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-f336925f79f0dd7b4a8d84fa8dcd6d89d5670753ac660bfae6756ac8811f5ef6 | What is the probability that exactly 3 out of 5 cards drawn simultaneously from a standard deck of 52 cards are queens? | [
"\\(\\frac{4512}{2598960}\\) or approximately \\(0.001739\\)"
] | [
"What is the probability that exactly 3 out of 5 cards drawn simultaneously from a standard deck of 52 cards are queens?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
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"cot": 16,
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omr-ws-f9d812ee122053352f6a130b2d50c814dbac55967323055c550e1d278afa35d4 | What is the probability that exactly 3 persons will be seated on the same seats as the initial seats when 6 people $A, B, C, D, E, F$ exchange their seats, with any arrangement being equally probable? | [
"\\(\\frac{1}{18}\\)"
] | [
"What is the probability that exactly 3 persons will be seated on the same seats as the initial seats when 6 people $A, B, C, D, E, F$ exchange their seats, with any arrangement being equally probable?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
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} | 46 | 0 | 0 |
omr-ws-e7f8fc1b96569e759c2a5f94776b74e338db8200519b8998f1ee0904a145b379 | What is the probability that exactly 4 out of 10 people select their own names from a box containing cards with each person's name? | [
"\\(\\frac{5}{144}\\)"
] | [
"What is the probability that exactly 4 out of 10 people select their own names from a box containing cards with each person's name?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bb17f2c75338c1b21c2283e48fe71d3eb494ba7ae398e0c5454b02d844a74e69 | What is the probability that exactly 4 out of 9 specific friends will be on the same team as you, given there are 5 possible teams to be on? | [
"\\(\\frac{4^5}{5^9}\\)"
] | [
"What is the probability that exactly 4 out of 9 specific friends will be on the same team as you, given there are 5 possible teams to be on?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9ba22d304ac7ea7839129682a2264d1f24c64adab4dd3258bc30a40b085e762a | What is the probability that exactly 4 people out of 10 are born in the same month? | [
"\\dfrac{30701825}{429981696}"
] | [
"What is the probability that exactly 4 people out of 10 are born in the same month?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-7dac110fafe05dad98767a4af4b4878448c8c11a3b755ad27c55555b6b4ccdb3 | What is the probability that exactly one cell is empty if ten identical balls are distributed randomly into five distinct cells? | [
"\\(\\frac{60}{143}\\)"
] | [
"What is the probability that exactly one cell is empty if ten identical balls are distributed randomly into five distinct cells?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-9e4eb0e0cd222d0d7e95f1348b6689bbc9ef4a9b0086f6c2b939fca2ed056580 | What is the probability that exactly one of Alex or Bernie finishes their service within 15 minutes, given that the time to get a shave is exponentially distributed with parameter $\lambda=4$ per hour and the time to get a haircut is exponentially distributed with parameter $\mu=3$ per hour? | [
"\\( e^{-0.75} + e^{-1} - 2e^{-1.75} \\)"
] | [
"What is the probability that exactly one of Alex or Bernie finishes their service within 15 minutes, given that the time to get a shave is exponentially distributed with parameter $\\lambda=4$ per hour and the time to get a haircut is exponentially distributed with parameter $\\mu=3$ per hour?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
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] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-512034f8fcf0bdbf1e6d8a67ab42fe2d1098972b428c9456214c8dfb558eaaa2 | What is the probability that exactly one of the three randomly chosen balls is red, given that there are 40 balls in total: 10 black, 10 white, 10 red, and 10 blue? | [
"\\(\\frac{435}{988}\\)"
] | [
"What is the probability that exactly one of the three randomly chosen balls is red, given that there are 40 balls in total: 10 black, 10 white, 10 red, and 10 blue?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-4b48a96f63f8aa5a24aeab12bd043cdc87b9b8271f163d6da96fdf2585e86eb9 | What is the probability that exactly one out of four distinguishable kids is born, given each has a $\frac{1}{2}$ chance of being born? | [
"$\\frac{1}{4}$"
] | [
"What is the probability that exactly one out of four distinguishable kids is born, given each has a $\\frac{1}{2}$ chance of being born?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
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] | {
"cot": 16,
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} | 37 | 0 | 0 |
omr-ws-010654d11f5b300f1ac9e69bd9b744f29e0bd1c2e5dec52c24960ced5f253261 | What is the probability that exactly two out of 14 people share the same birthday? | [
"\\(\\frac{\\binom{14}{2}\\cdot\\frac{365!}{352!}}{365^{14}}\\)"
] | [
"What is the probability that exactly two out of 14 people share the same birthday?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 18
} | 20 | 0 | 0 |
