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-15aeaf441f5f3263e626389b51cd3b69c859b7a065699bf9a55dcdc10fbe19dc | What is the probability that the series will end in four games? Five games? Six games? Seven games? Compare with your estimates from part (A). | [
"Probability of ending in four games: \\(\\frac{1}{8}\\), Probability of ending in five games: \\(\\frac{1}{4}\\), Probability of ending in six games: \\(\\frac{5}{16}\\), Probability of ending in seven games: \\(\\frac{5}{16}\\)"
] | [
"What is the probability that the series will end in four games? Five games? Six games? Seven games? Compare with your estimates from part (A)."
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-49c5e41acb95a0da1d277fe8f4d922696fd2a2e8fa70b3fd9ef6794c30e50f87 | What is the probability that the smallest number taken is 4 when 5 numbers are drawn without replacement from a bag containing the numbers 1 through 10? | [
"\\(\\frac{5}{84}\\)"
] | [
"What is the probability that the smallest number taken is 4 when 5 numbers are drawn without replacement from a bag containing the numbers 1 through 10?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-b3ee9e3b68e26be0e7df429c44ac929dda130acbeb808ebe404efa649615ba79 | What is the probability that the square root of a randomly selected two-digit whole number is less than eight? Express your answer as a common fraction. | [
"\\frac{3}{5}"
] | [
"What is the probability that the square root of a randomly selected two-digit whole number is less than eight? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-829c1cb15eeaf2026caf19951efeaaf7eb5620b3158768597dae693c44ff3f8e | What is the probability that the string "UNIVERSITY" occurs as a block in a sequence of 15 random draws, one at a time with replacement, from the set $\{A, B, C, \ldots, Y, Z\}$? | [
"\\( 6 \\times \\left(\\frac{1}{26}\\right)^{10} \\)"
] | [
"What is the probability that the string \"UNIVERSITY\" occurs as a block in a sequence of 15 random draws, one at a time with replacement, from the set $\\{A, B, C, \\ldots, Y, Z\\}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-2a9aad32610e766257235cb95764278062408d4363d50796816433142ac18218 | What is the probability that the string $MERRYCHRISTMAS$ occurs as a block in a sequence of 35 random draws, one at a time with replacement, from the set of the English alphabet {$A,B,C,...,X,Y,Z$}? | [
"\\dfrac{22}{26^{14}}"
] | [
"What is the probability that the string $MERRYCHRISTMAS$ occurs as a block in a sequence of 35 random draws, one at a time with replacement, from the set of the English alphabet {$A,B,C,...,X,Y,Z$}?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-ebfa0cdd5eb3b214adc87257cebfe176c377adca719500ee23edc47012fdb007 | What is the probability that the sum of 3 different integers chosen between 1 and 13 inclusive is divisible by 4? | [
"\\(\\frac{73}{286}\\)"
] | [
"What is the probability that the sum of 3 different integers chosen between 1 and 13 inclusive is divisible by 4?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-1ff41431fe2af10cd10d28e6660db7cefb42488b66316b1f267712afa0a33ef3 | What is the probability that the sum of 4 integers, each chosen randomly between 20 and 35 inclusive, will be greater than or equal to 100? | [
"\\(\\frac{56961}{65536}\\)"
] | [
"What is the probability that the sum of 4 integers, each chosen randomly between 20 and 35 inclusive, will be greater than or equal to 100?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-a95185b8761f7eebddc7ae3c755c2d6976339ed8c3e165d7ae4aee2ebae2e25b | What is the probability that the sum of 6 numbers chosen without replacement from the set \(\{11, 12, 13, 14, \ldots, 23\}\) is an even number? | [
"\\(\\frac{1484}{3003}\\)"
] | [
"What is the probability that the sum of 6 numbers chosen without replacement from the set \\(\\{11, 12, 13, 14, \\ldots, 23\\}\\) is an even number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 31
} | 79 | 0 | 0 |
omr-ws-81a4cf66e01c4d0a6b6b30161f8a8c6997fd49fed303dcfb97aa876be8321fb5 | What is the probability that the sum of 665 rolls of a 665-sided die (where each side is labeled from 1 to 665 and the probability of landing on a side is proportional to the number on it) is 4913? | [
"0"
] | [
"What is the probability that the sum of 665 rolls of a 665-sided die (where each side is labeled from 1 to 665 and the probability of landing on a side is proportional to the number on it) is 4913?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ae757cdcfea7f23985bbfce5dfdfb7b595751d183bb16214bb5f7e5354a0a3f | What is the probability that the sum of 7 rolls of a fair 6-sided die is equal to 15? | [
"\\(\\frac{2807}{6^7}\\)"
] | [
"What is the probability that the sum of 7 rolls of a fair 6-sided die is equal to 15?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-6261e80ca137dfb62cfe4b6f05ee787b87dd1768c29fbe93a17af545cf30c3e2 | What is the probability that the sum of Fred's dice (one 6-sided and one 7-sided) will be greater than the sum of Jeff's dice (one 5-sided and one 6-sided)? | [
"\\dfrac{5}{9}"
] | [
"What is the probability that the sum of Fred's dice (one 6-sided and one 7-sided) will be greater than the sum of Jeff's dice (one 5-sided and one 6-sided)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4546a902aff80bf4dbea4e0b427d41f3191c7ba1fb0adc62d04c1c9092003747 | What is the probability that the sum of any two different single-digit natural numbers is a prime number? | [
"\\(\\frac{5}{12}\\)"
] | [
"What is the probability that the sum of any two different single-digit natural numbers is a prime number?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cfcdc0ec843d011409ba5c273c8304e6e4701feaba4b4fbd9326e863620f6416 | What is the probability that the sum of repeatedly rolling a dice will be equal to a positive integer \( m \)? | [
