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-b3dbf7de4cffb215c56c6329f1ec5c7a8f07cf9a80a43b136c6a19f4a91e21bc | What is the probability that a randomly selected leap year will contain 53 Fridays and 54 Sundays? | [
"0\n\nGiven the forum discussions, it is clear that a leap year cannot contain 54 Sundays. Therefore, the probability that a randomly selected leap year will contain 53 Fridays and 54 Sundays is 0."
] | [
"What is the probability that a randomly selected leap year will contain 53 Fridays and 54 Sundays?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-80590a40574d53c4a090a0708b30a4f551cd3268aa342b0b511b5e50a3d77f3a | What is the probability that a randomly selected number from the set \(\{1, 2, \ldots, 100\}\) is a prime number given that it is odd? | [
"\\(\\frac{12}{25}\\)"
] | [
"What is the probability that a randomly selected number from the set \\(\\{1, 2, \\ldots, 100\\}\\) is a prime number given that it is odd?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-77362a1fe9eae69acd5fdef424f3f29e212c607454b9344e358164a20a7b8aba | What is the probability that a randomly selected stamp from a 10 by 10 sheet of 100 stamps is not one of the border stamps? | [
"\\(\\frac{16}{25}\\)"
] | [
"What is the probability that a randomly selected stamp from a 10 by 10 sheet of 100 stamps is not one of the border stamps?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-81bff7bdad527df630ad2db3e4155919bf35cbc4ca594df78eae8d9af5a702cc | What is the probability that a randomly selected three-digit number in base-10 has a three-digit representation in both base-9 and base-11? | [
"E"
] | [
"What is the probability that a randomly selected three-digit number in base-10 has a three-digit representation in both base-9 and base-11?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0ef591e6bc08729b7a70c48266b900d406e2cb8f175471a3317f2a628a0b96d4 | What is the probability that a reader guessing at random will match all three celebrities with their correct baby pictures? | [
"\\(\\frac{1}{6}\\)"
] | [
"What is the probability that a reader guessing at random will match all three celebrities with their correct baby pictures?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-0595b488e7a65752229f563975631d98f821070fa9cac06195e7583e8395968d | What is the probability that a secretary, who randomly places \( n \) letters into \( n \) envelopes, puts the 3rd and 7th letters into their correct envelopes? | [
"\\(\\frac{1}{n(n-1)}\\)"
] | [
"What is the probability that a secretary, who randomly places \\( n \\) letters into \\( n \\) envelopes, puts the 3rd and 7th letters into their correct envelopes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-5e123e9630d08b292520d86f9430b3d2d07f5817197e573868a9226a7252457d | What is the probability that a sequence \( S_n \) of \( n \) objects, where each object is a natural number less than or equal to \( n \), is monotonically non-decreasing? | [
"\\(\\frac{\\binom{2n-1}{n}}{n^n}\\)"
] | [
"What is the probability that a sequence \\( S_n \\) of \\( n \\) objects, where each object is a natural number less than or equal to \\( n \\), is monotonically non-decreasing?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-3698d65340036d90c89e5ca781e9e1eb6c49f82c38ea2f208555a600dba378ac | What is the probability that a specific group of six people will be selected when a committee of 6 is chosen from 16 candidates? | [
"\\(\\frac{1}{8008}\\)"
] | [
"What is the probability that a specific group of six people will be selected when a committee of 6 is chosen from 16 candidates?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-cc24e6a8c2a2578d02e0ff58f25434c2918ccde89e287f36203bf0cc9c6373d1 | What is the probability that a specific player will have 5 or more spades in their hand when each player is dealt 13 cards from a properly shuffled deck of 52 cards? | [
"0.176"
] | [
"What is the probability that a specific player will have 5 or more spades in their hand when each player is dealt 13 cards from a properly shuffled deck of 52 cards?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-fefd69ac35c9fb0b127ff4a2af84b119db8fefe3e135c19c94e7d2c1bb0482dd | What is the probability that a stick, when cut into three pieces at random, can form a triangle? | [
"1/4"
] | [
"What is the probability that a stick, when cut into three pieces at random, can form a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-2d24d0e92e6d47027459d8457b5b5fbcffe4610c008f2291dbb6b9368fc04256 | What is the probability that a tarot hand of 18 cards drawn at random from a 78-card tarot deck has a void, meaning that at least one suit is not present among the 18 cards? | [
"\\dfrac{4\\dbinom{64}{18} - 6\\dbinom{50}{18} + 4\\dbinom{36}{18} - \\dbinom{22}{18}}{\\dbinom{78}{18}}"
] | [
"What is the probability that a tarot hand of 18 cards drawn at random from a 78-card tarot deck has a void, meaning that at least one suit is not present among the 18 cards?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-96a086bb6b37231b228444ea6c395a06f6d61f47fd84872d83ca7a34f7e1b40d | What is the probability that a tetrahedron contains the center of the sphere? | [
"\\(\\frac{1}{8}\\)"
] | [
"What is the probability that a tetrahedron contains the center of the sphere?"