omr-ws-270a3e09ae3711775589440eda1221ce8df33398d499f3a07b964694f0d64eb8 | What is the probability that exactly two out of three cards dealt from a standard 52-card deck are face cards? | [
"\\(\\frac{132}{1105}\\)"
] | [
"What is the probability that exactly two out of three cards dealt from a standard 52-card deck are face cards?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-78a44b700e0abe8633207e35211be89f75ebdb07cdaa07231c794f930587b416 | What is the probability that exactly two out of three people have their birthdays in the same month? | [
"11/48"
] | [
"What is the probability that exactly two out of three people have their birthdays in the same month?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-4b6ac6a4a8b7e5aa5a15fea4f0967267b75bf5e7b9b16cb59e239d1144472840 | What is the probability that fewer than half of six tossed coins turn up heads? | [
"$\\dfrac{11}{32}$"
] | [
"What is the probability that fewer than half of six tossed coins turn up heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-c6d6ab7731880a0851ff312efe75536e66324a078550a47b11edaafc37e7161a | What is the probability that five random points selected on a circle form a convex pentagon? | [
"1"
] | [
"What is the probability that five random points selected on a circle form a convex pentagon?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-2682daa9802afb1e85f358f7058373909a24078256bdc86b25bb0c363d61c51f | What is the probability that for a 3 or 4 digit number, its reverse (drome) is greater than the number by at least 30 and not more than 250? | [
"\\(\\frac{19}{660}\\)"
] | [
"What is the probability that for a 3 or 4 digit number, its reverse (drome) is greater than the number by at least 30 and not more than 250?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-16eada081ae3d091ccdc6ea21dc37994bbd664c25108952732d7dfa2ab3c9218 | What is the probability that for random real numbers \(a\), \(b\), and \(c\), the inequality \((ab)^3 + (ac)^3 + (bc)^3 \ge abc(a^3 + b^3 + c^3)\) holds? | [
"\\dfrac{1}{2}"
] | [
"What is the probability that for random real numbers \\(a\\), \\(b\\), and \\(c\\), the inequality \\((ab)^3 + (ac)^3 + (bc)^3 \\ge abc(a^3 + b^3 + c^3)\\) holds?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a1ae64655bffbebf38a4abc1e68ec4d8ac04e8a81f6e8581e67c0ed45f08dea5 | What is the probability that four S's come together in the word "MISSISSIPPI"? | [
"\\(\\frac{4}{165}\\)"
] | [
"What is the probability that four S's come together in the word \"MISSISSIPPI\"?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-176e40a969e18002ff96154f631601ba909b50152842b00b277a6fdd58cbf9ee | What is the probability that four fair, six-sided dice sum to 18? | [
"\\(\\frac{5}{81}\\)"
] | [
"What is the probability that four fair, six-sided dice sum to 18?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-7893c8975c4f751373263613a7cfaf3928408c7694b3e33a143a444315e337df | What is the probability that four parts obtained by breaking a wooden stick at three random points can form the sides of a quadrilateral? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the probability that four parts obtained by breaking a wooden stick at three random points can form the sides of a quadrilateral?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 17
} | 30 | 0 | 0 |
omr-ws-e685d5a4ed67c5b465c18f9c01c0222833e2a51e180b0d807603267b7c81d13f | What is the probability that four randomly selected points on a sphere lie on the same hemisphere? | [
"1"
] | [
"What is the probability that four randomly selected points on a sphere lie on the same hemisphere?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d2477e8573b201985b6e775c6ab8c21695faa42e036c5650321b3a48ca3b8719 | What is the probability that heads does not occur three times in a row in 21 tosses? | [
"\\dfrac{51343}{262144}"
] | [
"What is the probability that heads does not occur three times in a row in 21 tosses?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5732cff0b490b3de1d11f29e03be0cf9c01a4525e65806d6ce57740bb3645733 | What is the probability that heads does not occur three times in a row when a coin is tossed 21 times? | [
"\\dfrac{51343}{262144}"
] | [
"What is the probability that heads does not occur three times in a row when a coin is tossed 21 times?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-13b244ba67736c9f055915da649ed8eb0910eb4ed5bb7ea9dde60521023f3e61 | What is the probability that if 4 people choose integers (which may be repeated) between 1 and 50 inclusive, the sum of the 4 numbers is greater than 50? | [
"\\(\\frac{60197}{62500}\\)"
] | [
"What is the probability that if 4 people choose integers (which may be repeated) between 1 and 50 inclusive, the sum of the 4 numbers is greater than 50?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-a4494a92656f755fb185242684330a2a6a930e463132c798ab9ce5e3a734ef62 | What is the probability that if a pair of dice is rolled twice, the total product of all four dice has exactly 12 positive factors? | [
"17/81",
"\\dfrac{65}{324}"
] | [
"What is the probability that if a pair of dice is rolled twice, the total product of all four dice has exactly 12 positive factors?"
