\( 4\\cdot \\frac{\\binom{39}{13}}{\\binom{52}{13}} - 6\\cdot \\frac{\\binom{26}{13}}{\\binom{52}{13}} + 4\\cdot \\frac{1}{\\binom{52}{13}} \\approx 0.051 \\)"
] | [
"What is the probability that a bridge hand is void in at least one suit?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 26
} | 47 | 0 | 0 |
omr-ws-741d69e727fe1da68ae13cdb12ff735abf4c33513492d35a7ce1eb7c8bcd3911 | What is the probability that a chain of coin flips, which terminates once one side appears 4 times, is alternating (i.e., two consecutive flips give different outcomes)? | [
"\\(\\frac{1}{56}\\)"
] | [
"What is the probability that a chain of coin flips, which terminates once one side appears 4 times, is alternating (i.e., two consecutive flips give different outcomes)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9cdb3b82154783722fa831f16260a95b74dd4999d18e15b488bcf4bce202807d | What is the probability that a class of 23 students contains at least two students with the same birthday? (Ignore leap years.) | [
"0.51"
] | [
"What is the probability that a class of 23 students contains at least two students with the same birthday? (Ignore leap years.)"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-a7586b4e283d8c51177210262ca6bc2072bc1ea9b905325ef21f04f474f51bdf | What is the probability that a coach randomly selects exactly 2 of her 4 students out of a group of 8 students (4 from each coach)? If the probability is expressed as a common fraction \( \frac{x}{y} \), what is \( x + y \)? | [
"53"
] | [
"What is the probability that a coach randomly selects exactly 2 of her 4 students out of a group of 8 students (4 from each coach)? If the probability is expressed as a common fraction \\( \\frac{x}{y} \\), what is \\( x + y \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 7
} | 27 | 0 | 0 |
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