"\\dfrac{2}{7}"
] | [
"What is the probability that the sum of repeatedly rolling a dice will be equal to a positive integer \\( m \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2fc205b63dcb1a2508e6f738c4539a7d7b986d75758083e6026024be411dd0dd | What is the probability that the sum of squares of three randomly selected natural numbers is divisible by a given prime number \( p \)? | [
"\\(\\frac{1}{p}\\)"
] | [
"What is the probability that the sum of squares of three randomly selected natural numbers is divisible by a given prime number \\( p \\)?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-fa015e1a5600ee6223ff70e9f5a95ecdcee478b979f983ef4c623b6aeea67373 | What is the probability that the sum of ten fair dice is 45? | [
"\\dfrac{23089}{1679616}"
] | [
"What is the probability that the sum of ten fair dice is 45?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-1ac2cece572a53fb2ec8856f74f3323bc971c210ceab63240ba2ec8f6c32743f | What is the probability that the sum of the digits of a 10-digit number is divisible by 10? | [
"1/10"
] | [
"What is the probability that the sum of the digits of a 10-digit number is divisible by 10?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-b56d7935fadf8de1f27a0e6a2d4a544d5d264152a443b6e656e012bc6dc77fa4 | What is the probability that the sum of the digits of a randomly generated 7-digit number (including zeros) is 37? | [
"\\dfrac{80661}{2000000}"
] | [
"What is the probability that the sum of the digits of a randomly generated 7-digit number (including zeros) is 37?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-e8a2d153ec308380d59ab98f0f4028f7f9448178f60b3a2859b37c11271e751c | What is the probability that the sum of the numbers Player A gets from throwing two dice will be higher than the sum of the numbers Player B gets from throwing two dice? | [
"\\(\\frac{575}{1296}\\)"
] | [
"What is the probability that the sum of the numbers Player A gets from throwing two dice will be higher than the sum of the numbers Player B gets from throwing two dice?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-c25a71dbe3bb22fcca8780a461879c2c5f36b9ae2b1d240e32f7630bbc9c8545 | What is the probability that the sum of the numbers is odd when you roll two dice, each with faces labeled by the first 6 prime numbers? | [
"\\(\\frac{5}{18}\\)"
] | [
"What is the probability that the sum of the numbers is odd when you roll two dice, each with faces labeled by the first 6 prime numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-71dbaaf31f6ce39b77b5f8519ee21ef2d95eea0b5540c4a8c8c08e87f053842b | What is the probability that the sum of the numbers obtained from throwing a fair 6-face die 3 times is a prime number? | [
"\\(\\frac{73}{216}\\)"
] | [
"What is the probability that the sum of the numbers obtained from throwing a fair 6-face die 3 times is a prime number?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-123604c71c5e583aa19cf80635f1309a012f54b84fe3d904fa2e04e432ec3979 | What is the probability that the sum of the numbers on 4 rolled dice is 18? | [
"\\(\\frac{5}{81}\\)"
] | [
"What is the probability that the sum of the numbers on 4 rolled dice is 18?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-deaada65120f98ac3aa864a32835dd1cf5c59fea3d92cb6d11df9792e062d7cf | What is the probability that the sum of the numbers on n dice, each with f faces marked from 1 to f, will be equal to p when the dice are thrown at random? | [
"\\frac{1}{f^n} \\sum_{k=0}^{\\left\\lfloor \\frac{p - n}{f} \\right\\rfloor} (-1)^k \\binom{n}{k} \\binom{p - k f - 1}{n - 1}"
] | [
"What is the probability that the sum of the numbers on n dice, each with f faces marked from 1 to f, will be equal to p when the dice are thrown at random?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 2
} | 17 | 0 | 0 |
omr-ws-df901e76ce937efabca04187ae133bdcee9ca51806bfb140676c16ce7a08350f | What is the probability that the sum of the numbers on the first two dice equals the number on the third die, given that the dice are distinguishable? | [
"\\(\\frac{5}{24}\\)"
] | [
"What is the probability that the sum of the numbers on the first two dice equals the number on the third die, given that the dice are distinguishable?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7ed41ff9d46513e80d5a17f6b3d8e888084addae4b85dbd972b8581b36924e9e | What is the probability that the sum of the numbers on the red die and the white die is equal to the number on the blue die when Rhoda Bones rolls all three dice? (nearest whole percent) | [
"42%"
] | [
"What is the probability that the sum of the numbers on the red die and the white die is equal to the number on the blue die when Rhoda Bones rolls all three dice? (nearest whole percent)"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-980b3566dbb4b93827e03b9048d28548ebd2abef5b09a6d3ce5aa5b94c9113d2 | What is the probability that the sum of the numbers on the upper faces is 7 when 5 fair dice are thrown simultaneously? | [
"\\(\\frac{5}{2592}\\)"
] | [
"What is the probability that the sum of the numbers on the upper faces is 7 when 5 fair dice are thrown simultaneously?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-311f71724af6c4a8bbfda85d972a909481ecc084c9aae77723685e244e4eda5a | What is the probability that the sum of the numbers on top of a fair six-sided die, rolled 10 times, adds up to 52? | [
"\\(\\frac{23760}{6^{10}}\\)"
] | [
"What is the probability that the sum of the numbers on top of a fair six-sided die, rolled 10 times, adds up to 52?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 23
} | 43 | 0 | 0 |
omr-ws-cd7643eeb63df90545d1d4c97817ba1040875f8bae1497236ab2c6e8c68f3259 | What is the probability that the sum of the numbers on two balls picked from a box of 101 balls (numbered 1 to 101) is 52? | [