] | [
"aops_c4_high_school_math",
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] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 9,
"tir": 21
} | 43 | 0 | 0 |
omr-ws-171b86fcfd77ac25e9e38cc00a7f5f6267b69f0f5916370fb9b3c531dade0be3 | What is the probability that if person A does not start the rumor, person A receives the third call, given that a person can pass the rumor on to anyone except the individual who just called? | [
"\\(\\frac{1}{8}\\)"
] | [
"What is the probability that if person A does not start the rumor, person A receives the third call, given that a person can pass the rumor on to anyone except the individual who just called?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-420f5e6d8bd9fbade2cc1494a7c5ee0ee3512e6c782d07f710234349fa00b9cc | What is the probability that if person A starts the rumor, person A receives the third call, given that a person can pass the rumor on to anyone except the individual who just called? | [
"\\(\\frac{1}{8}\\)"
] | [
"What is the probability that if person A starts the rumor, person A receives the third call, given that a person can pass the rumor on to anyone except the individual who just called?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c0577f7f3bf16d512e6c81b4fee5d1494ad3c172657473946c36edec27e23d3b | What is the probability that if the ski club elects 3 officers from 8 women and 7 men, all 3 of them will be women? | [
"\\(\\frac{8}{65}\\)"
] | [
"What is the probability that if the ski club elects 3 officers from 8 women and 7 men, all 3 of them will be women?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-afb03d4438f1010d0bdcbe9e7bb1179bca027cecdc2bedc027cd29a0cb5bb54f | What is the probability that if three points are chosen at random on the circumference of a circle, the triangle formed by connecting the three points does not have a side with length greater than the radius of the circle? | [
"\\(\\frac{1}{12}\\)"
] | [
"What is the probability that if three points are chosen at random on the circumference of a circle, the triangle formed by connecting the three points does not have a side with length greater than the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 3
} | 3 | 0 | 0 |
omr-ws-11115aa6e3077fedfbf3a34bbc474dbe14ef64dd2ffb9f0c8ad6a44af1e6f82f | What is the probability that if three points are chosen at random on the circumference of a circle, the triangle formed by connecting these points does not have a side with length greater than the radius of the circle? | [
"\\(\\frac{1}{12}\\)"
] | [
"What is the probability that if three points are chosen at random on the circumference of a circle, the triangle formed by connecting these points does not have a side with length greater than the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-b94176ebe98568d69ddfb53e9659dc3c0e2143d740ad5b6cd1cf848afa2b3a10 | What is the probability that if three points are chosen at random on the circumference of a circle, the triangle formed by connecting these three points does not have a side with a length greater than the radius of the circle? | [
"\\(\\frac{1}{12}\\)"
] | [
"What is the probability that if three points are chosen at random on the circumference of a circle, the triangle formed by connecting these three points does not have a side with a length greater than the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-cddda06869ce80521b25dfc78518b505a4e851a1d591e2a64c4e53b6b5c6e9ff | What is the probability that if three points are chosen at random on the circumference of a circle, the triangle formed by connecting these three points does not have a side with length greater than the radius of the circle? | [
"\\(\\frac{1}{12}\\)"
] | [
"What is the probability that if three points are chosen at random on the circumference of a circle, the triangle formed by connecting these three points does not have a side with length greater than the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-c8ac4131a24c6e22979bb36914ab09a6d34a2844458ae0ce164e30c73361a298 | What is the probability that if three points are chosen at random on the circumference of a circle, then the triangle formed by connecting the three points does not have a side length greater than the radius of the circle? | [
"\\dfrac{1}{12}"
] | [
"What is the probability that if three points are chosen at random on the circumference of a circle, then the triangle formed by connecting the three points does not have a side length greater than the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-499770cd9b119213708ffbac35f19a023714afb6e011854a2e5ea6466f829ef6 | What is the probability that if three points are chosen at random on the circumference of a circle, then the triangle formed by connecting the three points does not have a side with length greater than the radius of the circle? | [
"\\(\\frac{1}{12}\\)"
] | [
"What is the probability that if three points are chosen at random on the circumference of a circle, then the triangle formed by connecting the three points does not have a side with length greater than the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-ba17c5563ae1e82bff2f4ea2e3c7efd06a30c907888f40374a4859ae5dee72e3 | What is the probability that if two cards are selected at random from a deck of 52 cards, one will be a jack and one will be a red nine? | [
"\\(\\frac{4}{663}\\)"
] | [
"What is the probability that if two cards are selected at random from a deck of 52 cards, one will be a jack and one will be a red nine?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-9f31ba23ea42f91f1f7b9f63e3d1ae515b8707e3d67cdaa57f2e91dfdb781b72 | What is the probability that in $n$ coin flips, tails will not show $k < n$ times in a row? | [
"\\frac{a(n)}{2^n}"
] | [
"What is the probability that in $n$ coin flips, tails will not show $k < n$ times in a row?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-c6ce25bf23b8ebf85acf246626abe3dc03893a4fd874ba145c81e51e01c5966b | What is the probability that in 10 flips of a coin, there is at least one series of 3 consecutive heads? | [
"\\dfrac{65}{128}"
] | [
