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-b159ba5cb0106b29f209d31f682042251ff085950745911f6673d77a584a1373 | What is the probability of getting 2 consecutive heads before getting 3 tails? | [
"\\(\\frac{37}{64}\\)"
] | [
"What is the probability of getting 2 consecutive heads before getting 3 tails?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 12
} | 26 | 0 | 0 |
omr-ws-682918d0b5bf2cd726648bb9bca5855bde0e111d9c887c0518462bc7ea7bb4b5 | What is the probability of getting 2 sums of 7 before both a sum of 6 and a sum of 8 when two standard six-sided dice are rolled simultaneously for an indefinite number of times? | [
"\\(\\frac{3519}{7744}\\)"
] | [
"What is the probability of getting 2 sums of 7 before both a sum of 6 and a sum of 8 when two standard six-sided dice are rolled simultaneously for an indefinite number of times?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 12
} | 33 | 0 | 0 |
omr-ws-7e197160dcdf9969290057f70877d213feea9bf2d16878f68e827cd7f7630790 | What is the probability of getting 4 heads in a row when flipping 9 fair coins? | [
"\\( 1 - \\frac{401}{512} \\)"
] | [
"What is the probability of getting 4 heads in a row when flipping 9 fair coins?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 2,
"tir": 29
} | 41 | 0 | 0 |
omr-ws-88de7788c6becde6de76409855f8ebeafea46a8273d29e75c9c45fae1197b882 | What is the probability of getting 4 of a kind when rolling a standard die 5 times? (4 of a kind is defined as rolling 4 of one number and 1 of any other number, which can be the same as the first number.) | [
"\\(\\frac{150}{6^5}\\)"
] | [
"What is the probability of getting 4 of a kind when rolling a standard die 5 times? (4 of a kind is defined as rolling 4 of one number and 1 of any other number, which can be the same as the first number.)"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 27
} | 39 | 0 | 0 |
omr-ws-79b3421fcea046c2ea6d55e7447775eee1ef2c09dfb9145dd302f5f055087f8e | What is the probability of getting 5 Sundays in the month of January and December of the year 2012? | [
"1"
] | [
"What is the probability of getting 5 Sundays in the month of January and December of the year 2012?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 1
} | 12 | 0 | 0 |
omr-ws-73107ad3e935cf2a0847a5abcdefa0931838a632dbd8250aab629b01b31d7f63 | What is the probability of getting 8 correct answers out of 10 in a multiple-choice quiz with 4 answer choices per question if answers are chosen at random? | [
"\\(\\frac{405}{1048576}\\)"
] | [
"What is the probability of getting 8 correct answers out of 10 in a multiple-choice quiz with 4 answer choices per question if answers are chosen at random?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-b469b8f12f155b79b40cc5c4bdab6201747a729d953be790ddd47b179e50cd62 | What is the probability of getting HHHHH before TT in a sequence of coin flips? | [
"\\dfrac{3}{34}"
] | [
"What is the probability of getting HHHHH before TT in a sequence of coin flips?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-959fd74c252c554747f2f9bebe3e0c37758134923afa49bfa9602505e809ccfa | What is the probability of getting HT or TH in a row when flipping a coin? | [
"\\dfrac{1}{2}"
] | [
"What is the probability of getting HT or TH in a row when flipping a coin?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d82e8c8f3ccc540ee9469aa887d2e7af29f7615d899922765d3d4120214505d2 | What is the probability of getting a Yahtzee after two throws in the game of Yahtzee? | [
"\\(\\frac{569}{1296}\\)"
] | [
"What is the probability of getting a Yahtzee after two throws in the game of Yahtzee?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8fd4a383057140d320ea62f14771951acfa9a192cddd2e9ffa39a0b61723c94c | What is the probability of getting a Yahtzee on the first throw in the game of Yahtzee? | [
"\\(\\frac{1}{1296}\\)"
] | [
"What is the probability of getting a Yahtzee on the first throw in the game of Yahtzee?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-641aed18c7c3eed0c0eae330ade7de460f0431e156a8c3ad807945f9e27403ae | What is the probability of getting a certain combination of red and brown M&Ms if you choose 4 randomly from a bag containing M&Ms of six different colors (red, brown, green, yellow, orange, blue)? | [
"\\dfrac{1}{81}"
] | [
"What is the probability of getting a certain combination of red and brown M&Ms if you choose 4 randomly from a bag containing M&Ms of six different colors (red, brown, green, yellow, orange, blue)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-4cb05b28e11ab19181c36781521e605032eee4a53362d3aaef646ea4f8d0ecd9 | What is the probability of getting a full house from a hand of 13 cards? | [
"\\dfrac{506068992}{7937669495}"
] | [
"What is the probability of getting a full house from a hand of 13 cards?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-33cac9df602deed4f23a2f1f14dbbdbea20a7fe5ef53c69f44403a0ff5576d37 | What is the probability of getting a head and a tail in any order when flipping a coin? | [
"\\dfrac{1}{2}"
] | [
"What is the probability of getting a head and a tail in any order when flipping a coin?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-80b25718db7949d6feac3496882611277f511d9e2f277a3002b4a6081e947887 | What is the probability of getting a product of 6 when three dice are rolled? | [
"\\(\\frac{1}{24}\\)"
] | [