"\\(\\frac{1}{101}\\)"
] | [
"What is the probability that the sum of the numbers on two balls picked from a box of 101 balls (numbered 1 to 101) is 52?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-21f596c626ecf2a0154da67d46ebb715b00d89b456b0429582867acdc91a280b | What is the probability that the sum of the numbers rolled on 10 six-sided dice is 20? | [
"85228"
] | [
"What is the probability that the sum of the numbers rolled on 10 six-sided dice is 20?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7ba82b8d2b4a892c74f7977a56c422bed4e66ca1e29b4abd07c0181bb7e44cd6 | What is the probability that the sum of the numbers shown on five rolled regular dice is equal to 14? | [
"\\(\\frac{5}{72}\\)"
] | [
"What is the probability that the sum of the numbers shown on five rolled regular dice is equal to 14?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-720bc88e517aee73d69eae2666f0afedeaacd9ff868d508107acac9d7ec6af1f | What is the probability that the sum of the numbers shown when a regular die is rolled $n$ times is a multiple of $5$? | [
"\\dfrac{1}{5} + \\dfrac{2}{5 \\cdot 6^n} \\left( \\cos \\dfrac{2\\pi n}{5} + \\cos \\dfrac{4\\pi n}{5} \\right)"
] | [
"What is the probability that the sum of the numbers shown when a regular die is rolled $n$ times is a multiple of $5$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-d577aabc8412a630aaf46018976ef8f077c76e213786cb993bc69c5daf962db7 | What is the probability that the sum of the pips on the top faces is 9 when six dice are thrown once? | [
"\\(\\frac{7}{5832}\\)"
] | [
"What is the probability that the sum of the pips on the top faces is 9 when six dice are thrown once?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-056ab5089033d5516e67e5e72ffb8bb526bc729eb4b0f0d1dc8c6e7d49946ef3 | What is the probability that the sum of the points obtained from rolling 2022 honest dice simultaneously is a multiple of 5? | [
"\\dfrac{6^{2022} - 1}{5 \\cdot 6^{2022}}"
] | [
"What is the probability that the sum of the points obtained from rolling 2022 honest dice simultaneously is a multiple of 5?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-06f94fc6b47c44c584afe07099a52b5d0af729ecc205316235fddd5bfe5cde55 | What is the probability that the sum of the rolls is 20 when ten dice are rolled? | [
"\\(\\frac{85228}{6^{10}}\\) or approximately \\(0.0014\\)"
] | [
"What is the probability that the sum of the rolls is 20 when ten dice are rolled?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-5aa51d0a0c9bcc15cebcf931596877c983c41b8f9880ac46813d1912d0891df5 | What is the probability that the sum of the rolls of ten dice is 20? Solve using generating functions. | [
"\\(\\frac{85228}{6^{10}}\\)"
] | [
"What is the probability that the sum of the rolls of ten dice is 20? Solve using generating functions."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-e051f05893182f9f2aea668d68a274c3d03f24f2ed4dba43d3ace4f4a9077262 | What is the probability that the sum of the rolls of ten dice is 20? Solve using the complement method. | [
"\\(\\frac{89518}{6^{10}}\\)"
] | [
"What is the probability that the sum of the rolls of ten dice is 20? Solve using the complement method."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-2fe34e359995d4bd8a23592383b593a55c82a494b5243506374620b329323443 | What is the probability that the sum of the scores from throwing a cubic die 4 times is equal to 10? | [
"\\dfrac{5}{81}"
] | [
"What is the probability that the sum of the scores from throwing a cubic die 4 times is equal to 10?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-006fc5a4296115639e95d7e12dba0d510de3307288d1d96f7b0b150349e54189 | What is the probability that the sum of the squares of the dots from three rolls of a fair six-sided die is divisible by 3? | [
"\\(\\frac{1}{3}\\)"
] | [
"What is the probability that the sum of the squares of the dots from three rolls of a fair six-sided die is divisible by 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-3bdc23869e969ad6e39b5076bb58b6ff1a98b248844f7bfa100808f17e8ecb6b | What is the probability that the sum of the squares of two integers is a multiple of $5$? | [
"\\(\\frac{9}{25}\\)"
] | [
"What is the probability that the sum of the squares of two integers is a multiple of $5$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-311de4d0f8398fb717119b420378bf0e24021344752ce8a464b49937fab92ab0 | What is the probability that the sum of the values of four fair six-sided dice is greater than or equal to 13? | [
"\\dfrac{287}{432}"
] | [
"What is the probability that the sum of the values of four fair six-sided dice is greater than or equal to 13?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-26bed18b52559dca9703c480dbee3d31397c0dcd4a7ab978425e26d1af3f399f | What is the probability that the sum of the values of the top faces of six fair 6-sided dice is divisible by 7? | [
"\\(\\frac{1111}{7776}\\)"
] | [
"What is the probability that the sum of the values of the top faces of six fair 6-sided dice is divisible by 7?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-9360178cf2c6676988fc766f0ebfce66a98da897d14145f16a583d6c00d878fb | What is the probability that the sum of the values rolled on 100 dice is divisible by 7? | [
"\\(\\frac{1}{7} - \\frac{1}{7 \\cdot 6^{100}}\\)"
] | [
"What is the probability that the sum of the values rolled on 100 dice is divisible by 7?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 15
} | 29 | 0 | 0 |
omr-ws-b2ddb7f4d0f4617333dde19e58018be4c5f0be5f640d0ae93727baf8726586e8 | What is the probability that the sum of three dice is 16? | [
"\\(\\frac{1}{36}\\)"
] | [
"What is the probability that the sum of three dice is 16?"