"What is the probability that in 10 flips of a coin, there is at least one series of 3 consecutive heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ba9588aa3817cc59e0014a8dd77c2d39c8391315efb76ecffeb342f3b4db13ba | What is the probability that in 100 coin tosses, a sequence of 4 heads in a row ("HHHH") appears? | [
"1 - \\frac{a_{100}}{2^{100}}"
] | [
"What is the probability that in 100 coin tosses, a sequence of 4 heads in a row (\"HHHH\") appears?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-1eebc3c25609b99537fd98cd6b582f9d2229a94f8165d96235c9374da13c7fe8 | What is the probability that in 1000 spins there is no 17 consecutive spins that are losses? | [
"0.9925"
] | [
"What is the probability that in 1000 spins there is no 17 consecutive spins that are losses?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-8bdb4d92d61abe9c9136b2a0b2f11bae7e02c780ae88b64f2c6909a198972423 | What is the probability that in a family of 7 people, no person's birthday is the arithmetic mean of any two other members' birthdays? Assume there are 365 days in a year, and if the arithmetic mean of two birthdays is an odd number, take the ceiling of that number. | [
"\\dfrac{364 \\times 362 \\times 359 \\times 355 \\times 350}{365^5}"
] | [
"What is the probability that in a family of 7 people, no person's birthday is the arithmetic mean of any two other members' birthdays? Assume there are 365 days in a year, and if the arithmetic mean of two birthdays is an odd number, take the ceiling of that number."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-827a32d60cadd967f1c7159f2d977d8f9fbe587702e33058335f2778090f796b | What is the probability that in a group of k people, there are at least two persons who have birthdays on the same date or on consecutive dates (considering January 1 is consecutive to December 31)? | [
"1 - \\frac{\\left( \\binom{365 - k}{k} + \\binom{365 - k - 1}{k - 1} \\right) k!}{365^k}"
] | [
"What is the probability that in a group of k people, there are at least two persons who have birthdays on the same date or on consecutive dates (considering January 1 is consecutive to December 31)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-06d7e5f1bc7b002ac4e830cf4b901d197f7197564c99a427087881b4c8983be7 | What is the probability that in a random permutation of the numbers from $1$ to $7$, the image of $1$ is not $2$, the image of $2$ is not $3$, the image of $3$ is not $4$, the image of $4$ is not $5$, the image of $5$ is not $6$, and the image of $6$ is always $1$? | [
"\\(\\frac{103}{1680}\\)"
] | [
"What is the probability that in a random permutation of the numbers from $1$ to $7$, the image of $1$ is not $2$, the image of $2$ is not $3$, the image of $3$ is not $4$, the image of $4$ is not $5$, the image of $5$ is not $6$, and the image of $6$ is always $1$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 3,
"tir": 20
} | 31 | 0 | 0 |
omr-ws-29d2dc1d634b5eaf7dcadf343ca3638b8f4c5f6dfb84fc8480d0860c32616d69 | What is the probability that in a sequence of 16 coin flips, where each flip has an equal probability of landing heads or tails, there will never be 2 consecutive heads or 3 consecutive tails? | [
"\\(\\frac{151}{65536}\\)"
] | [
"What is the probability that in a sequence of 16 coin flips, where each flip has an equal probability of landing heads or tails, there will never be 2 consecutive heads or 3 consecutive tails?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-6c6a6926ebfb4dddf8a6e7540c5dfc43a68ac16f4bbdec0076887cd6370db1aa | What is the probability that in a shuffled 52-card deck, no two adjacent cards have the same value? | [
"0"
] | [
"What is the probability that in a shuffled 52-card deck, no two adjacent cards have the same value?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-47aed2642cb2cc8e95a54c359b437e54de209377b7694ec7376202cd5f7f30e3 | What is the probability that in eight coin flips, there are no three consecutive heads? | [
"\\(\\frac{149}{256}\\)"
] | [
"What is the probability that in eight coin flips, there are no three consecutive heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-341a0917e480a42ca92a80c0011d3ca8519a2d048806d507e2b80956a63f28ae | What is the probability that in five rolls of a six-sided die (with two sides having a 5 and no 6), each number from 1 to 5 comes up exactly once, in any order? | [
"\\(\\frac{5}{162}\\)"
] | [
"What is the probability that in five rolls of a six-sided die (with two sides having a 5 and no 6), each number from 1 to 5 comes up exactly once, in any order?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-ab925f635bbfe7c710f13c90b5014fdf3659b0f64178816a2a97923ebdb39f77 | What is the probability that in flipping coins and writing down the results, the sequence HH is seen before the sequence TH? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the probability that in flipping coins and writing down the results, the sequence HH is seen before the sequence TH?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a601e197d1f946a390868f82b4dde84d7a709fd5d4e551479a190c11f7e7ed1a | What is the probability that in k throws of a die, each of the numbers 1 through 6 will appear at least once? | [
"\\(\\frac{1}{6^k}\\sum_{j=0}^{5}(-1)^j\\binom{6}{j}(6-j)^k\\)"
] | [
"What is the probability that in k throws of a die, each of the numbers 1 through 6 will appear at least once?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 20
} | 34 | 0 | 0 |
omr-ws-162f9d1bae2c802fd8b1ff358024bf8047048b4b5e7c7d70c21eeea68e0427e7 | What is the probability that in n flips of a coin, there is at least one series of 3 consecutive heads? | [
"1 - \\frac{a_n}{2^n}"
] | [
"What is the probability that in n flips of a coin, there is at least one series of 3 consecutive heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9e9e6c9628def75d7963107f702027903c23ac4d85f43d95073b57075b074f10 | What is the probability that in ten coin flips, there are no instances of three consecutive heads? | [
"\\(\\frac{63}{128}\\)"
] | [
"What is the probability that in ten coin flips, there are no instances of three consecutive heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-f689185c2c3847685c2e88d36da3352dc6ed5dbc8ec8820dfb6ea353d70b1f95 | What is the probability that it takes exactly \( n \) rolls to mark down each number from 1 to 6? | [
