Exact probability of dice sums, coin-flip heads and streaks, drawing special cards from a deck and each 5-card poker hand, as a fraction and 1 in N.
Rolling 2 dice with 6 sides, the sum is exactly 7 in 6 of the 36 equally likely outcomes: a 16.67% chance.
Counts come from convolving one die's distribution N − 1 times in exact integers (each new die adds a sliding window of S counts).
Dice sums are counted exactly: one die's outcomes are convolved N − 1 times, so all Sᴺ rolls of N dice with S sides are counted rather than simulated. Heads in coin flips follow the binomial distribution, and runs of heads use Feller's recurrence for success runs. Card draws use the hypergeometric distribution, because each card leaves the deck, and poker hands are counted over all 2,598,960 five-card hands.
The default, two six-sided dice summing to 7, is the most likely total: 6 of the 36 rolls, a chance of 1/6. With three dice, a sum of at least 11 comes up exactly half the time, 108 of 216 rolls, and 3 or more heads in a row turn up in 50.8% of 10-flip sequences.
The results assume fair dice, independent flips and a well-shuffled deck; for a biased coin, enter its chance of heads. Answers are exact fractions unless the numbers grow too large.
Checked against: 6 of the 36 ordered rolls give 7 (standard two-dice table)
Checked against: Python brute force over all 216 rolls of 3d6: 108 have sum ≥ 11 (symmetry about 10.5)
Checked against: Python brute force over all 1296 rolls of 4d6
Checked against: Python brute force over all 2¹⁰ sequences: 520 contain HHH
7, which 6 of the 36 equally likely rolls produce (1+6, 2+5, 3+4 and their reverses), a probability of 1/6 or 16.67%. The extremes 2 and 12 each need one exact roll, 1/36 or 2.78%. With three dice the peak shifts to 10 and 11, each 27 of 216 rolls, or 12.5%.
1 in 1,024 with a fair coin, because each flip halves the chance: (1/2)¹⁰ = 1/1,024, about 0.098%. After 9 heads in a row, the 10th flip is still 50% heads, since flips are independent; expecting tails to be "due" is the gambler's fallacy. Any other specific sequence of 10 flips, such as HTHTHTHTHT, is exactly as unlikely.
Of the 2,598,960 five-card hands, 4 are royal flushes (1 in 649,740), 624 are four of a kind (1 in 4,165) and 3,744 are full houses (about 1 in 694). One pair turns up in 42.26% of deals, and 50.12% of hands hold nothing better than a high card. These are standard counts for the first five cards dealt, with no draws or community cards.
4/52 = 1/13, about 7.69%, for one card from a full 52-card deck. For at least one ace in a 5-card hand, subtract the chance of none: 1 − C(48, 5)/C(52, 5) = 34.12%. Cards drawn without replacement follow the hypergeometric distribution, so each ace drawn lowers the chance that the next card is an ace.
More likely than most people guess. In 10 fair flips there is a 65/128 = 50.78% chance of at least one run of 3 or more heads, because a run can start at any of the 8 positions from flip 1 to flip 8. Long real sequences therefore contain streaks, and a streak alone is weak evidence that a coin is biased.
Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 9 worked examples whose answers come from independent sources; for example, “Two dice sum to 7 (defaults)” is checked against 6 of the 36 ordered rolls give 7 (standard two-dice table).
Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. — §XIII.7 (success runs) and §II.6 (hypergeometric); Wikipedia — Poker probability: frequency of 5-card poker hands; NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.18 Binomial distribution.
9 worked examples with independently sourced answers ship with this calculator. They run in the test suite; you can run them here too.
Binomial distribution calculator: exact P(X = k), P(X ≤ k), tails and ranges for binomial, Poisson, geometric, negative binomial and hypergeometric.
Calculate nCr and nPr exactly: combinations and permutations with or without repetition, for poker hands, lottery odds and PIN codes.
Calculate P(A or B), P(A and B) and P(A | B) for two events, or apply Bayes' theorem to a test result, with a probability tree and Venn diagram.
Calculate expected value E[X], variance and standard deviation for a bet, game or decision from its outcomes and their probabilities or weights.
Convert odds between fractional, decimal and American (moneyline) formats and get the implied probability, worked in exact fractions.
Generate random numbers from a seed: integers in a range, decimals, lottery-style draws without repeats and dice rolls, repeatable with xoshiro256**.
Allow optional Google Analytics to measure page visits? Calculators work either way. Privacy and choices
Optional analytics: off.