"What is the probability of getting a product of 6 when three dice are rolled?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-3e60347cf4abcbd312fb74d2a945084da39e31588cca11bc5bbc0665f533e0cb | What is the probability of getting a royal flush in Texas Hold'em (7 cards)? A royal flush consists of an ace, king, queen, jack, and 10 of the same suit. | [
"\\(\\frac{1}{30940}\\)"
] | [
"What is the probability of getting a royal flush in Texas Hold'em (7 cards)? A royal flush consists of an ace, king, queen, jack, and 10 of the same suit."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-baa1316eaa91167e2b27051c9bc94cea9048a9b285281dc71dfc4ffd95633f6b | What is the probability of getting a set of 4 random cards from a deck of 52 cards (where each card has a value from 1 to 13 and there are 4 of each number) such that it is impossible to form the number 24 using addition, subtraction, multiplication, and division, and brackets? | [
"0.19543"
] | [
"What is the probability of getting a set of 4 random cards from a deck of 52 cards (where each card has a value from 1 to 13 and there are 4 of each number) such that it is impossible to form the number 24 using addition, subtraction, multiplication, and division, and brackets?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-94c54882754e4665ade0a58f53339eb7cdbd65f6ec147139843ee7c73de9b6c7 | What is the probability of getting a sum greater than 9 but less than 16 when three dice are rolled? | [
"\\(\\frac{125}{216}\\)"
] | [
"What is the probability of getting a sum greater than 9 but less than 16 when three dice are rolled?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-88c69213de36630301428c4e66fa25026aa4a4d0013a7fb4b1fab3c4ae781c7a | What is the probability of getting a sum that is a multiple of 5 when a die is thrown \( m \) times? | [
"\\(\\begin{cases} \n\\frac{6^m - 1}{5 \\cdot 6^m}, & \\text{if } m \\not\\equiv 0 \\pmod{5} \\\\\n\\frac{6^m + 4}{5 \\cdot 6^m}, & \\text{if } m \\equiv 0 \\pmod{5}\n\\end{cases}\\)"
] | [
"What is the probability of getting a sum that is a multiple of 5 when a die is thrown \\( m \\) times?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 1,
"tir": 16
} | 21 | 0 | 0 |
omr-ws-4ad8eadbb09a45653b845beab5eae2a7ce61103fcdc0c5cb847f04c0e38cd8d2 | What is the probability of getting a three-of-a-kind in poker? | [
"54912"
] | [
"What is the probability of getting a three-of-a-kind in poker?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3787b84fd0c562b92afbd2bb08fab7150f0cce417c9ff292a6f1b1e5e5662999 | What is the probability of getting a total between 5 and 16 (inclusive) when three dice are thrown? | [
"\\dfrac{26}{27}"
] | [
"What is the probability of getting a total between 5 and 16 (inclusive) when three dice are thrown?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-856f7c7ee372f4efa1b59e977f823d4e353313f59e91a2f9a0953ad992d3d298 | What is the probability of getting a two pair hand in 5 card poker? | [
"0.04754"
] | [
"What is the probability of getting a two pair hand in 5 card poker?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-e8910d4405fe98f6fe3ed29a7d7383b93d8870ced9398a4fa71644a00824b218 | What is the probability of getting all sixes in exactly k rolls when rolling 5 dice, keeping the sixes, and re-rolling the others, as a function of k? | [
"\\( (1-(5/6)^k)^5 - (1-(5/6)^{k-1})^5 \\)"
] | [
"What is the probability of getting all sixes in exactly k rolls when rolling 5 dice, keeping the sixes, and re-rolling the others, as a function of k?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 37,
"tir": 10
} | 47 | 0 | 0 |
omr-ws-23a9273ceb58e33cd690b22e685641519c2fe87688f85d1c2c77798b60b682af | What is the probability of getting all three different figures when purchasing five meals, given that each meal comes with one of three different figures? | [
"\\(\\frac{1}{7}\\)"
] | [
"What is the probability of getting all three different figures when purchasing five meals, given that each meal comes with one of three different figures?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e25f16d6c68a5ea8803110a6be04d3bd9e9cb630f41c4193e455f5002e21816f | What is the probability of getting an even number of heads in 50 throws of a biased coin, where the probability of landing on heads is \( \frac{2}{3} \)? | [
"\\(\\frac{1}{2}\\left(1 + \\frac{1}{3^{50}}\\right)\\)"
] | [
"What is the probability of getting an even number of heads in 50 throws of a biased coin, where the probability of landing on heads is \\( \\frac{2}{3} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 27
} | 73 | 0 | 0 |
omr-ws-b89e6f4e78747cbe5d63664e78743acd87404573f50d11d40a2a417199d26fb6 | What is the probability of getting at least $m$ dollars, starting with 0 dollars, if you can click a button 10 times, where each click has a $\frac{1}{3}$ probability of adding 20 dollars, a $\frac{1}{3}$ probability of deducting 20 dollars, and a $\frac{1}{3}$ probability of doubling your current amount? | [
"\\sum_{k=\\lceil m/20 \\rceil}^{512} P(10, k)"
] | [
"What is the probability of getting at least $m$ dollars, starting with 0 dollars, if you can click a button 10 times, where each click has a $\\frac{1}{3}$ probability of adding 20 dollars, a $\\frac{1}{3}$ probability of deducting 20 dollars, and a $\\frac{1}{3}$ probability of doubling your current amount?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-a2ea9173a8393bc9d8204fb9570db7cad76978e06430ced5e5f9234e1678563b | What is the probability of getting at least 1 card in each color in a set of 7 cards from an Uno deck with no wild cards, assuming there are the same number of cards in each color? | [