"\\dfrac{6! \\cdot S(n-1, 5)}{6^n}"
] | [
"What is the probability that it takes exactly \\( n \\) rolls to mark down each number from 1 to 6?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-5de8bf01bb8946278afab4d2bb88c1826445b76b75677e2a2499125878486aab | What is the probability that it takes exactly six rolls for all six faces of a die to show up? | [
"\\(\\frac{5}{324}\\)"
] | [
"What is the probability that it takes exactly six rolls for all six faces of a die to show up?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-2a6fe4b71d1eff8aeeabd2950ed7fe2358b9134aa38cf2a73442a4d2223eb51e | What is the probability that it takes exactly three throws of a die for the sum of all points rolled to exceed 9? Answer in the form of an ordinary fraction. | [
"\\(\\frac{35}{216}\\)"
] | [
"What is the probability that it takes exactly three throws of a die for the sum of all points rolled to exceed 9? Answer in the form of an ordinary fraction."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-16432fa5999619f581be0d4962a00310f4154f06de901bd213f35d412dc9f3f8 | What is the probability that n points on a sphere lie in the same hemisphere? | [
"\\dfrac{n^2 - n + 2}{2^{n}}"
] | [
"What is the probability that n points on a sphere lie in the same hemisphere?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-54699829e2269427c85b8ffef9a3a3b87884c6f587d0fbc7ec237227cc499661 | What is the probability that no ant will encounter another, either en route or at the next vertex, when ants at each vertex of a cube simultaneously crawl along an edge to the next vertex, each choosing its path randomly? (Express your answer as a reduced fraction.) | [
"\\dfrac{4}{243}"
] | [
"What is the probability that no ant will encounter another, either en route or at the next vertex, when ants at each vertex of a cube simultaneously crawl along an edge to the next vertex, each choosing its path randomly? (Express your answer as a reduced fraction.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-4f17586841eae6ada88cb4efba68b48723df1b6de9a6853cbccc7cdb30c7620d | What is the probability that no ant will encounter another, either en route or at the next vertex, when ants at each vertex of a dodecahedron simultaneously crawl along an edge to the next vertex, each choosing its path randomly? (Express your answer as a reduced fraction.) | [
"\\dfrac{2^{20}}{3^{20}}"
] | [
"What is the probability that no ant will encounter another, either en route or at the next vertex, when ants at each vertex of a dodecahedron simultaneously crawl along an edge to the next vertex, each choosing its path randomly? (Express your answer as a reduced fraction.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-aca10d38361435b8e3acec5133e20e9b165f6d0bed583bab8b2c487feda4b8c2 | What is the probability that no ant will encounter another, either en route or at the next vertex, when ants at each vertex of a tetrahedron simultaneously crawl along an edge to the next vertex, each choosing its path randomly? (Express your answer as a reduced fraction.) | [
"\\dfrac{2}{27}"
] | [
"What is the probability that no ant will encounter another, either en route or at the next vertex, when ants at each vertex of a tetrahedron simultaneously crawl along an edge to the next vertex, each choosing its path randomly? (Express your answer as a reduced fraction.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-571b7231236ffeb9d1ceeb9ef5fe10af374581be9dd7ac2406ef129bb20deb80 | What is the probability that no ant will encounter another, either en route or at the next vertex, when ants at each vertex of an icosahedron simultaneously crawl along an edge to the next vertex, each choosing its path randomly? (Express your answer as a reduced fraction.) | [
"\\dfrac{2}{244140625}"
] | [
"What is the probability that no ant will encounter another, either en route or at the next vertex, when ants at each vertex of an icosahedron simultaneously crawl along an edge to the next vertex, each choosing its path randomly? (Express your answer as a reduced fraction.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-dc84ff0ac5888fec81527b111ab4559fd2c50e42e162686a0de0a373c4e36b09 | What is the probability that no ant will encounter another, either en route or at the next vertex, when ants at each vertex of an octahedron simultaneously crawl along an edge to the next vertex, each choosing its path randomly? (Express your answer as a reduced fraction.) | [
"\\dfrac{3}{1024}"
] | [
"What is the probability that no ant will encounter another, either en route or at the next vertex, when ants at each vertex of an octahedron simultaneously crawl along an edge to the next vertex, each choosing its path randomly? (Express your answer as a reduced fraction.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-e87528dec38ffb3d56aa688115d8df516a53ef817af22139f088bcddf8acc3f3 | What is the probability that no boys are sitting next to each other when 5 boys and 5 girls are sitting at a table? | [
"\\(\\frac{1}{126}\\)"
] | [
"What is the probability that no boys are sitting next to each other when 5 boys and 5 girls are sitting at a table?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-0dc755f2aab2d9e146d133366b58a5239561dd12b6e290c5be06f38a294c58d0 | What is the probability that no husband sits next to his wife when 4 married couples are arranged in a row? | [
"\\dfrac{12}{35}"
] | [
"What is the probability that no husband sits next to his wife when 4 married couples are arranged in a row?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-648a5a4f956a5aef1d42884d9c4f9e89d48ff51d1815efb767bff97cfb00a65d | What is the probability that no husband-wife pair sits next to each other when 15 husband-wife pairs (30 people) are seated at a circular table? | [
"\\(1 + \\sum_{k=1}^{15} (-1)^k \\binom {15} k \\frac{(30-k-1)! 2^{k}}{29!}\\)"
] | [
"What is the probability that no husband-wife pair sits next to each other when 15 husband-wife pairs (30 people) are seated at a circular table?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-1a05a250b8b0ac15bbd9661f5cf746f460a5a4c25b05925b52392e8dbf450860 | What is the probability that no more than 4 radios need to be tested to locate both of the defective radios? | [