] | [
"c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-c686a4c04e8b9ba3d14cac3388eb20dcee73c6f6636b7daff92577c8a492e92d | What is the probability that a tetrahedron formed by 4 arbitrarily selected points on a sphere goes through the center of the sphere? | [
"\\(\\frac{1}{8}\\)"
] | [
"What is the probability that a tetrahedron formed by 4 arbitrarily selected points on a sphere goes through the center of the sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 21
} | 42 | 0 | 0 |
omr-ws-18d8f87b5f65a20bee5d23e4796929541858ba9259baac4952ff6189210e1ec0 | What is the probability that a third-degree polynomial \( P(x) \) has two extreme points? | [
"\\dfrac{1}{2}"
] | [
"What is the probability that a third-degree polynomial \\( P(x) \\) has two extreme points?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-54c5dac7bb52954750374ba90d68038d8bbff6ac22d17a4967b310250ae94347 | What is the probability that a three-number lock combination forms a geometric sequence in order, where each number is between 0 and 39? | [
"\\dfrac{47}{32000}"
] | [
"What is the probability that a three-number lock combination forms a geometric sequence in order, where each number is between 0 and 39?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-6390adbf1195172b6c41b78f8ab5b89c2881471f572ac07c28f8f9e4ae41bdc9 | What is the probability that a three-number lock combination forms an arithmetic sequence in order, where each number is between 0 and 39? | [
"\\dfrac{1}{80}"
] | [
"What is the probability that a three-number lock combination forms an arithmetic sequence in order, where each number is between 0 and 39?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d1fe29150fd76f3ab34f52e2f5a35f36ed41fc57950c8a2e6ebcf81c5b702273 | What is the probability that a triangle formed by three randomly chosen points on the perimeter of a square contains the center of the square? | [
"\\dfrac{1}{4}"
] | [
"What is the probability that a triangle formed by three randomly chosen points on the perimeter of a square contains the center of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-e1874bce55b633d3b94bae8eaa5c114c63ec55a8245454851ae2ef54fc84a3bd | What is the probability that a triangle formed by three randomly selected points on the surface of a cube is equilateral? | [
"0"
] | [
"What is the probability that a triangle formed by three randomly selected points on the surface of a cube is equilateral?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-daee02e97240c823b052fb973d2e9df7079f68633097aafb0d5b8725371ce2c8 | What is the probability that a triangle is formed when a stick is broken at two random places? | [
"\\(\\frac{1}{4}\\)"
] | [
"What is the probability that a triangle is formed when a stick is broken at two random places?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-6d75f7beb0ecb5c09ae4e52bee67c7b2a179a110581f10966590371204590be3 | What is the probability that a unit circle centered at a randomly chosen point P in the coordinate plane contains exactly two lattice points in its interior? | [
"\\( 4 - \\frac{2\\pi}{3} - \\sqrt{3} \\)"
] | [
"What is the probability that a unit circle centered at a randomly chosen point P in the coordinate plane contains exactly two lattice points in its interior?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-d579c1967d1c187657c1e42f819098af4ea26656667ca8137bbf4c94f56b20f6 | What is the probability that a unit circle with a randomly chosen center $P$ in the coordinate plane contains exactly two lattice points in its interior? | [
"\\(\\frac{4}{3}\\pi - 2\\sqrt{3}\\)"
] | [
"What is the probability that a unit circle with a randomly chosen center $P$ in the coordinate plane contains exactly two lattice points in its interior?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1add09487442775fd7c541749af8338e83d2c7241a3d6b1e5dea73744c0ff690 | What is the probability that a woman with three children has more than one girl given that she has at least one girl? | [
"\\(\\frac{4}{7}\\)"
] | [
"What is the probability that a woman with three children has more than one girl given that she has at least one girl?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-9d0036aca0026ebee132ec89b349ca099110925dddcbb0d063119ab036cd7682 | What is the probability that a word formed by arranging the letters of the word $\bf{TITANIC}$ starts either with $\bf{T}$ or a vowel? | [
"$\\frac{5}{7}$"
] | [
"What is the probability that a word formed by arranging the letters of the word $\\bf{TITANIC}$ starts either with $\\bf{T}$ or a vowel?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-0b331fe7abf3d4d4ec33962bcf56962ca38ef3db46c67d2bcdfc706e66b77f8e | What is the probability that after pressing the TELEPORT button three times, Belinda will be 2 AU closer to Pigeonia than she was when she started? | [
"\\( 3 p^2 (1-p) \\)"
] | [
"What is the probability that after pressing the TELEPORT button three times, Belinda will be 2 AU closer to Pigeonia than she was when she started?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b0b1ae2b999d97774533cf70b66b5ac62390ed309ff5a6782ec2561491af7a48 | What is the probability that after pressing the TELEPORT button twice, Belinda will be 1 AU closer to Pigeonia than she was when she started? | [
"\\( 2p(1 - p) \\)"
] | [
"What is the probability that after pressing the TELEPORT button twice, Belinda will be 1 AU closer to Pigeonia than she was when she started?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4a0d3d1d2fb337505ec1f97760e7bda32bc7758e0650b07b0b52657dbb818b0c | What is the probability that after rolling a $k$-times $n$-sided die, the sum of the outcomes is congruent to $m$ modulo $t$, where $k, n, m, t \in \mathbb{Z}$ and $n+1 \leq t$? | [