"\\(\\frac{1}{6}\\)"
] | [
"What is the probability of getting at least 1 card in each color in a set of 7 cards from an Uno deck with no wild cards, assuming there are the same number of cards in each color?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-464a6e2604509768950e8f665f801932fc53fb3913bff0999776aed168a96314 | What is the probability of getting at least 1 queen when 7 cards are randomly selected from a standard deck of 52 cards? | [
"\\(\\frac{\\binom{52}{7} - \\binom{48}{7}}{\\binom{52}{7}}\\)"
] | [
"What is the probability of getting at least 1 queen when 7 cards are randomly selected from a standard deck of 52 cards?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-ab10e4c78d8c0fed76684bca617a52ebbef14d1fe46fd9345e9f1199e2ad38a9 | What is the probability of getting at least 200 dollars, starting with 0 dollars, if you can click a button 10 times, where each click has a $\frac{1}{3}$ probability of adding 20 dollars, a $\frac{1}{3}$ probability of deducting 20 dollars, and a $\frac{1}{3}$ probability of doubling your current amount? | [
"\\dfrac{347}{19683}"
] | [
"What is the probability of getting at least 200 dollars, starting with 0 dollars, if you can click a button 10 times, where each click has a $\\frac{1}{3}$ probability of adding 20 dollars, a $\\frac{1}{3}$ probability of deducting 20 dollars, and a $\\frac{1}{3}$ probability of doubling your current amount?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-e68eec7cc8fd5303d07a6fc1d13a5e2bc04ba4b2caf3b795dea38cf6d16dab17 | What is the probability of getting at least 200 heads in 300 tosses of a fair coin? | [
"0.00000040074"
] | [
"What is the probability of getting at least 200 heads in 300 tosses of a fair coin?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 24
} | 47 | 0 | 0 |
omr-ws-fce427ade24494f9ba28d90e2f146b76d8c15861188d29bc69f905d2866c05ff | What is the probability of getting at least 30% of the questions correct on a 100-question multiple-choice test with 4 choices per question if answers are selected randomly? | [
"0.124106705"
] | [
"What is the probability of getting at least 30% of the questions correct on a 100-question multiple-choice test with 4 choices per question if answers are selected randomly?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fb2eb7f1da1154fb2abb80020de9a7d1bf724534a48de0bdf09a42738ea3c2b1 | What is the probability of getting at least 4 consecutive heads when a fair coin is tossed 10 times? | [
"\\(\\frac{251}{1024}\\)"
] | [
"What is the probability of getting at least 4 consecutive heads when a fair coin is tossed 10 times?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 25
} | 47 | 0 | 0 |
omr-ws-56b1f8fbf705d2255e41176827629f702439fb98d2a31727c56c1b07a393710c | What is the probability of getting at least a string of 9 or more heads when flipping a coin 15 times? | [
"\\(\\frac{1}{128}\\)"
] | [
"What is the probability of getting at least a string of 9 or more heads when flipping a coin 15 times?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-0dd8e1731a9bb8e730bf2bb7afa7dd68c478757035d8813f8c17dffb0d25abf9 | What is the probability of getting at most one jack when 4 cards are drawn from a standard deck with replacement? | [
"\\(\\frac{1744}{28561}\\)"
] | [
"What is the probability of getting at most one jack when 4 cards are drawn from a standard deck with replacement?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e7b0191bc0dcaefe0fbe3526c83c5320e2d860a8405a3327671750386c0230b | What is the probability of getting consecutive two tails when four coins are flipped one after the other? | [
"\\(\\frac{3}{8}\\)"
] | [
"What is the probability of getting consecutive two tails when four coins are flipped one after the other?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5e8518df2b76cc73013c89a9ddb7a9ab32252bcf47002a638df78d6d3e2c3665 | What is the probability of getting exactly 4 clubs when 5 cards are dealt from a standard deck? | [
"\\dfrac{143}{13328}"
] | [
"What is the probability of getting exactly 4 clubs when 5 cards are dealt from a standard deck?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b5b60ad1a938ed7c834c9e096225a58d8ceb8435b8cdd5f08c67129efd8784c1 | What is the probability of getting heads twice in a row when a fair coin is flipped 3 times? | [
"\\(\\frac{3}{8}\\)"
] | [
"What is the probability of getting heads twice in a row when a fair coin is flipped 3 times?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-55180030ccc8ffa67f3fb4a2f7a03275f44d988634d3f0619af4b4b887b6b940 | What is the probability of getting no 3 consecutive heads or tails in 10 tosses of a fair coin? | [
"\\dfrac{89}{512}"
] | [
"What is the probability of getting no 3 consecutive heads or tails in 10 tosses of a fair coin?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dfc7589f968775f9eb66391d7d1caa0fbe794092fc7a8d9b368eb5f48b541f21 | What is the probability of getting one ace, or an ace and a king when six cards are dealt from a pack of 52 cards? | [
"$\\frac{C(48,5)C(4,1)}{C(52,6)};\\ \\frac{C(44,4)C(4,1)C(4,1)}{C(52,6)}$"
] | [
"What is the probability of getting one ace, or an ace and a king when six cards are dealt from a pack of 52 cards?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d15a250552ae9db929ef966f424a1629a683b52fe6fa14bd29ef03cadccff23f | What is the probability of getting three heads in a row before getting two tails when flipping a fair coin? | [
"\\(\\frac{15}{64}\\)"
] | [