"\\(\\frac{2}{7}\\)"
] | [
"What is the probability that no more than 4 radios need to be tested to locate both of the defective radios?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-82120e10241b510eb1f3090a8a08a6debb67a5cef590e26d960ad554068079ec | What is the probability that no one sits next to his/her spouse when 3 married couples are randomly seated around a round table? | [
"\\dfrac{4}{15}"
] | [
"What is the probability that no one sits next to his/her spouse when 3 married couples are randomly seated around a round table?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-cfde99fbb0217f4b838d38309caead7aa8021439e11bd3dfd0de13aa449e6f34 | What is the probability that no one will be seated on the same seats as the initial seats when 6 people $A, B, C, D, E, F$ exchange their seats, with any arrangement being equally probable? | [
"\\(\\frac{53}{144}\\)"
] | [
"What is the probability that no one will be seated on the same seats as the initial seats when 6 people $A, B, C, D, E, F$ exchange their seats, with any arrangement being equally probable?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-41ffe490fa774e07ae90aeef2c4f8d5d6a0691825e58fd618af4c7561e57538a | What is the probability that no person gets their own hat back when N people each randomly pick a hat from a collection of N different hats? | [
"\\(\\frac{1}{e}\\)"
] | [
"What is the probability that no person gets their own hat back when N people each randomly pick a hat from a collection of N different hats?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 5
} | 6 | 0 | 0 |
omr-ws-7d111bfa5af27daadc8e9cd20abeba16c699e7c9050fe4732d43a5b4741b866f | What is the probability that no three people from the same team are next to each other when 10 people from team red and 10 people from team blue are combined and randomly shuffled in a row? | [
"\\dfrac{2049}{46189}"
] | [
"What is the probability that no three people from the same team are next to each other when 10 people from team red and 10 people from team blue are combined and randomly shuffled in a row?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-43255a6cf4c1fb453a4dd5ee7f8e42268d418588ac5192da2cde60db2ee65f6f | What is the probability that no two beads of the same color are adjacent when arranging 3 red, 2 white, and 1 blue bead in a straight line? | [
"\\(\\frac{1}{6}\\)"
] | [
"What is the probability that no two beads of the same color are adjacent when arranging 3 red, 2 white, and 1 blue bead in a straight line?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 5,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-44d2397839e537e70493a94c728bd62b0ba9654d9e46bfb77dc1fb7e23620360 | What is the probability that no two consecutive cards in a shuffled deck of 52 cards (13 types, 4 suits each) are of the same type? | [
"$\\frac{2}{13 \\cdot 17 \\cdot 125}$"
] | [
"What is the probability that no two consecutive cards in a shuffled deck of 52 cards (13 types, 4 suits each) are of the same type?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0363dc50b23b5530fc55999493174b2734934c7485da4468f977ac36ab633efa | What is the probability that no two consecutive green faces occur in the three throws when a fair coin is flipped to decide between two dice: die A (4 red faces, 2 green faces) and die B (2 red faces, 4 green faces). | [
"\\(\\frac{5}{8}\\)"
] | [
"What is the probability that no two consecutive green faces occur in the three throws when a fair coin is flipped to decide between two dice: die A (4 red faces, 2 green faces) and die B (2 red faces, 4 green faces)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-12801efb4648d55f5c58ee1eb8f9e8742a5cf96aac38b7b473884d0dec780910 | What is the probability that no two of five chords with distinct endpoints drawn on a circle intersect in the circle's interior? | [
"\\(\\frac{2}{45}\\)"
] | [
"What is the probability that no two of five chords with distinct endpoints drawn on a circle intersect in the circle's interior?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-a22d3044c4bb012f574cdba250b46eb5abfd9a50ab20ba8423d05b439f15763d | What is the probability that no two out of 5 people sitting randomly in 20 chairs arranged in a circle are next to each other? | [
"\\(\\frac{1001}{3876}\\)"
] | [
"What is the probability that no two out of 5 people sitting randomly in 20 chairs arranged in a circle are next to each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-aae251ef682e94f42a2de55e6fee51aab02611c0e82f385eb9a8619439e94e79 | What is the probability that no woman gets her own card back when 10 women are seated at a restaurant and the waiter returns their credit cards in a random order? | [
"\\(\\frac{16481}{44800}\\)"
] | [
"What is the probability that no woman gets her own card back when 10 women are seated at a restaurant and the waiter returns their credit cards in a random order?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 28
} | 45 | 0 | 0 |
omr-ws-b7aac82bb92afc5d2f4e850fcedbf08b7f2549e6ea1d05f59a520c13b09e2032 | What is the probability that none of the $n$ mails will reach its correct destination when $n$ mails are randomly placed into $n$ envelopes? | [
"\\( \\frac{1}{e} \\)"
] | [
"What is the probability that none of the $n$ mails will reach its correct destination when $n$ mails are randomly placed into $n$ envelopes?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 9
} | 21 | 0 | 0 |
omr-ws-259d815404e05649b455bea33d6486907104f70194ed6604c5cb2ca94f0d8c48 | What is the probability that none of the 10 people select their own names from a box containing cards with each person's name? | [
"\\(\\frac{1}{e}\\)"
] | [
"What is the probability that none of the 10 people select their own names from a box containing cards with each person's name?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-c9e387a297c7266d0ee0627a291eed0e2cd51e29bb8b84e76453f23de8685d9d | What is the probability that none of the 3 randomly chosen knights from a round table of 25 knights are sitting next to each other? | [