] | [
"aops_c4_high_school_math",
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"cot": 16,
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omr-ws-ff21e4f8b0e043797550f3117f1aeb9556fada6de94b69e4f47c3e2b77b737ec | What is the probability that the sum of three distinct numbers chosen at random from $1, 2, 3, \ldots, 1000$ is divisible by 3? | [
"\\(\\frac{1}{3}\\)"
] | [
"What is the probability that the sum of three distinct numbers chosen at random from $1, 2, 3, \\ldots, 1000$ is divisible by 3?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
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"cot": 1,
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omr-ws-06935d252d5765ce90f013ac1b54f189083ba33e983e93f3834e8bf7ca2c0695 | What is the probability that the sum of three numbers selected from the set $\{1, 2, \ldots, 15\}$ is divisible by $3$? | [
"\\(\\frac{31}{91}\\)"
] | [
"What is the probability that the sum of three numbers selected from the set $\\{1, 2, \\ldots, 15\\}$ is divisible by $3$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
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"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-708658c8153a2ddd0bd9bf54e016d14eb81821b30a8abff51c35adea622490b7 | What is the probability that the sum of three randomly chosen numbers from the set \(\{1, 2, \ldots, 15\}\) is divisible by 3? | [
"\\(\\frac{1}{3}\\)"
] | [
"What is the probability that the sum of three randomly chosen numbers from the set \\(\\{1, 2, \\ldots, 15\\}\\) is divisible by 3?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-08b0eaeb43d8e282dfd583260bb653220b98d9a01d9018852e3312a30ee2df40 | What is the probability that the sum of three randomly selected numbers from the set $\{1, 2, 3, \ldots, 100\}$ is odd? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the probability that the sum of three randomly selected numbers from the set $\\{1, 2, 3, \\ldots, 100\\}$ is odd?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
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"cot": 16,
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omr-ws-ba1e24258b954b1b7641f25e8b11b55980399ef45cec046a2be5b87b1d09b76c | What is the probability that the sum of two eight-sided dice will be divisible by 3? | [
"\\dfrac{11}{32}"
] | [
"What is the probability that the sum of two eight-sided dice will be divisible by 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
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"cot": 16
} | 16 | 0 | 0 |
omr-ws-191c3a32abc3a9b8f35a316d39ce5df2c6e2b22b1accf137d9a8e2ae49c2cff3 | What is the probability that the sum of two integers selected without replacement from the set of integers between 1 and 1000 inclusive is odd? | [
"\\(\\frac{500}{999}\\)"
] | [
"What is the probability that the sum of two integers selected without replacement from the set of integers between 1 and 1000 inclusive is odd?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-1c3b72d3f0bd494204396c50df58d87edb9cf4f3d6e9f587d8e4912ac562b732 | What is the probability that the sum of two randomly chosen positive divisors of $52!$ is even? | [
"\\(\\frac{1201}{1250}\\)"
] | [
"What is the probability that the sum of two randomly chosen positive divisors of $52!$ is even?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
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"cot": 13,
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} | 36 | 0 | 0 |
omr-ws-5038178985dd8e31318e2f11fe5a38283ca93f30208c2558460fca6c131666e9 | What is the probability that the sum of two real numbers chosen randomly in the interval $(0,1)$ is greater than $1$, while the sum of their cubes is less than $1$? | [
"\\dfrac{1}{4}"
] | [
"What is the probability that the sum of two real numbers chosen randomly in the interval $(0,1)$ is greater than $1$, while the sum of their cubes is less than $1$?"