"What is the probability of getting three heads in a row before getting two tails when flipping a fair coin?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-fb74a8ebb113f056cf897d1e5f378ee3cae063af0b3a3e8deef8a250a577b002 | What is the probability of getting two consecutive tails when four coins are flipped one after another? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the probability of getting two consecutive tails when four coins are flipped one after another?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-045984d33266dae381550711a63cc904532107f184ba0e20c113fa2c9d5fb114 | What is the probability of getting two heads in a row when flipping a coin? | [
"\\dfrac{1}{4}"
] | [
"What is the probability of getting two heads in a row when flipping a coin?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1bcc31ef39a634ae00424209678ebbad504023c49fbf9d33a2394f592e74bcd3 | What is the probability of having 2 adjacent white balls or 2 adjacent blue balls in a random arrangement of 3 red, 2 white, and 2 blue balls? | [
"\\(\\frac{10}{21}\\)"
] | [
"What is the probability of having 2 adjacent white balls or 2 adjacent blue balls in a random arrangement of 3 red, 2 white, and 2 blue balls?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 26
} | 38 | 0 | 0 |
omr-ws-83871196c91e5710b5d23eb32dcd1a7b2229e8aed9e4d105f442b4037d129143 | What is the probability of having 4 boys and 4 girls in a family of 8 children? | [
"\\(\\frac{35}{128}\\)",
"\\dfrac{35}{128}"
] | [
"What is the probability of having 4 boys and 4 girls in a family of 8 children?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 27
} | 59 | 0 | 0 |
omr-ws-90c4f7809e5ee7676e359835d1a4d80898dab0f78f32dcac329b7b5d5688bdda | What is the probability of having \( k \) heads in 10 tosses of a biased coin where the probability of a head in each toss is \( \frac{1}{3} \)? | [
"\\(\\binom{10}{k} \\left( \\frac{1}{3} \\right)^k \\left( \\frac{2}{3} \\right)^{10-k}\\)"
] | [
"What is the probability of having \\( k \\) heads in 10 tosses of a biased coin where the probability of a head in each toss is \\( \\frac{1}{3} \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-728a86f74b7a2e353e0391031089f4b7828b2df04344ee9613035dc686937344 | What is the probability of having at least one pair of girls when $n$ pairs are formed from $n$ girls and $n$ boys at random? For which $n$ is this probability greater than $0.9$? | [
"\\( n \\geq 6 \\)"
] | [
"What is the probability of having at least one pair of girls when $n$ pairs are formed from $n$ girls and $n$ boys at random? For which $n$ is this probability greater than $0.9$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 15
} | 62 | 0 | 0 |
omr-ws-0d5c037f2f8194670164de14183e66451a65770b62a174da83e64e39254ff388 | What is the probability of having at most five different digits in a sequence of seven randomly chosen digits, where each digit can be any of $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$? | [
"\\( 1 - \\frac{6 \\cdot \\binom{10}{6} \\cdot \\frac{7!}{2!} + \\binom{10}{7} \\cdot 7!}{10^7} \\)"
] | [
"What is the probability of having at most five different digits in a sequence of seven randomly chosen digits, where each digit can be any of $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-153396ecbd15128ce36599a770f8d6bb63310eb2ac3095bb5898e0bb2647b253 | What is the probability of having no $B$ before the first $A$ in a random permutation of the word $\text{ABRAKADABRA}$? | [
"\\(\\frac{5}{7}\\)"
] | [
"What is the probability of having no $B$ before the first $A$ in a random permutation of the word $\\text{ABRAKADABRA}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-2e8485624cb5a059818528b723c4f44a7f233d8e8b691545902fdd469cee7c58 | What is the probability of knocking down $m$ consecutive hurdles out of $n$ hurdles, given that the probability of knocking down any hurdle is $p$? | [
"\\((n-m+1) \\cdot p^m \\cdot (1-p)^{n-m}\\)"
] | [
"What is the probability of knocking down $m$ consecutive hurdles out of $n$ hurdles, given that the probability of knocking down any hurdle is $p$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-78737cc0d498667661b810399baa80c0d56b6fe7a0272d16850563da6c15aca8 | What is the probability of making 8 coin flips, given that the game ends as soon as 2 consecutive tails appear and the maximum number of flips is 8? | [
"\\(\\frac{1}{16}\\)"
] | [
"What is the probability of making 8 coin flips, given that the game ends as soon as 2 consecutive tails appear and the maximum number of flips is 8?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40c3af337d95f3f77226a29cce3651a13200ac3f0db6f3e4d9c5dd6377eb86a9 | What is the probability of making a Type II error in the given hypothesis test? The null hypothesis (Ho) is that the population mean is 1.5, and the alternative hypothesis (Ha) is that the population mean is greater than 1.5. The test rejects Ho when the sample mean is greater than 1.85. Assume a normal distribution. | [
"0.0037"
] | [
"What is the probability of making a Type II error in the given hypothesis test? The null hypothesis (Ho) is that the population mean is 1.5, and the alternative hypothesis (Ha) is that the population mean is greater than 1.5. The test rejects Ho when the sample mean is greater than 1.85. Assume a normal distributi... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-15770cacdf6aea0e1461a223699da21899909fa4e7650f8aa3c1b8622f94064c | What is the probability of missing a target that consists of a circular disk with a probability of hitting $.35$, a ring with a probability of hitting $.3$, and another ring with a probability of hitting $.25$? | [