"\\dfrac{35}{46}"
] | [
"What is the probability that none of the 3 randomly chosen knights from a round table of 25 knights are sitting next to each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d5f0d4cfbef4566ecfd7e6d3905fbcc21cba9bba398e003e9ed5e3d061f374fa | What is the probability that none of the four vertices selected uniformly at random from the twelve vertices of a regular dodecagon are consecutive? | [
"\\(\\frac{7}{33}\\)"
] | [
"What is the probability that none of the four vertices selected uniformly at random from the twelve vertices of a regular dodecagon are consecutive?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-c8ed8188820ea5d7286c38b123fcd354b2c3f933e179133cd936ef1b3abb1a29 | What is the probability that on 6 consecutive school days, there were 2 seniors, 2 juniors, 1 sophomore, and 1 freshman selected to read the announcements, given that there are 7 seniors, 9 juniors, 10 sophomores, and 6 freshmen on the math team? Express the probability as a decimal rounded to 4 significant figures. | [
"0.0399"
] | [
"What is the probability that on 6 consecutive school days, there were 2 seniors, 2 juniors, 1 sophomore, and 1 freshman selected to read the announcements, given that there are 7 seniors, 9 juniors, 10 sophomores, and 6 freshmen on the math team? Express the probability as a decimal rounded to 4 significant figure... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 2,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-dd26279758c1f579badc8d0d86c5d832ddef7d84a81cc4d8903fcc5ed40183d6 | What is the probability that one can travel from any town to any other town, given that the probability of a road existing between any two towns is 1/2? (Note: The two questions are identical, so they are restated as one problem.) | [
"$\\frac{19}{32}$"
] | [
"What is the probability that one can travel from any town to any other town, given that the probability of a road existing between any two towns is 1/2? \n\n(Note: The two questions are identical, so they are restated as one problem.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 7,
"tir": 18
} | 46 | 0 | 0 |
omr-ws-3a7df1a025fe916d38a62806b1f9aa5b0695bc12dbb41b4bc63fef44c406f01a | What is the probability that one card is a heart and the other is a queen when two cards are randomly drawn from a standard 52-card deck? | [
"\\dfrac{1}{26}"
] | [
"What is the probability that one card is a heart and the other is a queen when two cards are randomly drawn from a standard 52-card deck?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6f15c723a73104ed9301a25096d1955cc3f0b3475454953f359fb277dcd32092 | What is the probability that one person receives both the ace of hearts and the three of clubs when 52 cards are dealt evenly to 4 people? | [
"4/17"
] | [
"What is the probability that one person receives both the ace of hearts and the three of clubs when 52 cards are dealt evenly to 4 people?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-7d7a24be8074e96b03639249b08033956f7d7ef635dd021153b3552b6dfd8755 | What is the probability that one randomly selected energy efficient light bulb lasts between 10 years and 16 years, given that the lifetime of the bulbs follows a Normal model with a mean of 15 years and a standard deviation of 4 years? | [
"0.4931"
] | [
"What is the probability that one randomly selected energy efficient light bulb lasts between 10 years and 16 years, given that the lifetime of the bulbs follows a Normal model with a mean of 15 years and a standard deviation of 4 years?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-975ecf135bf1a1d3b0379e3ed0fdf1cae303f0749107be2e16119d25db313af0 | What is the probability that one randomly selected energy efficient light bulb lasts less than 10 years, given that the lifetime of the bulbs follows a Normal model with a mean of 15 years and a standard deviation of 4 years? | [
"0.1056"
] | [
"What is the probability that one randomly selected energy efficient light bulb lasts less than 10 years, given that the lifetime of the bulbs follows a Normal model with a mean of 15 years and a standard deviation of 4 years?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3b1a45350211ccad6d1a7b93e2834d883188e6f37637c10454d9210b021b366d | What is the probability that only one of the 70 applicants will appear on all four lists, assuming the lists were created independently and randomly? | [
"\\( q(1)=\\left[ {70 \\choose 10} \\right]^{-3}\\sum_{j=1}^{10} j {70-j \\choose 9} \\sum_{i=j}^{10} {i \\choose j} {70-i \\choose 10-j} {10 \\choose i} { 60 \\choose 10-i} \\)"
] | [
"What is the probability that only one of the 70 applicants will appear on all four lists, assuming the lists were created independently and randomly?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4319cfb2b1b8d47a55d8ad18294db4b7bdd72ab0b24c3f2dfc76b3eb91122682 | What is the probability that out of 8 randomly selected people from 20 couples, there are exactly 2 couples? | [
"\\(\\frac{C_{20}^{2}C_{18}^{4}2^4}{C_{40}^{8}}\\)"
] | [
"What is the probability that out of 8 randomly selected people from 20 couples, there are exactly 2 couples?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 8
} | 9 | 0 | 0 |
omr-ws-0b005ba5dc01ecc23d5f10622f32ce65849b9eec2913a812623e9ee47a6669e9 | What is the probability that out of the first 20 days, the event happened at least 15 days? | [
"\\dfrac{5425}{262144}"
] | [
"What is the probability that out of the first 20 days, the event happened at least 15 days?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3c370f292dc1be72a7f8cd061fdd9bec074757a65b4ceae1cf275a9db1861762 | What is the probability that person A and person B will be on the same team when 20 students are split into 5 groups of 4 students each? | [
"\\(\\frac{3}{19}\\)"
] | [
"What is the probability that person A and person B will be on the same team when 20 students are split into 5 groups of 4 students each?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-ac65812c6aec055b24021d613d27f1930bfdadb48a9bd25c3e2f2128e9b7fc1b | What is the probability that person A will sit next to person B (including diagonally) in a 4 by 5 grid of seats? | [