"\\dfrac{1}{t}"
] | [
"What is the probability that after rolling a $k$-times $n$-sided die, the sum of the outcomes is congruent to $m$ modulo $t$, where $k, n, m, t \\in \\mathbb{Z}$ and $n+1 \\leq t$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-218929a499d2cd94b42d9e29a61cec684f981043b6912380d25ee9c09a362abb | What is the probability that after servicing 10 out of 15 phones (5 cellular, 5 cordless, and 5 corded), only 2 of the 3 types of phones remain to be serviced? | [
"\\(\\frac{250}{1001}\\)"
] | [
"What is the probability that after servicing 10 out of 15 phones (5 cellular, 5 cordless, and 5 corded), only 2 of the 3 types of phones remain to be serviced?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 27
} | 48 | 0 | 0 |
omr-ws-06cfb49755dacef0d463621dfe0fffe2fe6513216b06ca439a99e5d9ff81c090 | What is the probability that all 10 identical particles moving in random motion with equal speeds within a closed hollow sphere are located in one half of the sphere at a particular instant, given that all collisions with the sphere's walls and among themselves are elastic? | [
"\\dfrac{1}{512}"
] | [
"What is the probability that all 10 identical particles moving in random motion with equal speeds within a closed hollow sphere are located in one half of the sphere at a particular instant, given that all collisions with the sphere's walls and among themselves are elastic?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-d1063416d8be91fbe1e34b94d0502ca70488200cc9fedde8f42f297aa88a834d | What is the probability that all 3 balls drawn from a bag containing 5 red balls, 4 white balls, and 3 blue balls have different colors? | [
"\\(\\frac{1}{11}\\)"
] | [
"What is the probability that all 3 balls drawn from a bag containing 5 red balls, 4 white balls, and 3 blue balls have different colors?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-70d3c9e095da3c4f45ca01d83873fa137f1651e1c3baa4cf42363169255f8e9c | What is the probability that all 5 cordless phones are among the first 10 to be serviced out of 15 phones (5 cellular, 5 cordless, and 5 corded)? | [
"\\(\\frac{252}{3003}\\)"
] | [
"What is the probability that all 5 cordless phones are among the first 10 to be serviced out of 15 phones (5 cellular, 5 cordless, and 5 corded)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-7923f2fc3a5c55b083f52ec7e0fb5b9451ff4f3cdd3b5874c63927d545a5d508 | What is the probability that all 6 faces of a die appear when 10 dice are thrown? | [
"\\dfrac{38045}{139968}"
] | [
"What is the probability that all 6 faces of a die appear when 10 dice are thrown?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-74dcd02b8b341035599acc508b7223c77a174eac726bf40b733670155b4656bc | What is the probability that all 6 points placed at random in a circle of circumference 2 are covered by some arc of length 1? | [
"\\dfrac{3}{16}"
] | [
"What is the probability that all 6 points placed at random in a circle of circumference 2 are covered by some arc of length 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1b78a60421ab11ac0ca50eb537eadb26cbb0f381971b2214d0e5d3688f73a590 | What is the probability that all 8 friends have the same number of cards when 64 cards are dealt to them? | [
"\\(\\dfrac{1}{\\binom{71}{7}}\\)"
] | [
"What is the probability that all 8 friends have the same number of cards when 64 cards are dealt to them?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-a084cf59f1754391b4a8d3396eadc9214837d57786fdf54eb107928636172434 | What is the probability that all angles of a triangle are acute? | [
"\\dfrac{1}{4}"
] | [
"What is the probability that all angles of a triangle are acute?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-69ea232c289c00d8aca0fa3bdfc7b1954d80b63f8f952c2326afeaa26aa2b31e | What is the probability that all balls of the same color are adjacent when 2 blue, 2 red, and 2 white balls are arranged randomly around a circle? | [
"\\(\\frac{1}{15}\\)"
] | [
"What is the probability that all balls of the same color are adjacent when 2 blue, 2 red, and 2 white balls are arranged randomly around a circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-30f43cc36c73c64d344cc95fbf61971584f2a5ae38481c82404ae42ef20c7124 | What is the probability that all of the numbers 1-6 appear at least once when 8 dice are rolled? | [
"\\(\\frac{7}{429}\\)"
] | [
"What is the probability that all of the numbers 1-6 appear at least once when 8 dice are rolled?"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-5eb51ee2352748489e56f992447463b5211d72f9d13412f4d18c9e3bd0af2bef | What is the probability that all three darts hit the same hemisphere of a ball, given that the darts are equally likely to hit any part of the ball and all three darts hit the ball? | [
"1"
] | [
"What is the probability that all three darts hit the same hemisphere of a ball, given that the darts are equally likely to hit any part of the ball and all three darts hit the ball?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-e508eada2840e59e83c29ab32f2bb81c57e3bb39bd0c7b38003f17c02cf902ce | What is the probability that among 40 people sitting on a bus, at least two of them share a birthday? | [
"\\(1-\\frac{365}{365}*\\frac{364}{365}*\\frac{363}{365}*…*\\frac{326}{365}\\)"
] | [
"What is the probability that among 40 people sitting on a bus, at least two of them share a birthday?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-8362d41eb8beb325994d8faa7a9e22391346472ace64b6ae0d234cd7a51ea88e | What is the probability that an Ace will be next to a King in a randomly shuffled deck of 52 playing cards? | [