"0.1"
] | [
"What is the probability of missing a target that consists of a circular disk with a probability of hitting $.35$, a ring with a probability of hitting $.3$, and another ring with a probability of hitting $.25$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-d9a381d900a5ca9d12dc5dff478372db0d33ba4b4b5a59937142c2ae50bebc0d | What is the probability of no customers arriving in a five-minute interval at an ATM where customers arrive at a rate of 30 customers per hour? | [
"0.0821"
] | [
"What is the probability of no customers arriving in a five-minute interval at an ATM where customers arrive at a rate of 30 customers per hour?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 2,
"tir": 29
} | 41 | 0 | 0 |
omr-ws-3797ac87289684a3f00014658d9b2f901e6e33940051b52dc6e515f42ba07d00 | What is the probability of not getting 3 heads or 2 tails in a row when flipping a coin 8 times? | [
"\\(\\frac{1}{16}\\)"
] | [
"What is the probability of not getting 3 heads or 2 tails in a row when flipping a coin 8 times?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-d3455c4a244409dbed8dc060831f58a669b03098282df9448d998bdd12ac3c2e | What is the probability of not getting the subsequence "HTHH" when flipping a coin 8 times? | [
"\\dfrac{181}{256}"
] | [
"What is the probability of not getting the subsequence \"HTHH\" when flipping a coin 8 times?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-4ab04c893057f2c56beebf18b20082218b67bf51461553f5844ed6e50057132a | What is the probability of observing the sequence \(HHTTHTHTH\) in an infinite sequence of experiments, where each experiment has two possible outcomes \(H\) and \(T\)? | [
"1"
] | [
"What is the probability of observing the sequence \\(HHTTHTHTH\\) in an infinite sequence of experiments, where each experiment has two possible outcomes \\(H\\) and \\(T\\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c2bbb940ff9d2540eb30e373a45ccb56cbd5657f1ce23ffe63a1f4125ae1721e | What is the probability of obtaining a straight in five-card poker? A straight consists of a set of 5 cards that can be arranged in a consecutive sequence (aces high), but are not necessarily of the same suit. | [
"\\(\\frac{10 \\cdot 4^5 - 40}{\\binom{52}{5}}\\)"
] | [
"What is the probability of obtaining a straight in five-card poker? A straight consists of a set of 5 cards that can be arranged in a consecutive sequence (aces high), but are not necessarily of the same suit."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-b22bd99e333f95f165a9dfec5677f9ab1d00d876d62ad79c97cdc278083193ab | What is the probability of obtaining a sum of 15 with three throws of a single six-sided die? | [
"\\(\\frac{5}{108}\\)"
] | [
"What is the probability of obtaining a sum of 15 with three throws of a single six-sided die?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-af534daa1ed63902fc60b0a778614d430835672ec104e572775b8d68a4928f37 | What is the probability of picking 5 balls from a set of balls with 33 consecutive numbers when 5 balls are selected from an urn containing 50 balls numbered from 1 to 50? | [
"\\(\\frac{106082}{264845}\\)"
] | [
"What is the probability of picking 5 balls from a set of balls with 33 consecutive numbers when 5 balls are selected from an urn containing 50 balls numbered from 1 to 50?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 14
} | 35 | 0 | 0 |
omr-ws-0a5321c3905dd405f4b722ddcfd274f744513164f49f69e4d0223969b839da0e | What is the probability of picking 5 cards from a standard deck of 52 cards such that 4 of the cards have the same value? | [
"\\(\\frac{1}{4165}\\)"
] | [
"What is the probability of picking 5 cards from a standard deck of 52 cards such that 4 of the cards have the same value?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-ac22581a8bf03942e8c7b8df318858cab5fa24266d3d6be239848c6b52e48b52 | What is the probability of picking 5 consecutive numbers when randomly selecting 5 cards from a deck of 52 cards, where each number from 1 to 13 appears on 4 cards? | [
"\\(\\frac{192}{54145}\\)"
] | [
"What is the probability of picking 5 consecutive numbers when randomly selecting 5 cards from a deck of 52 cards, where each number from 1 to 13 appears on 4 cards?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 18
} | 32 | 0 | 0 |
omr-ws-82f510ecc42f2cb2b548049a9d37c772b0588fca4a963feb08fda9ec1793e79c | What is the probability of picking a matching pair of shoes when 4 shoes are randomly selected from a set of 4 pairs (8 shoes in total)? | [
"\\(\\frac{27}{35}\\)"
] | [
"What is the probability of picking a matching pair of shoes when 4 shoes are randomly selected from a set of 4 pairs (8 shoes in total)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-27eb90aee8f043634014d0d6d9835674b7ac5f1a486a246becae229ac28ec947 | What is the probability of picking at random three points on a circle of radius one so that all three lie in a semicircle? | [
"\\dfrac{3}{4}"
] | [
"What is the probability of picking at random three points on a circle of radius one so that all three lie in a semicircle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-fc84d7a1de505ce1bb9c2d305338e88910f69925056fdf12c7903439ea93985d | What is the probability of picking exactly 3 out of 6 numbers correctly in a lottery game where you need to pick 6 different numbers from 1 to 50? | [