] | [
"aops_c7_college_math"
] | [
"cot"
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"cot": 3
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omr-ws-214b8e3ef3548c96f0c929d3c9825cbfbe2d41e7445c7704e1d6660b0837cbec | What is the probability that the third and seventh letters are placed in the correct envelopes when $n$ letters are placed in $n$ envelopes at random? | [
"\\(\\frac{1}{n^2 - n}\\)"
] | [
"What is the probability that the third and seventh letters are placed in the correct envelopes when $n$ letters are placed in $n$ envelopes at random?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
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} | 33 | 0 | 0 |
omr-ws-d79a3c2159020ebf4c346d54fc37ce94e05a7d5468b617e117bf64f3fc97d491 | What is the probability that the third die (with 4 dots on 4 sides, 3 dots on 1 side, and 2 dots on 1 side) will roll a higher number than a normal die? | [
"\\dfrac{5}{12}"
] | [
"What is the probability that the third die (with 4 dots on 4 sides, 3 dots on 1 side, and 2 dots on 1 side) will roll a higher number than a normal die?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-76952e4fb4896307333cde25f04a78bf77d10dcac7699a16a9995255f79f77fa | What is the probability that the third flip is heads and there are more tails than heads when a coin is flipped six times? | [
"\\(\\frac{3}{32}\\)"
] | [
"What is the probability that the third flip is heads and there are more tails than heads when a coin is flipped six times?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
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"cot": 15,
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} | 47 | 0 | 0 |
omr-ws-e5cc6074780edc5aaa5e04105c6a43ff41720f2ede3b9a1dc2fc6d20cd3f78bc | What is the probability that the three elements in a randomly chosen three-element subset of the set \(\{-10, -9, -8, \ldots, 8, 9, 10\}\) sum to zero? | [
"\\(\\frac{5}{133}\\)"
] | [
"What is the probability that the three elements in a randomly chosen three-element subset of the set \\(\\{-10, -9, -8, \\ldots, 8, 9, 10\\}\\) sum to zero?"
] | [
"aops_c4_high_school_math",
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"cot",
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"cot": 15,
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} | 48 | 0 | 0 |
omr-ws-a3310a1e9cd857f2866faeef2fa9484f72366065e1e9cc8e203f65b323c8e448 | What is the probability that the top and bottom cards of a well-shuffled deck of 52 cards have the same face value? | [
"\\(\\frac{1}{17}\\)"
] | [
"What is the probability that the top and bottom cards of a well-shuffled deck of 52 cards have the same face value?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
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} | 42 | 0 | 0 |
omr-ws-90b8deb1e12a8ebe5740c09cc0a8d5c42ba9a7d5998786dd815eabf8d5a3b419 | What is the probability that the top two cards in a shuffled deck are an Ace (of any suit) and a Spade? The Ace of Spades can be used to satisfy both requirements. | [
"\\(\\frac{23}{442}\\)"
] | [
"What is the probability that the top two cards in a shuffled deck are an Ace (of any suit) and a Spade? The Ace of Spades can be used to satisfy both requirements."
] | [
"aops_c7_college_math"
] | [
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13109519b2f5aecfe9234f9e34c2d325fd424ddd84d861463e9dbab49970990d | What is the probability that the top two cards in a shuffled deck are consecutive cards of the same suit? The Ace can be considered either high or low. | [
"\\(\\frac{2}{51}\\)"
] | [
"What is the probability that the top two cards in a shuffled deck are consecutive cards of the same suit? The Ace can be considered either high or low."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-dd6f325e84e022fb94c22265080e9a77e8a7aacef0d14fa82440aa8e01c27706 | What is the probability that the total number of eyes rolled is a multiple of 5 when a regular 6-sided die is rolled $n$ times? | [
"\\(\\frac{6^n - 1}{5 \\cdot 6^n}\\) if \\(5 \\nmid n\\), \\(\\frac{6^n + 4}{5 \\cdot 6^n}\\) if \\(5 \\mid n\\)"
] | [
"What is the probability that the total number of eyes rolled is a multiple of 5 when a regular 6-sided die is rolled $n$ times?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
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] | {
"cot": 5,
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"tir": 15
} | 21 | 0 | 0 |
omr-ws-8ad12587573d91c3c799217b29e50bf14ea19901d8a7483e51750000eb65c371 | What is the probability that the triangle $ABC$ does not intersect triangle $DEF$ when 6 distinct points are chosen on a circle and named $A, B, C, D, E, F$? | [
"0.3"
] | [
"What is the probability that the triangle $ABC$ does not intersect triangle $DEF$ when 6 distinct points are chosen on a circle and named $A, B, C, D, E, F$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-47c4de58855c67d380e48ba770f9f345a5da7ad789cf3bacc6b2a58c57622d44 | What is the probability that the twentieth patient tested will be the one with the third infection, given that the infection rate is 20%? | [
"\\((0.2)^3(0.8)^{17}\\binom{19}{2}\\)"
] | [
"What is the probability that the twentieth patient tested will be the one with the third infection, given that the infection rate is 20%?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 9
} | 12 | 0 | 0 |
omr-ws-c4021cb47c2200bd313135b4cc3ef618058b79e1809b017390b204df7d7a5887 | What is the probability that the two page numbers showing sum to a multiple of 6 when a 2010-page dictionary is opened to a random page? | [
"0"
] | [
"What is the probability that the two page numbers showing sum to a multiple of 6 when a 2010-page dictionary is opened to a random page?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ce605f489c9a111e61882b9187b6242fda133e0a8871e676928a43a73c828f0d | What is the probability that the two tallest girls are in different groups when $2n$ girls are randomly divided into two subgroups of $n$ girls each? | [
"\\(\\frac{n}{2n-1}\\)"
] | [