"\\(\\frac{8}{13}\\)"
] | [
"What is the probability that an Ace will be next to a King in a randomly shuffled deck of 52 playing cards?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-4944d8fc745b7bdbae5d73452993ebc6c601439de318b2587dc2e974fb38d5b8 | What is the probability that an ace appears before a face card when drawing cards one at a time with replacement from an ordinary deck of 52 cards? | [
"\\dfrac{1}{4}"
] | [
"What is the probability that an ace appears before a face card when drawing cards one at a time with replacement from an ordinary deck of 52 cards?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-31c250919f7f6f1d690b6833cd4325551e6646ee7fd3f3982d4da0fc5ba3bff7 | What is the probability that an even number of 101 fair 6-sided dice come up even? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the probability that an even number of 101 fair 6-sided dice come up even?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-f191a4aa9e523196e9fbca6e81fa67c4f5f8d48e5e900bbbe524f8a8941fcfcb | What is the probability that an event will occur at least once within six years if it has a 20% chance of occurring each year? | [
"$1 - \\left(\\frac{4}{5}\\right)^6$"
] | [
"What is the probability that an event will occur at least once within six years if it has a 20% chance of occurring each year?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-6625ae7a8382697a27ecd2276fae13f968749bbca4020cf0c44ee3ee741a684b | What is the probability that an integer in the set $\{1, 2, 3, \ldots, 100\}$ is divisible by 2 and not divisible by 3? | [
"\\(\\frac{17}{50}\\)"
] | [
"What is the probability that an integer in the set $\\{1, 2, 3, \\ldots, 100\\}$ is divisible by 2 and not divisible by 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-f6336a94431ba53d236884cf5829ee845f643dc75f41793f29025263864e5e99 | What is the probability that any given 3 cards form a set, where each card is uniquely defined by four properties: color (red, blue, green), number (1, 2, 3), shape (circle, triangle, square), and shading (filled, partially shaded, empty)? | [
"\\(\\frac{3}{79}\\)"
] | [
"What is the probability that any given 3 cards form a set, where each card is uniquely defined by four properties: color (red, blue, green), number (1, 2, 3), shape (circle, triangle, square), and shading (filled, partially shaded, empty)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e7b31b130dae5a826afe860f2c25b6b059bdddad8b614ab849ae389c1c3d22b4 | What is the probability that at least $k$ out of $n$ tossed coins land on heads? | [
"\\frac{1}{2^n} \\sum_{m=k}^{n} \\binom{n}{m}"
] | [
"What is the probability that at least $k$ out of $n$ tossed coins land on heads?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a61e7e3e5d405f37ec4e014a5e66f5610e7fe9837759ae80f44b38eadf31d935 | What is the probability that at least 2 knights are sitting next to each other at a round table? | [
"\\dfrac{4}{5}"
] | [
"What is the probability that at least 2 knights are sitting next to each other at a round table?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-4e948795d87535b1cde7f91b518480bf8d46399a8acbfdecdd597cbf69183fe1 | What is the probability that at least 2 out of 30 people share the same birthday? | [
"\\( 1 - \\left( \\frac{365}{365} \\times \\frac{364}{365} \\times \\frac{363}{365} \\times \\cdots \\times \\frac{336}{365} \\right) \\)"
] | [
"What is the probability that at least 2 out of 30 people share the same birthday?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-3108d93556bfa49592e9376fac6d1e5e69004f72a5e4eb25fa6e82358596671b | What is the probability that at least 2 people out of 10 have the same birthday? | [
"11.7%\n\nThe first post in the forum discussion directly addresses the problem and provides the probability as 11.7%. Therefore, the answer to the problem \"What is the probability that at least 2 people out of 10 have the same birthday?\" is 11.7%."
] | [
"What is the probability that at least 2 people out of 10 have the same birthday?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-b435cdc5bc4ccbb7e9a2a477387a1bf60e2644a2466857b1bd56748a0504e2cc | What is the probability that at least 2 rooms will have at least 3 guests staying in them, given that 12 friends are staying in a hotel with 6 distinct rooms (painted red, green, blue, yellow, purple, and white respectively)? | [
"\\( 1 - \\frac{521261106}{6^{12}} \\)"
] | [
"What is the probability that at least 2 rooms will have at least 3 guests staying in them, given that 12 friends are staying in a hotel with 6 distinct rooms (painted red, green, blue, yellow, purple, and white respectively)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 8,
"tir": 16
} | 27 | 0 | 0 |
omr-ws-739f776c2c7e86fffd02601ce4d4e578072c1c2875f900b86bb17674f4d7398a | What is the probability that at least 3 of the 5 cards drawn from a standard deck of 52 cards are queens? | [
"\\(\\frac{4560}{2598960}\\) or approximately \\(0.00175455\\)"
] | [
"What is the probability that at least 3 of the 5 cards drawn from a standard deck of 52 cards are queens?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-516180a7c57787f0d10c8f7ea96debfe6b14b5a4b76637b08ab03a51a3496af2 | What is the probability that at least 3 out of 5 standard 6-sided dice show a six? | [
"\\(\\frac{276}{6^5}\\)",
"\\dfrac{23}{648}"
] | [
"What is the probability that at least 3 out of 5 standard 6-sided dice show a six?"