"\\( \\frac{\\binom{6}{3} \\binom{44}{3}}{\\binom{50}{6}} \\)"
] | [
"What is the probability of picking exactly 3 out of 6 numbers correctly in a lottery game where you need to pick 6 different numbers from 1 to 50?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-6d8fddd8ad0d49c99cfe5fb3692bb0612610d63ffbebfacbdf89557e1cf587f2 | What is the probability of picking exactly two clubs when dealt 5 cards from a standard 52-card deck? | [
"\\(\\frac{\\binom{13}{2} \\cdot \\binom{39}{3}}{\\binom{52}{5}}\\)"
] | [
"What is the probability of picking exactly two clubs when dealt 5 cards from a standard 52-card deck?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 1,
"tir": 11
} | 20 | 0 | 0 |
omr-ws-2b37fbbd2b10b8a6102bc4e122e51fdffb90afb0903481690adebe40326ef35f | What is the probability of picking out the word "enraged" when the letters of "angered" are shuffled and picked out one-by-one? | [
"\\(\\frac{1}{2520}\\)"
] | [
"What is the probability of picking out the word \"enraged\" when the letters of \"angered\" are shuffled and picked out one-by-one?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-642bb4208074447e812afea142b3eff47e4e0a613e115b5c9af76ccd606edf82 | What is the probability of picking the number 5 and not picking the number 6 when choosing three numbers from the set $\{1,2,3,4,5,6,7,8,9\}$? | [
"\\(\\frac{1}{4}\\)"
] | [
"What is the probability of picking the number 5 and not picking the number 6 when choosing three numbers from the set $\\{1,2,3,4,5,6,7,8,9\\}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-6457aa86ba6f3a9853babf6494d6dde48c358dc330f7b1c5c0eded2863c6763e | What is the probability of picking two cards from a standard deck of 52 cards such that one card is black and one is red, and the value of the red card is greater than the value of the black card? | [
"\\(\\frac{6}{13}\\)"
] | [
"What is the probability of picking two cards from a standard deck of 52 cards such that one card is black and one is red, and the value of the red card is greater than the value of the black card?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 2,
"tir": 16
} | 41 | 0 | 0 |
omr-ws-1cd43d69ea51f0a822e3cccee6da0eed4a8f078f38ad2dccc18929c89e832793 | What is the probability of picking two consecutive integers if you pick 8 integers from a set of 52? | [
"\\dfrac{917923}{1286390}"
] | [
"What is the probability of picking two consecutive integers if you pick 8 integers from a set of 52?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-39bd2ed54ac4c35b1d003978b415131c55641bbbbb64d5122f41e716df4b0f2c | What is the probability of picking two yellow balls and one red ball when selecting three balls from a set containing 5 red balls and 7 yellow balls? | [
"\\(\\frac{21}{88}\\)"
] | [
"What is the probability of picking two yellow balls and one red ball when selecting three balls from a set containing 5 red balls and 7 yellow balls?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-39803fce9a0331d852875bae51c9ae55f2e677d18a7f40abc5b2d648a18ff9dc | What is the probability of reaching at least $m$ dollars, starting with $d$ dollars, and clicking the button $i$ times, where each click has a $\frac{1}{3}$ probability of adding 20 dollars, a $\frac{1}{3}$ probability of deducting 20 dollars, and a $\frac{1}{3}$ probability of doubling your current amount? | [
"P(d, i)"
] | [
"What is the probability of reaching at least $m$ dollars, starting with $d$ dollars, and clicking the button $i$ times, where each click has a $\\frac{1}{3}$ probability of adding 20 dollars, a $\\frac{1}{3}$ probability of deducting 20 dollars, and a $\\frac{1}{3}$ probability of doubling your current amount?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-2170263cb24dfdbafdc61a2cbb70d1ee63ce5a9c7f54ccf26f2b32d8582c9c4f | What is the probability of reaching at least $m$ dollars, starting with $d$ dollars, if you can click a button 10 times, where each click has a $\frac{1}{3}$ probability of adding 20 dollars, a $\frac{1}{3}$ probability of deducting 20 dollars, and a $\frac{1}{3}$ probability of doubling your current amount? | [
"P(10, d)"
] | [
"What is the probability of reaching at least $m$ dollars, starting with $d$ dollars, if you can click a button 10 times, where each click has a $\\frac{1}{3}$ probability of adding 20 dollars, a $\\frac{1}{3}$ probability of deducting 20 dollars, and a $\\frac{1}{3}$ probability of doubling your current amount?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-d08b7fcb0cef651e2cd50ea9eb7964cf21d89a8fe84f135653e5170d50cab819 | What is the probability of receiving exactly 3 kings in a 5-card draw? | [
"\\(\\frac{4512}{2598960}\\) or approximately \\(0.001736\\)"
] | [
"What is the probability of receiving exactly 3 kings in a 5-card draw?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 32
} | 49 | 0 | 0 |
omr-ws-de7553bfca2dae4106fa834f3b839b6c5effcb46ebd8938fcb2b8dae8fc8e487 | What is the probability of rolling a consonant, then a vowel, then a consonant in that exact order on a die with the first 12 letters of the alphabet, and then flipping a coin to get heads, such that the letters spell a word? | [
"\\dfrac{9}{128}"
] | [
"What is the probability of rolling a consonant, then a vowel, then a consonant in that exact order on a die with the first 12 letters of the alphabet, and then flipping a coin to get heads, such that the letters spell a word?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-9a7c093e391ec65d26e0a430c6ed92827cc6c492e1c4b0dd6dcf79918ecbd722 | What is the probability of rolling a die 4 times such that the product of the numbers rolled is a perfect square? | [