"What is the probability that the two tallest girls are in different groups when $2n$ girls are randomly divided into two subgroups of $n$ girls each?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
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} | 33 | 0 | 0 |
omr-ws-067f1cde6b4200f839b02a98e93e29ddd4c8c687ba70a6e3bc6ee8fed5c88d70 | What is the probability that there are at least 20 left-handed students in a school of 200 students, given that 12 percent of the population is left-handed? | [
"0.8078"
] | [
"What is the probability that there are at least 20 left-handed students in a school of 200 students, given that 12 percent of the population is left-handed?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0ecd49e9a78e5c40bdc4e2ff56666844580a861c6041196b818fc1553096ea9a | What is the probability that there are at most 19 red balls in a sample of 400 balls chosen from a collection of 100000 balls, where 4000 of the balls are red and the rest are blue? | [
"0.814"
] | [
"What is the probability that there are at most 19 red balls in a sample of 400 balls chosen from a collection of 100000 balls, where 4000 of the balls are red and the rest are blue?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 2
} | 15 | 0 | 0 |
omr-ws-0dedb4da9a650d616e5c1bf316c64d7006eeb6831f3c3cde34c2a481efadad3d | What is the probability that there are exactly \( k \) pairs of a man and his hat with the same number, given \( N \) men and \( N \) hats, each numbered from 1 to \( N \) randomly? | [
"\\(\\frac{1}{k! e}\\)"
] | [
"What is the probability that there are exactly \\( k \\) pairs of a man and his hat with the same number, given \\( N \\) men and \\( N \\) hats, each numbered from 1 to \\( N \\) randomly?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-b2cb8696354c0a6f68774a3e294c1c2575e773c231de3dd7795215588f1ae0bc | What is the probability that there are more customers needing a connection than there are modems? Provide both an exact solution and an approximation using the Poisson PMF. | [
"\\sum_{k=m+1}^\\infty \\frac{(np)^k e^{-np}}{k!}"
] | [
"What is the probability that there are more customers needing a connection than there are modems? Provide both an exact solution and an approximation using the Poisson PMF."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-a1cd26ac03a33dc4797dc738f0dee94f94ed057d9785f63c1ac3cf34444ea31d | What is the probability that there exists at least one pair of couples where both husbands share the same birthday and both wives share the same birthday, given that 10 couples are invited to a party, each person's birthday is independent, and any day of the year (365 days) is equally likely to be a person's birthday? | [
"\\(0.00033772543\\)"
] | [
"What is the probability that there exists at least one pair of couples where both husbands share the same birthday and both wives share the same birthday, given that 10 couples are invited to a party, each person's birthday is independent, and any day of the year (365 days) is equally likely to be a person's birth... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-1be830f0becb356742a9a7c86c4c9c76157ff22a1273958ecc2f746a35ed24ad | What is the probability that there is a 2-way tie for the winner given $n$ people are playing? In other words, what's the probability that 2 people get the first BINGO at the same time? | [
"\\dfrac{n(n - 1)}{2(2^n - 1)}"
] | [
"What is the probability that there is a 2-way tie for the winner given $n$ people are playing? In other words, what's the probability that 2 people get the first BINGO at the same time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-9f17a4ca8d695904548682e0008de4043c3f741e212aab8b96a9fa7ffb286bcd | What is the probability that there is a group of three people who share the same birthday among 36 people, disregarding February 29? | [
"\\(\\frac{{36 \\choose 3}}{365^2}\\)"
] | [
"What is the probability that there is a group of three people who share the same birthday among 36 people, disregarding February 29?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aa057c73f9306c3e2aa47f0e6f74dc50be0dd60bae207fd4bb17ef245b8b5dd1 | What is the probability that there is a group of three people who share the same birthday among 36 people, taking February 29 into account? | [
"0.052"
] | [
"What is the probability that there is a group of three people who share the same birthday among 36 people, taking February 29 into account?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-4e828f7f4307bae8a99f2f837019008f367e8fe0fcab51cc1576b97352d13c2b | What is the probability that there is a group of three people who share the same birthday, and no other group of two people share the same birthday among 36 people, disregarding February 29? | [
"\\(\\frac{{36 \\choose 3} \\cdot 365 \\cdot 364! / 331!}{365^{36}}\\)"
] | [
"What is the probability that there is a group of three people who share the same birthday, and no other group of two people share the same birthday among 36 people, disregarding February 29?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
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] | {
"cot": 10,
"genselect": 2,
"tir": 23
} | 35 | 0 | 0 |
omr-ws-f035238d0961dada1fe63b5874484983c4f0c8cd7405a51bc696f3775ab6fcbd | What is the probability that there is a pile with no hearts when a standard deck of 52 cards is shuffled and split into four piles of 13 cards each? | [
"0.0511"
] | [
"What is the probability that there is a pile with no hearts when a standard deck of 52 cards is shuffled and split into four piles of 13 cards each?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-1081c222828d7e241180657ddf366450c71c8040b0bf028835c3a6aed6c87c23 | What is the probability that there is a string of 5 heads before a string of 2 tails when a fair coin is repeatedly flipped? | [
"\\(\\frac{3}{34}\\)"
] | [
"What is the probability that there is a string of 5 heads before a string of 2 tails when a fair coin is repeatedly flipped?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-8c4a497081903d3f4762adbbdc6f8c6ddf8c97feb6e26a960304a7765e2bb708 | What is the probability that there is at least one accident this week? Calculate this probability using both the random variable \( T \) (time between accidents) and the random variable \( Y \) (number of accidents per week). | [