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 33,
"tir": 30
} | 63 | 0 | 0 |
omr-ws-59aa447aba9d36ffe425daa355a247ec964d242728b3c5cf5db23782fe4d8087 | What is the probability that at least one card will not be a king or at least one will be red when 2 cards are drawn from a standard deck of cards? | [
"\\(\\frac{1325}{1326}\\)"
] | [
"What is the probability that at least one card will not be a king or at least one will be red when 2 cards are drawn from a standard deck of cards?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-ba58691412493173f3143b5c35d5ba341a44a98ad7c9f8e72e6c479d6b2b9b9f | What is the probability that at least one in a group of 15 individuals, each with an annual income exceeding $200,000$ dollars, will be audited by the IRS, given that the IRS audits 5% of such individuals? | [
"0.5367"
] | [
"What is the probability that at least one in a group of 15 individuals, each with an annual income exceeding $200,000$ dollars, will be audited by the IRS, given that the IRS audits 5% of such individuals?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-eb9e261c05c847e6480d779fb147cf1707d5442d27943b091a576ab53fe34aa6 | What is the probability that at least one of the 10 people grabs their own hat when each person randomly picks a hat from a hat rack containing all 10 hats? | [
"\\( 1 - \\frac{1334961}{3628800} \\) or approximately \\( 0.632121 \\)"
] | [
"What is the probability that at least one of the 10 people grabs their own hat when each person randomly picks a hat from a hat rack containing all 10 hats?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-3cc0b2200fe221cc1087a26d541cbb806dd2ca3627807fcc8e171e8216e3d863 | What is the probability that at least one of the 4 randomly selected people from a group of 40 (where 10 are bad guys) is a bad guy? | [
"\\(\\frac{12797}{18278}\\)"
] | [
"What is the probability that at least one of the 4 randomly selected people from a group of 40 (where 10 are bad guys) is a bad guy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-2d1726eea9f1cac523afbf46b57764143d2a55d9d01875e70fe04cd75f242773 | What is the probability that at least one of the letters is in the envelope with the correct address when \( n \) letters are randomly placed into \( n \) envelopes, each with a different address? What happens to this probability as \( n \) approaches infinity? | [
"\\( 1 - \\exp(-1) \\)"
] | [
"What is the probability that at least one of the letters is in the envelope with the correct address when \\( n \\) letters are randomly placed into \\( n \\) envelopes, each with a different address? What happens to this probability as \\( n \\) approaches infinity?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-abb885faa2f7d1df8dbd7cf3417e91aa52837ffee45c5b78e36014b55dcc01f9 | What is the probability that at least one of the numbers thrown is a prime number given that the sum of the scores of three distinct regular six-faced unbiased dice is odd? | [
"\\dfrac{95}{108}"
] | [
"What is the probability that at least one of the numbers thrown is a prime number given that the sum of the scores of three distinct regular six-faced unbiased dice is odd?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-46ce3d4c7782d88e6acdbccb0a2a79fb860bcaf3ebf57eda7919a832622655dd | What is the probability that at least one of the two cards drawn from a deck of 52 cards is not a king or at least one is red? | [
"\\(\\frac{1325}{1326}\\)"
] | [
"What is the probability that at least one of the two cards drawn from a deck of 52 cards is not a king or at least one is red?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-73ef8e0d09dde136aff737305bb2136717a4b98c43f29a0c8d97d9910abba6e9 | What is the probability that at least one of three randomly picked natural numbers is divisible by $7$? | [
"\\(\\frac{127}{343}\\)"
] | [
"What is the probability that at least one of three randomly picked natural numbers is divisible by $7$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-307e28ad6b5b7edecd325229e39180a498a985937f3ba334fa949eac13e8eb61 | What is the probability that at least one of two cards drawn from an ordinary deck of 52 cards is not a king or at least one is red? | [
"\\(\\frac{1325}{1326}\\)"
] | [
"What is the probability that at least one of two cards drawn from an ordinary deck of 52 cards is not a king or at least one is red?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-223865edf135abfde53f7b58f8b5b8e6ec84ff215bd38f4751ff9c8661ea8df0 | What is the probability that at least one of two randomly spun line segments makes an angle less than 45 degrees with the x-axis (measured in the anti-clockwise direction)? | [
"\\(\\frac{15}{64}\\)"
] | [
"What is the probability that at least one of two randomly spun line segments makes an angle less than 45 degrees with the x-axis (measured in the anti-clockwise direction)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 26
} | 38 | 0 | 0 |
omr-ws-b398dd91f18997aa9087a0b6adb130ef41c2f7ee9ec8f15a82e501c4c2195ed4 | What is the probability that at least one run of N or more answers of the same type occur in a multiple choice test with M questions, each with P possible answers, given that all answers are chosen at random with equal chance of occurring? | [
"1 - \\frac{S(M)}{P^M}"
] | [
"What is the probability that at least one run of N or more answers of the same type occur in a multiple choice test with M questions, each with P possible answers, given that all answers are chosen at random with equal chance of occurring?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-ad9bdce296b5ed4309c1163c29c50ae5cb8a91868f92fee232d44d9a9e6d1930 | What is the probability that at least one side of a dice is not rolled when the dice is rolled 7 times? | [