"\\dfrac{25}{162}"
] | [
"What is the probability of rolling a die 4 times such that the product of the numbers rolled is a perfect square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-59e31e91461bb72bbcc3564568a2c688d03a6da15eda561e05a8e99d028e69ff | What is the probability of rolling a die 5 times such that each subsequent roll results in a higher number than the previous roll? | [
"\\(6^{-4}\\)"
] | [
"What is the probability of rolling a die 5 times such that each subsequent roll results in a higher number than the previous roll?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-5b1bfd49de2e27870e15e8706d9b271183a86d509159e856666477b7f5696008 | What is the probability of rolling a double when two dice are rolled? Consider both methods: (1) where order matters and (2) where order does not matter. | [
"\\(\\frac{1}{6}\\)"
] | [
"What is the probability of rolling a double when two dice are rolled? Consider both methods: (1) where order matters and (2) where order does not matter."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 12
} | 15 | 0 | 0 |
omr-ws-b0511a3409e610e07ebb37b049ab26efacd77a0a04f09f68b7edc0ab7e513c3a | What is the probability of rolling a sum of $n$ using $x$ $y$-sided dice? | [
"\\frac{1}{y^x} \\sum_{k=0}^{\\left\\lfloor \\frac{n - x}{y} \\right\\rfloor} (-1)^k \\binom{x}{k} \\binom{n - 1 - ky}{x - 1}"
] | [
"What is the probability of rolling a sum of $n$ using $x$ $y$-sided dice?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-32ed62bee9edd4e5f65d27fb1aea53619073e20661d3f16df2176f0d0038f6a5 | What is the probability of rolling a sum of 15 when three dice are rolled together? | [
"5/108"
] | [
"What is the probability of rolling a sum of 15 when three dice are rolled together?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-b016f4ba0fc174846ad7c2d00c1b68acc7896de8f538aaf24468f63b4e4728f8 | What is the probability of rolling at least a "5" six times when a fair die is tossed six times? | [
"\\(\\frac{1}{729}\\)"
] | [
"What is the probability of rolling at least a \"5\" six times when a fair die is tossed six times?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-67918147e271c622040cc07f9eaa7b249cee9d6c089eee9d4d0b1dbe2887a37f | What is the probability of rolling at least a five at least five times when a fair die is rolled six times? | [
"\\(\\frac{13}{729}\\)"
] | [
"What is the probability of rolling at least a five at least five times when a fair die is rolled six times?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 60
} | 91 | 0 | 0 |
omr-ws-af124e9e8bb7e054e10ddb0dd40cc67d6237a8367062c78f2ee4224b7bb0b844 | What is the probability of rolling at least one $6-6$ and at least one $1-2$ in 24 throws with two dice? | [
"0.362"
] | [
"What is the probability of rolling at least one $6-6$ and at least one $1-2$ in 24 throws with two dice?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-d066c3b0df39703c2f43d7854c7f76c9a4866ff4e29cbab81dea4a0be7fbf106 | What is the probability of rolling four 1's before seeing the first six on a standard die? | [
"\\(\\frac{1}{16}\\)"
] | [
"What is the probability of rolling four 1's before seeing the first six on a standard die?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 24
} | 71 | 0 | 0 |
omr-ws-b17a7580de4afc653cfa9b0f05a19cc71eefc772be9d98d2f3ee23b3eafd04a8 | What is the probability of rolling six standard, six-sided dice and getting six distinct numbers? Express your answer as a common fraction. | [
"\\frac{5}{324}"
] | [
"What is the probability of rolling six standard, six-sided dice and getting six distinct numbers? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7adb441a91fd8dfd2d002f4dc0fd55337c3113b082f311d9637c419aff779aa2 | What is the probability of rolling three dice and getting a sum of 10? | [
"\\(\\frac{27}{216}\\)"
] | [
"What is the probability of rolling three dice and getting a sum of 10?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f433df0b372e1bce09a392e41303c46baf9104836fd3fcc748215e215bebbcd4 | What is the probability of rolling two $6-6$ outcomes in 24 throws with two dice? | [
"\\({{24}\\choose 2}\\left(\\frac{1}{36}\\right)^2 \\left(\\frac{35}{36}\\right)^{22}\\)"
] | [
"What is the probability of rolling two $6-6$ outcomes in 24 throws with two dice?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 2
} | 5 | 0 | 0 |
omr-ws-c5f2bc30497842119c9678d129316d88b06e7f457fb28fe8c8baecd0b1623b60 | What is the probability of rolling two 1's and one 6 in three separate rolls of a 7-faced die, where the order of the rolls does not matter? | [
"$\\frac{3}{343}$"
] | [
"What is the probability of rolling two 1's and one 6 in three separate rolls of a 7-faced die, where the order of the rolls does not matter?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-05fb4327c35d50a44a314f627176ca5e1f3ee7fbbae23e20f7905960671a2ac1 | What is the probability of rolling two dice and getting a 1 and an even number? | [
"\\dfrac{1}{6}"
] | [
"What is the probability of rolling two dice and getting a 1 and an even number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e56ac66bc79c2eec96cf66ec934dec28033d5077bab51c24587019a7dad0ef42 | What is the probability of seeing at least 2 bus A while waiting for bus B, given that bus A runs every 10 minutes and bus B runs every 15 minutes? | [