"\\(\\frac{13}{95}\\)"
] | [
"What is the probability that person A will sit next to person B (including diagonally) in a 4 by 5 grid of seats?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ec846c3c28a9a46985e6fda47953fd80885188c79b5d08a1958f3bb2417568c0 | What is the probability that person A will sit next to person B (next to, in front of, or behind them) in a 4 by 5 grid of seats? | [
"\\(\\frac{8}{19}\\)"
] | [
"What is the probability that person A will sit next to person B (next to, in front of, or behind them) in a 4 by 5 grid of seats?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3da653f86d6a2447c947c5226d40bff8f128c3a30f0ac16f552ebcafbccaae85 | What is the probability that person A wins 5 games before person B wins one, given that each has a 1/24 chance of winning a game? | [
".0280640206"
] | [
"What is the probability that person A wins 5 games before person B wins one, given that each has a 1/24 chance of winning a game?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e5745d278bf1ddbeb179f5165e94055f991b6f0df5e943ccaa1b210dd4ab33a9 | What is the probability that person A wins 5 games before person B wins one, given that each has a 1/6 chance of winning a game? | [
"0.0194037913"
] | [
"What is the probability that person A wins 5 games before person B wins one, given that each has a 1/6 chance of winning a game?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-beb84b4dab3bce01eb473052aa3f906e7027234d9df48187da5fc7df571adf77 | What is the probability that player $A$ receives $0$ aces from a poker deck when $13$ cards are drawn for each of the players $A$, $B$, $C$, and $D$? | [
"\\(\\frac{{48 \\choose 13}}{{52 \\choose 13}}\\)"
] | [
"What is the probability that player $A$ receives $0$ aces from a poker deck when $13$ cards are drawn for each of the players $A$, $B$, $C$, and $D$?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 14
} | 15 | 0 | 0 |
omr-ws-a57a784b0eaae0e563bdb045dde70f149fbfdab7e382d6a30a373c33b6d2e0e5 | What is the probability that players $A$ and $B$ together receive $15$ red cards in total from a poker deck when $13$ cards are drawn for each of the players $A$, $B$, $C$, and $D$? | [
"\\(\\frac{\\binom{26}{15} \\binom{26}{11}}{\\binom{52}{26}}\\)"
] | [
"What is the probability that players $A$ and $B$ together receive $15$ red cards in total from a poker deck when $13$ cards are drawn for each of the players $A$, $B$, $C$, and $D$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 2
} | 6 | 0 | 0 |
omr-ws-a8b3bb4de4b222a2ad8cbf9a024c06183896926a1eafe99e538b5a24fddf60dd | What is the probability that selecting 3 points on a circle with radius 2010 will form a right triangle? | [
"0"
] | [
"What is the probability that selecting 3 points on a circle with radius 2010 will form a right triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9883a02a6918a5414594e5452c89526f01940700c45f3030edb9072477963109 | What is the probability that six persons have their birthdays in any 2 months of the year, assuming all months have an equal number of days? | [
"\\(\\frac{11}{864}\\)"
] | [
"What is the probability that six persons have their birthdays in any 2 months of the year, assuming all months have an equal number of days?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2605e6fa948d3b0e134e8870d87f4137599b31c6fda564f7bd96705583e818d6 | What is the probability that some number comes up at least six times when a 12-sided die is rolled 20 times? | [
"0.05559948"
] | [
"What is the probability that some number comes up at least six times when a 12-sided die is rolled 20 times?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 17
} | 19 | 0 | 0 |
omr-ws-1dc4f9f877f838719496c325b36e5559f54d7501f471674122349060ab9527e2 | What is the probability that some two girls will sit next to each other when 5 boys and 5 girls are seated at random around a 10-seat circular table? | [
"\\(\\frac{125}{126}\\)"
] | [
"What is the probability that some two girls will sit next to each other when 5 boys and 5 girls are seated at random around a 10-seat circular table?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-120beae1a0d4f78fdda5f725331b4b3acc7696acac3f4747ad541433d47c880f | What is the probability that somebody walking past the geysers will see geyser A erupt first, given that geyser A erupts every hour, geyser B erupts every 2 hours, and geyser C erupts every 3 hours? | [
"\\(\\frac{1}{3}\\)"
] | [
"What is the probability that somebody walking past the geysers will see geyser A erupt first, given that geyser A erupts every hour, geyser B erupts every 2 hours, and geyser C erupts every 3 hours?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 22
} | 29 | 0 | 0 |
omr-ws-75a8e0541f6f13896ab3155930710f382c6ce3e750204def875edebb9faff414 | What is the probability that the 3 balls drawn from a bag containing 5 red balls, 4 white balls, and 3 blue balls have 2 different colors? | [
"\\(\\frac{29}{44}\\)"
] | [
"What is the probability that the 3 balls drawn from a bag containing 5 red balls, 4 white balls, and 3 blue balls have 2 different colors?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-cc4452cb3f37f699dc6ac6fc4c12bb6ba27c847d95c18e75a6e1035a18ab043d | What is the probability that the 3D object formed by the convex hull of $n$ points chosen uniformly randomly on a sphere contains the center of the sphere? | [
"\\( 1 - \\frac{n^2 - n + 2}{2^n} \\)"
] | [
"What is the probability that the 3D object formed by the convex hull of $n$ points chosen uniformly randomly on a sphere contains the center of the sphere?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 28,
"genselect": 6,
"tir": 6
} | 40 | 0 | 0 |
omr-ws-ee5a8d73cac0d1b380790eaecaa64787476d7d943739be7a98e9042737f8f310 | What is the probability that the 4 points chosen by Keith do not contain the center $O$ when selecting 4 points out of 15 points on a circle? | [
"\\(\\frac{98}{273}\\)"
] | [
"What is the probability that the 4 points chosen by Keith do not contain the center $O$ when selecting 4 points out of 15 points on a circle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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