"} 1 - \\frac{7!}{2 \\times 6^6} \\]"
] | [
"What is the probability that at least one side of a dice is not rolled when the dice is rolled 7 times?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 3
} | 15 | 0 | 0 |
omr-ws-01883b3003b5cfedadda2b979fbf5fc4f2a7740552655546150cc11e6fde6360 | What is the probability that at least one table has exactly one student, given that each of the $n$ students independently decides to sit at table $X$ with probability $p$ or table $Y$ with probability $1 - p$? | [
"\\( n p (1-p)^{n-1} + n (1-p) p^{n-1} \\)"
] | [
"What is the probability that at least one table has exactly one student, given that each of the $n$ students independently decides to sit at table $X$ with probability $p$ or table $Y$ with probability $1 - p$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-0351ba728f9218d0b4f0992478574d669f51f2610624dd5b0dfb3ee4003b2595 | What is the probability that at least three of the 1340 chosen numbers from the set $\{1, 2, 3, \ldots, 2007\}$ are consecutive? | [
"1"
] | [
"What is the probability that at least three of the 1340 chosen numbers from the set $\\{1, 2, 3, \\ldots, 2007\\}$ are consecutive?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-06753df224f63aaf0243c196330c20e992a52f4d74054834a03c7532710b3dbf | What is the probability that at least three of the four fair six-sided dice show the same value? | [
"\\(\\frac{7}{72}\\)"
] | [
"What is the probability that at least three of the four fair six-sided dice show the same value?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-4200ead85465ef973c115612f51d36e501473860de57f92a1cc3435279905b79 | What is the probability that at least three out of N=23 people have their birthdays on the same day? | [
"0.014"
] | [
"What is the probability that at least three out of N=23 people have their birthdays on the same day?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-d487a3708bc7a35c026c6bfc73acbaf3c04fda2643e34f1478cec5525dd5a305 | What is the probability that at least three out of five distinct dice show a six when rolled? | [
"\\dfrac{23}{648}"
] | [
"What is the probability that at least three out of five distinct dice show a six when rolled?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d1aa9a05e26440bf5ef875e7ad35289ca0b8cfbf1865e8a7e3e6b21670287ca5 | What is the probability that at least three out of six standard six-sided dice show a 6? Solve using the principle of inclusion and exclusion. | [
"\\dfrac{1453}{23328}"
] | [
"What is the probability that at least three out of six standard six-sided dice show a 6? Solve using the principle of inclusion and exclusion."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-278de9a27cc5fd0e6230cef2cd2e1f8cede4b81e55025c7ff2fa08cdb35737e0 | What is the probability that at least two of the three randomly chosen knights from a group of 25 knights sitting around a circular table are sitting next to each other? | [
"\\(\\frac{11}{46}\\)"
] | [
"What is the probability that at least two of the three randomly chosen knights from a group of 25 knights sitting around a circular table are sitting next to each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-e53e37e00ac3a6ba38e5337879d4daa4e0e35d0313bf83f595ec59aae33c5edf | What is the probability that at least two of three randomly chosen people from a group of 25 sitting around a circular table had been sitting next to each other? | [
"\\(\\frac{11}{46}\\)"
] | [
"What is the probability that at least two of three randomly chosen people from a group of 25 sitting around a circular table had been sitting next to each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-83f4f16e3e113f1c8b2d03ba4c84569452cc5196d553c10ed5af622d6c2073ad | What is the probability that at least two out of 25 people have the same birthday, assuming a year has 365 days? | [
"\\( 1 - \\frac{365 \\cdot 364 \\cdot 363 \\cdot \\cdots \\cdot 341}{365^{25}} \\)"
] | [
"What is the probability that at least two out of 25 people have the same birthday, assuming a year has 365 days?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-d232558657f274c08814c72ea064438e2117f4c076a73368bd422068694433ff | What is the probability that at least two people out of $n$ people share a birthday today? | [
"\\(1 - \\left(1 - \\frac{1}{365}\\right)^n - n \\left(\\frac{1}{365}\\right) \\left(1 - \\frac{1}{365}\\right)^{n-1}\\)"
] | [
"What is the probability that at least two people out of $n$ people share a birthday today?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 12
} | 26 | 0 | 0 |
omr-ws-513290738fbd456bb7c8e8af25d9c2bc80f52c2d589c2cdcf07f1599ebe77ce4 | What is the probability that at least two people out of a team of 10 will have their birthday on the same day of the week? | [
"1"
] | [
"What is the probability that at least two people out of a team of 10 will have their birthday on the same day of the week?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-b01053081edf7d557e699ad1ef71edcdd5b499a6d9afda0ec860526f77fa0a26 | What is the probability that at least two people share a birthday in a room of fifty people? | [
"0.970374"
] | [
"What is the probability that at least two people share a birthday in a room of fifty people?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-dda793127071c9134c179a01c033bd436c3ded5d83c5ce156066d09f8b76d5a8 | What is the probability that at some point, the number of tails will be greater than twice the number of heads when a coin is tossed repeatedly? | [
"\\dfrac{\\sqrt{5} - 1}{2}"
] | [