"\\( 1 - e^{-3} \\)"
] | [
"What is the probability that there is at least one accident this week? Calculate this probability using both the random variable \\( T \\) (time between accidents) and the random variable \\( Y \\) (number of accidents per week)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2925609482e747410b96e31bd8e42d467cc6f50ab71c887846b9fac934f256b3 | What is the probability that there is at least one common birthday (month/day) in a class of 30 students? | [
"0.7063"
] | [
"What is the probability that there is at least one common birthday (month/day) in a class of 30 students?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-d6c98674bf95e54ebf6021e76ab8da805143ea4ccb2539dafcd287329a7db5fc | What is the probability that there is at least one run of ten or more face cards in an eight-deck shoe (stack of cards), assuming a random, fair shuffle? | [
"0.00012"
] | [
"What is the probability that there is at least one run of ten or more face cards in an eight-deck shoe (stack of cards), assuming a random, fair shuffle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-b96f03e4697f5e54345c7c8cf0510a86cd4a1022efa99053a24637745114abce | What is the probability that there is no accident for a period of two weeks? Calculate this probability using both the random variable \( T \) (time between accidents) and the random variable \( Y \) (number of accidents per week). | [
"\\( e^{-6} \\approx 0.00248 \\)"
] | [
"What is the probability that there is no accident for a period of two weeks? Calculate this probability using both the random variable \\( T \\) (time between accidents) and the random variable \\( Y \\) (number of accidents per week)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-03d6836ce56e34b388837d762b0ed1365700d0a90c707d03c10e1b13ba4cc74f | What is the probability that there is no couple in the same group when 16 people (8 couples) are divided into 4 groups, with each group consisting of 2 males and 2 females? | [
"\\dfrac{12}{35}"
] | [
"What is the probability that there is no couple in the same group when 16 people (8 couples) are divided into 4 groups, with each group consisting of 2 males and 2 females?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-fd67663c0f72ff669fc46104d3dca5165bda060fee7bc5e4cfddef740192d216 | What is the probability that there is one defective phone in each box, given that there are 2 boxes each containing 100 distinct phones and exactly 2 of them are defective? | [
"\\(\\frac{100}{199}\\)"
] | [
"What is the probability that there is one defective phone in each 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": 29
} | 45 | 0 | 0 |
omr-ws-cfd97390bd63937fbe6310cc26a626c6a523f1d6d4a319e1faa9c7661d08ee84 | What is the probability that there will be at least one floor such that both elevators stop at that floor, given that there are 5 people in each of two elevators and they leave the elevators randomly on any of the 11 floors above the ground floor? | [
"\\dfrac{2115060901}{2357947691}"
] | [
"What is the probability that there will be at least one floor such that both elevators stop at that floor, given that there are 5 people in each of two elevators and they leave the elevators randomly on any of the 11 floors above the ground floor?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-67bd17d83555081b5aff2bb28e311862cfa9147ee0e02c55c34544f1591b494d | What is the probability that three cards drawn from a standard 52-card deck are all of the same rank? | [
"\\(\\frac{1}{425}\\)"
] | [
"What is the probability that three cards drawn from a standard 52-card deck are all of the same rank?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-bca14801f741b3a6933131540354952e378b7b22636234b5ac60a1474769d39c | What is the probability that three consecutive prime numbers will be produced when a graphing calculator generates a number between 1 and 20? | [
"\\(\\frac{1}{1000}\\)\n\nExplanation:\nThe primes between 1 and 20 are \\(2, 3, 5, 7, 11, 13, 17, 19\\). The consecutive primes in this range are \\(2, 3\\), \\(3, 5\\), \\(5, 7\\), \\(11, 13\\), and \\(17, 19\\). There are no sets of three consecutive primes in this range. Therefore, the probability of generating... | [
"What is the probability that three consecutive prime numbers will be produced when a graphing calculator generates a number between 1 and 20?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-00f1fcba20e9543ec51da452d83fd3ee7b122b9d52c8f20b65ce796966353789 | What is the probability that three distinct numbers chosen from the set {3, 4, 5, 6, 7, 8} can be the lengths of the sides of a non-degenerate triangle? | [
"\\(\\frac{17}{20}\\)"
] | [
"What is the probability that three distinct numbers chosen from the set {3, 4, 5, 6, 7, 8} can be the lengths of the sides of a non-degenerate triangle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-39d90eef94df06db0a36547750822c7b829a0b9246fccad426602da95f4000f5 | What is the probability that three draws are required to get a specific outcome from a set of 5 distinct items, where the draws are without replacement and the order of draws matters? | [
"\\(\\frac{1}{5}\\)"
] | [
"What is the probability that three draws are required to get a specific outcome from a set of 5 distinct items, where the draws are without replacement and the order of draws matters?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-8e56b0aa1f6d5aa7b16a7f14afb6ca4f998079181a845c82c9f53adddfe17fd6 | What is the probability that three gold, two silver, and two bronze chocolate coins arranged in a line have no consecutive coins of the same color? | [
"\\(\\frac{38}{210}\\)"
] | [
"What is the probability that three gold, two silver, and two bronze chocolate coins arranged in a line have no consecutive coins of the same color?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 2,
"tir": 23
} | 27 | 0 | 0 |