"\\(\\frac{1}{3}\\)"
] | [
"What is the probability of seeing at least 2 bus A while waiting for bus B, given that bus A runs every 10 minutes and bus B runs every 15 minutes?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-01eabb503ef666708d41159f1fed1d3189641eb90ac2f472d64f06da0df9bbc6 | What is the probability of seeing each number at least once after $n$ rolls of a die? | [
"\\( 1 - 6\\left(\\frac{5}{6}\\right)^n + 15\\left(\\frac{4}{6}\\right)^n - 20\\left(\\frac{3}{6}\\right)^n + 15\\left(\\frac{2}{6}\\right)^n - 6\\left(\\frac{1}{6}\\right)^n + \\left(\\frac{0}{6}\\right)^n \\)"
] | [
"What is the probability of seeing each number at least once after $n$ rolls of a die?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 23
} | 44 | 0 | 0 |
omr-ws-9b620c4059de68b86c7a30d8c35e5b0dae52a1407425eecd334d420adc0ccf63 | What is the probability of selecting 1 green chip and 1 red chip without replacement from a basket containing 10 green chips and 4 red chips? | [
"\\(\\frac{40}{91}\\)"
] | [
"What is the probability of selecting 1 green chip and 1 red chip without replacement from a basket containing 10 green chips and 4 red chips?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-70c4ec97081b37359bb6f761ce96ee272866e5e9da3ae18d6defe6ab7dcd10e5 | What is the probability of selecting 2 green chips without replacement from a basket containing 10 green chips and 4 red chips? | [
"\\(\\frac{45}{91}\\)"
] | [
"What is the probability of selecting 2 green chips without replacement from a basket containing 10 green chips and 4 red chips?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-61cbb82a6ef8b337d6b1aada77bd302f2bf8dbb8a794d4432ace513e78641703 | What is the probability of selecting 3 cards of one type and 2 cards of another type when 5 cards are randomly selected from a standard deck of 52 cards? | [
"\\(\\frac{6}{4165}\\)"
] | [
"What is the probability of selecting 3 cards of one type and 2 cards of another type when 5 cards are randomly selected from a standard deck of 52 cards?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-5a7bb834294684fc509580c046ca3374d6604e52fcce3c9bd235f6ecd103bf3e | What is the probability of selecting 4 E's when selecting 4 tiles from a set of 100 tiles, given that there are 12 E's among the tiles? | [
"\\(\\frac{3}{23765}\\)"
] | [
"What is the probability of selecting 4 E's when selecting 4 tiles from a set of 100 tiles, given that there are 12 E's among the tiles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 27
} | 48 | 0 | 0 |
omr-ws-c135aec742a1cbeaf8492b4e80aa8c538867b895df1aa0e5866c315720f576e8 | What is the probability of selecting a jack and a heart when drawing 2 cards from a 52-card deck? | [
"\\(\\frac{29}{442}\\)"
] | [
"What is the probability of selecting a jack and a heart when drawing 2 cards from a 52-card deck?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-83d45e55d5d328ff236f1cb8565ab5b54ba2105712b1dfc247e6277df8a9b306 | What is the probability of selecting a king, queen, and jack of the same suit when 3 cards are drawn from a standard deck of 52 playing cards? | [
"\\(\\frac{1}{5525}\\)"
] | [
"What is the probability of selecting a king, queen, and jack of the same suit when 3 cards are drawn from a standard deck of 52 playing cards?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-3fea4291a94cacc0515d5fbebc1f53ef7ddd611bedfd0316d9aadd439cc2ca22 | What is the probability of selecting a vowel from the set {M, A, T, H, C, O, U, N, T, S}? Express your answer as a common fraction. | [
"\\(\\frac{3}{10}\\)"
] | [
"What is the probability of selecting a vowel from the set {M, A, T, H, C, O, U, N, T, S}? Express your answer as a common fraction."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 6
} | 20 | 0 | 0 |
omr-ws-74699f01b8e74d3793ac552cc8981a21aeac5878c1445b5090bd17b4b648cb27 | What is the probability of selecting three random points on each side of an equilateral triangle such that they form an acute triangle? | [
"\\dfrac{1}{4}"
] | [
"What is the probability of selecting three random points on each side of an equilateral triangle such that they form an acute triangle?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9eefad6b64b0ca7393968b2bc609b96a5a23913428efd015ec2b14ba6154616b | What is the probability of selecting two diagonals in a convex polygon with $n$ sides that intersect each other? | [
"\\(\\frac{(n-1)(n-2)}{3(n^2 - 3n - 2)}\\)"
] | [
"What is the probability of selecting two diagonals in a convex polygon with $n$ sides that intersect each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-d138ba5311ead6618358be9ffd2109411b90a8fdbec7c4b2847f80ae49d3d888 | What is the probability of the letters AABBCD appearing in that exact order when randomly drawing from a set of 6 letters (AABBCD)? | [
"\\(\\frac{1}{180}\\)"
] | [
"What is the probability of the letters AABBCD appearing in that exact order when randomly drawing from a set of 6 letters (AABBCD)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-552dc9bb3c76196f045841aefd8e174866fb76034b508bf5d702e689d967192f | What is the probability of the world series going to seven games, assuming that the two teams are evenly matched? (The first team that wins four games wins the series.) | [
"\\(\\frac{5}{16}\\)"
] | [
"What is the probability of the world series going to seven games, assuming that the two teams are evenly matched? (The first team that wins four games wins the series.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.