"What is the probability that at some point, the number of tails will be greater than twice the number of heads when a coin is tossed repeatedly?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b2c16cbcebd853045b6fcbf363dfb5e85c17bf1499e886cf6547f75b4f8563af | What is the probability that balls of the same color are adjacent when 2 red, 2 green, and 2 blue identical balls are placed around a circular table? | [
"$\\tfrac{2}{15}$"
] | [
"What is the probability that balls of the same color are adjacent when 2 red, 2 green, and 2 blue identical balls are placed around a circular table?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 4
} | 12 | 0 | 0 |
omr-ws-e13f5adf75d81269cd1dc676df6886b656f0cb3849d19703558029be761e7ff2 | What is the probability that both Alex and Bernie finish their services within 30 minutes, 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? | [
"\\(0.671\\)"
] | [
"What is the probability that both Alex and Bernie finish their services within 30 minutes, 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"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 29
} | 32 | 0 | 0 |
omr-ws-699c885150e2cc58b730539ae8317ac9eb1812071d66b6dfaa00a6963a990497 | What is the probability that both Tuba player A and Tuba player B play the correct note, given that Tuba player A has a 3/4 chance of getting the note right and Tuba player B has a 4/5 chance of getting the note right? | [
"\\(\\frac{12}{13}\\)"
] | [
"What is the probability that both Tuba player A and Tuba player B play the correct note, given that Tuba player A has a 3/4 chance of getting the note right and Tuba player B has a 4/5 chance of getting the note right?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0cc12372c7e04d9e94327ffd8c85f0e48d81fb4d910c66837c4cbc78d9c58d07 | What is the probability that both cards drawn from a deck of 4 jacks and 4 queens are jacks, given that one of the cards is a red jack? | [
"$\\frac{1}{3}$"
] | [
"What is the probability that both cards drawn from a deck of 4 jacks and 4 queens are jacks, given that one of the cards is a red jack?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af8e46f825b0c5ba3d67e03b58a51ddcc2973c1bf962f0a5c596be75a661e6d8 | What is the probability that both cards drawn from a deck of 4 jacks and 4 queens are jacks, given that one of the cards is the jack of hearts? | [
"\\(\\frac{3}{7}\\)"
] | [
"What is the probability that both cards drawn from a deck of 4 jacks and 4 queens are jacks, given that one of the cards is the jack of hearts?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-69d675e2499f5206d9e3e234a87a72b7160311ad4645fa12823d2974487cdd68 | What is the probability that both defective phones are placed in the same box, given that there are 2 boxes each containing 100 distinct phones and exactly 2 of them are defective? | [
"\\(\\frac{99}{199}\\)"
] | [
"What is the probability that both defective phones are placed in the same box, given that there are 2 boxes each containing 100 distinct phones and exactly 2 of them are defective?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-5f594dddc6996adc7c0a453f14354c3f53bac9a1c048583c434050f9bb183ea7 | What is the probability that each drawn pair is of different colors (one Black and one White) when drawing two balls at a time from a bag containing 4 Black balls and 4 White balls until the bag is empty? | [
"\\(\\frac{8}{35}\\)"
] | [
"What is the probability that each drawn pair is of different colors (one Black and one White) when drawing two balls at a time from a bag containing 4 Black balls and 4 White balls until the bag is empty?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-ee5b3afaa2078b3f89f6841f2000fde269886c6d6c0059aba52b059cc56238c7 | What is the probability that each drawn pair is of the same color when drawing two balls at a time from a bag containing 4 Black balls and 4 White balls until the bag is empty? | [
"\\(\\frac{3}{35}\\)"
] | [
"What is the probability that each drawn pair is of the same color when drawing two balls at a time from a bag containing 4 Black balls and 4 White balls until the bag is empty?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-869cbcb4031c9ca2ee1351414ef55e9b98ddd4609b8fb5ac717d526472e8f57a | What is the probability that each of 3 children receives at least 1 piece of candy when a teacher distributes 10 pieces of candy randomly among them? | [
"\\(\\frac{6}{11}\\)"
] | [
"What is the probability that each of 3 children receives at least 1 piece of candy when a teacher distributes 10 pieces of candy randomly among them?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 1,
"tir": 1
} | 12 | 0 | 0 |
omr-ws-e975de1f92f59b459d62d67ce9b6508a5e359f2a2f306f846c0f96327b7b5f61 | What is the probability that each of the 4 boxes contains exactly 1 red ball when 16 white and 4 red balls (not identical) are distributed randomly into 4 boxes, with each box containing at most 5 balls? | [
"\\(\\frac{5^4}{\\binom{20}{4}}\\)"
] | [
"What is the probability that each of the 4 boxes contains exactly 1 red ball when 16 white and 4 red balls (not identical) are distributed randomly into 4 boxes, with each box containing at most 5 balls?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 42,
"genselect": 5,
"tir": 28
} | 75 | 0 | 0 |
omr-ws-695d2142859d1111b461b84d3dd2e53dcb339adba7dd4d5bc832325dbf10c376 | What is the probability that each of two distinguishable boxes will contain one ball if two indistinguishable balls are randomly placed into them? | [
"1/2"
] | [
"What is the probability that each of two distinguishable boxes will contain one ball if two indistinguishable balls are randomly placed into them?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-b87322e585ae2a10f1499a235e890c57300938af1b13b996e7158177c40d2e74 | What is the probability that each player has a hand containing at least one ace when a standard deck of 52 cards is dealt to four players, with each player receiving 13 cards? | [