omr-ws-9932ef18f3830be214880beacee06ab244b43363cc2c31423a37cdf55e99d6ce | What is the probability that three numbers belonging to the interval $(0,1)$ are the side-lengths of a triangle? | [
"$\\frac{1}{2}$"
] | [
"What is the probability that three numbers belonging to the interval $(0,1)$ are the side-lengths of a triangle?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 29
} | 43 | 0 | 0 |
omr-ws-708c6e1abafec74d3c62fd9fdc744ae02156bc3d98533c95ee4c383ba41285bb | What is the probability that three numbers chosen from the set $\{1, 2, 3, \ldots, 10\}$, not necessarily distinct, are in non-decreasing order (i.e., $x_1 \leq x_2 \leq x_3$)? | [
"\\(\\frac{11}{50}\\)"
] | [
"What is the probability that three numbers chosen from the set $\\{1, 2, 3, \\ldots, 10\\}$, not necessarily distinct, are in non-decreasing order (i.e., $x_1 \\leq x_2 \\leq x_3$)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-8029c121e4975757685c65f07c680f3697ddf5f60cf92350fa1dd50f0157d343 | What is the probability that three numbers selected at random from the interval [0,1] can form the lengths of the sides of a triangle? | [
"\\dfrac{1}{2}"
] | [
"What is the probability that three numbers selected at random from the interval [0,1] can form the lengths of the sides of a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-22b4bde55967f044937f2ddb6c64c3cb15b3416ef9e01801846faa800ab4ff98 | What is the probability that three numbers selected at random from the numbers 1 to 100 (one from each of three cards) can form the sides of a right-angled triangle, with no two sides being similar? | [
"0.000096"
] | [
"What is the probability that three numbers selected at random from the numbers 1 to 100 (one from each of three cards) can form the sides of a right-angled triangle, with no two sides being similar?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-84a7551b28202a70533d73ca9b1931df4099d89e2a84c9d0e6577db08c720d0d | What is the probability that three particular people sit together when $n$ people are sitting in a circle? | [
"\\(\\frac{6}{(n-1)(n-2)}\\)"
] | [
"What is the probability that three particular people sit together when $n$ people are sitting in a circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 1,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-3710eab89239581dc246e55175b7391126e80e2b5c3dc3bfb8626ce6a3e939e3 | What is the probability that three parts obtained by breaking a stick into three pieces can form a triangle? | [
"\\(\\frac{1}{4}\\)"
] | [
"What is the probability that three parts obtained by breaking a stick into three pieces can form a triangle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-011c3c94f5c31c1e4f731ed592c0f470ef91debd924e76d7d0231f703f0391f7 | What is the probability that three pieces of rope, obtained by making two random cuts on a rope, can form a triangle? | [
"\\(\\frac{1}{4}\\)"
] | [
"What is the probability that three pieces of rope, obtained by making two random cuts on a rope, can form a triangle?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5f46511904451449804bb44cc46a0d03de6742d864f282c3f1dcadc64ee3378d | What is the probability that three randomly chosen (not necessarily distinct) positive integers can form the side lengths of a triangle? | [
"\\dfrac{1}{2}"
] | [
"What is the probability that three randomly chosen (not necessarily distinct) positive integers can form the side lengths of a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-86c7bd3286ee2521b01f216f53c8fa5b32fa1864385572f6eebfa43ec19772b0 | What is the probability that three randomly chosen numbers between 0 and 1 can be the sides of a triangle? | [
"\\(\\frac{1}{4}\\)"
] | [
"What is the probability that three randomly chosen numbers between 0 and 1 can be the sides of a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e2e30f6e99858b7483cef3e9c769cf14e445ba8e9bcbde4e34dc68df3832f2b6 | What is the probability that three randomly chosen numbers from the interval $[0,1]$ can form a triangle? | [
"$\\frac{1}{2}$"
] | [
"What is the probability that three randomly chosen numbers from the interval $[0,1]$ can form a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-c95e3cfef44a8a7e4c242b149f1ea6e3827c602adc0ca87802b453606d4a1727 | What is the probability that three randomly chosen points from among 2010 points evenly spaced on a circle form the vertices of a right triangle? | [
"\\(\\frac{3}{2009}\\)"
] | [
"What is the probability that three randomly chosen points from among 2010 points evenly spaced on a circle form the vertices of a right triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-00b5aeaa12f91f68189a8cb74e44e249642191b1abc85039f5ed74a88c889b3a | What is the probability that three randomly chosen points on a circle form an obtuse triangle? | [
"\\(\\frac{3}{4}\\)"
] | [
"What is the probability that three randomly chosen points on a circle form an obtuse triangle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-abf03aa8ee1f07b4a8576613939506549a20e3a19b66e961fc0cad5666f1f854 | What is the probability that three randomly chosen vertices from a regular 7-sided polygon form the vertices of an isosceles triangle? | [
"\\(\\frac{3}{5}\\)"
] | [
"What is the probability that three randomly chosen vertices from a regular 7-sided polygon form the vertices of an isosceles triangle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-8e42554a9f90033fd825b0c237abcee124c5d554fc79a2b32cc4c1f938b3a7b5 | What is the probability that three randomly picked 1x1 squares on a chessboard will form an L-shaped tromino? | [
"\\dfrac{7}{1488}"
] | [
"What is the probability that three randomly picked 1x1 squares on a chessboard will form an L-shaped tromino?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9f9c757dfd25643c19f27324a8263ad20b22c82fad533c11166be8432ffe0984 | What is the probability that three randomly selected $1 \times 1$ squares from a chessboard form an L shape? | [
"\\(\\frac{196}{\\binom{64}{3}}\\)"
] | [
"What is the probability that three randomly selected $1 \\times 1$ squares from a chessboard form an L shape?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
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