"\\(\\frac{24 \\times 13^4}{52 \\times 51 \\times 50 \\times 49}\\)"
] | [
"What is the probability that each player has a hand containing at least one ace when a standard deck of 52 cards is dealt to four players, with each player receiving 13 cards?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-c9fe2b9f7b4bfd1dcad45f2f4fe9bd1186243059892350e0cdf7163293d50496 | What is the probability that each player receives a hand containing an ace when a standard deck of 52 cards is dealt to four players in a game of bridge? | [
"\\(\\frac{3}{32}\\)"
] | [
"What is the probability that each player receives a hand containing an ace when a standard deck of 52 cards is dealt to four players in a game of bridge?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6a02e361736ed7826407d0c27ba45a0c71740480d2876d062fe44bb56ada6554 | What is the probability that either John or Sara will be on a committee of 3 people selected randomly from 6 different people? | [
"\\(\\frac{3}{5}\\)"
] | [
"What is the probability that either John or Sara will be on a committee of 3 people selected randomly from 6 different people?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a99cae67d2248909b734cfbb7487725894dceb267f0be8cf41d37f9965260732 | What is the probability that every cookie has at least one chocolate chip when 8 chocolate chips are randomly distributed among 4 cookies? | [
"\\(\\frac{7}{33}\\)"
] | [
"What is the probability that every cookie has at least one chocolate chip when 8 chocolate chips are randomly distributed among 4 cookies?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 2
} | 5 | 0 | 0 |
omr-ws-54e575ce583ea1190e8cbb06d972bc005c0cd58f27721d2c5b405a1421e1b3b6 | What is the probability that every man will sit next to his wife when three married couples sit at a round table with twelve chairs? | [
"\\(\\frac{4}{495}\\)"
] | [
"What is the probability that every man will sit next to his wife when three married couples sit at a round table with twelve chairs?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 7
} | 25 | 0 | 0 |
omr-ws-aae4608d2a078fffde263602982a811c1d0c24a8d5905d890dad3a2f88b33ced | What is the probability that every man will sit next to his wife when three married couples sit on a row with twelve chairs? | [
"\\(\\frac{1}{165}\\)"
] | [
"What is the probability that every man will sit next to his wife when three married couples sit on a row with twelve chairs?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-8cc2f21e9aeb044b64b6c93aac74ddf7ddc25c80b5df17f7611d757c4e4f6722 | What is the probability that every person will get another person's gift when the boxes are given back randomly? | [
"\\(\\frac{11}{30}\\)"
] | [
"What is the probability that every person will get another person's gift when the boxes are given back randomly?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f71f097f45ac751f2740dfc8935bfdb201147a45aafd6b896dbd6377336005fc | What is the probability that every student ends up with a cap which is not their own when a bunch of students throw their caps in the air and each student catches a cap? | [
"\\( e^{-1} \\)"
] | [
"What is the probability that every student ends up with a cap which is not their own when a bunch of students throw their caps in the air and each student catches a cap?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 27
} | 31 | 0 | 0 |
omr-ws-d9a10cf7315258da94a1887bd8129e5e9f597c1ed946677f2d75ce53be2f01de | What is the probability that exactly $r$ white balls are selected when $k$ balls are chosen at random from a set containing $n$ identical white balls and $m$ black balls? | [
"\\(\\frac{{m \\choose {k-r}}}{{m+n \\choose k}}\\)"
] | [
"What is the probability that exactly $r$ white balls are selected when $k$ balls are chosen at random from a set containing $n$ identical white balls and $m$ black balls?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72fb043cb8b19a91f4dcbc296989baa8c3d16f31380286b33eff4c872c9c7355 | What is the probability that exactly 1 out of 3 balls drawn blindly from a box containing 14 black and 7 white balls is black? | [
"\\(\\frac{21}{95}\\)"
] | [
"What is the probability that exactly 1 out of 3 balls drawn blindly from a box containing 14 black and 7 white balls is black?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-50a0a93c562ff082440b965433d4c831b4197774bb14e0bd943f1356e8311fd7 | What is the probability that exactly 2 boys are between Billy and George when 7 boys are seated in a row? | [
"\\dfrac{4}{21}"
] | [
"What is the probability that exactly 2 boys are between Billy and George when 7 boys are seated in a row?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0ffeed504743a565bcdb15386f8a0448c616d7e641788ecb6f3cd9dacf946132 | What is the probability that exactly 20 coins will come up heads when a rigged coin is flipped 200 times, where on the $n$th flip, the coin has a $\frac{1}{\lceil{n/5}\rceil}$ chance of coming up heads? | [
"0"
] | [
"What is the probability that exactly 20 coins will come up heads when a rigged coin is flipped 200 times, where on the $n$th flip, the coin has a $\\frac{1}{\\lceil{n/5}\\rceil}$ chance of coming up heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-f193bf94415bbf3e27eb473a19205b126ca277809c31f8346b3ebf1a22d84bf5 | What is the probability that exactly 3 out of 4 randomly chosen railway stations from a line of 12 stations are consecutive? | [
"\\(\\frac{8}{55}\\)"
] | [
"What is the probability that exactly 3 out of 4 randomly chosen railway stations from a line of 12 stations are consecutive?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
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