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About this tool
Work out and/or/not and conditional probability, at-least-once odds, binomial P(X = k) with a chart, and exact dice, coin and card probabilities.
This probability calculator covers the questions people ask most, each on its own tab. Two events applies the addition and multiplication rules to P(A) and P(B) — for independent, mutually exclusive or overlapping events — and draws a Venn diagram. At least once uses 1 − (1 − p)ⁿ and tells you how many tries a 95% chance needs. Binomial computes P(X = k), P(X ≤ k) and P(X ≥ k) for up to 100,000 trials with a bar chart of the distribution. Dice, Coins and Cards count outcomes exactly with whole-number arithmetic, so answers come out as true fractions such as 1/6 or 63/256. Every result is shown as a decimal, percentage, fraction and 1-in-N, and all of it runs in your browser.
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Pick a tab — Two events, At least once, Binomial, Dice, Coins or Cards (keys 1–6 also switch).
Enter probabilities as 0.25, 25% or 1/4, or counts such as Number of dice, Flips (n) or Cards drawn (n).
Read the Results table: each probability appears as a decimal, percent, fraction and 1-in-N, and the bar chart highlights X = k, X ≤ k or X ≥ k.
Press Copy to put the whole results table on your clipboard as plain text.
Two-event rules, repeated trials, binomial, dice, coins and cards each have their own inputs, results table and chart, and the page remembers what you last entered.
Dice, coin and card answers are counted with BigInt arithmetic, so 3 dice summing to 10 shows as exactly 1/8 rather than a rounded decimal.
Type a probability as 0.25, 25%, 1/4 or “1 in 4”. Results are given in all four forms, plus odds for and against.
Bar charts show the full binomial, dice-sum, coin and card distributions, with the bars that make up your answer highlighted.
| Sum | Ways (out of 36) | Probability | Percent |
|---|---|---|---|
| 2 | 1 | 1/36 | 2.78% |
| 3 | 2 | 1/18 | 5.56% |
| 4 | 3 | 1/12 | 8.33% |
| 5 | 4 | 1/9 | 11.11% |
| 6 | 5 | 5/36 | 13.89% |
| 7 | 6 | 1/6 | 16.67% |
| 8 | 5 | 5/36 | 13.89% |
| 9 | 4 | 1/9 | 11.11% |
| 10 | 3 | 1/12 | 8.33% |
| 11 | 2 | 1/18 | 5.56% |
| 12 | 1 | 1/36 | 2.78% |
7 is the most likely total (6 ways, 1/6). Use the Dice tab for other numbers of dice, other die types and 'at least' or 'at most' totals.
For equally likely outcomes, probability = favourable outcomes ÷ total outcomes. Rolling a 4 on a fair die is 1 ÷ 6 ≈ 0.1667. A probability is always between 0 (impossible) and 1 (certain), and can be written as a decimal, a percentage or a fraction. For combined events you use the addition and multiplication rules instead of counting directly.
If A and B are independent, multiply: P(A and B) = P(A) × P(B), so 0.5 × 0.4 = 0.2. If they are mutually exclusive, P(A and B) = 0. Otherwise use the conditional form P(A and B) = P(A) × P(B | A), or enter a known joint probability in the “I know P(A and B)” option.
Use the addition rule: P(A or B) = P(A) + P(B) − P(A and B). Subtracting the overlap stops it being counted twice. For mutually exclusive events the overlap is zero, so P(A or B) is simply P(A) + P(B). For independent events with P(A) = 0.5 and P(B) = 0.4 the answer is 0.5 + 0.4 − 0.2 = 0.7.
Independent events do not affect each other — knowing A happened leaves the chance of B unchanged, so P(A and B) = P(A) × P(B). Mutually exclusive events cannot happen together, so P(A and B) = 0. Two events with non-zero probabilities cannot be both: if they are mutually exclusive, one happening tells you the other did not.
The probability of A given B is P(A | B) = P(A and B) ÷ P(B), provided P(B) is not zero. For example, if 40% of days are rainy and 10% of days are both rainy and cold, P(cold | rainy) = 0.10 ÷ 0.40 = 0.25. The calculator shows both P(A | B) and P(B | A).
Work out the chance it never happens and subtract from 1: P(at least once) = 1 − (1 − p)ⁿ for n independent tries with chance p each. Rolling at least one six in four rolls is 1 − (5/6)⁴ ≈ 0.5177. Adding the single-try chances (4 × 1/6) is a common mistake that can exceed 100%.
Solve 1 − (1 − p)ⁿ ≥ 0.95 for n, which gives n = ln(0.05) ÷ ln(1 − p), rounded up. With a 1-in-20 chance per try you need 59 tries for 95%, and about 90 for 99%. The At least once tab calculates this for any target percentage.
For n independent trials each with success chance p, the probability of exactly k successes is P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ, where C(n, k) counts the ways to choose which trials succeed. Cumulative probabilities like P(X ≤ k) add these terms. The mean is np and the variance is np(1 − p).
There are 36 equally likely outcomes for two dice and 6 of them total 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), so the probability is 6/36 = 1/6 ≈ 16.67%. Seven is the most likely sum; 2 and 12 are the least likely at 1/36 each.
A standard 52-card deck has 4 aces, so one card is an ace with probability 4/52 = 1/13 ≈ 7.69%. In a 5-card hand the chance of at least one ace is about 34.1%, found with the hypergeometric formula because cards are drawn without replacement. The Cards tab handles any deck size and target count.
Odds in favour compare successes to failures: a probability of 0.25 means 1 success for every 3 failures, or odds of 1 : 3 (3 : 1 against). To go back, divide: odds of a : b in favour give probability a ÷ (a + b). The calculator shows probability as a decimal, a percentage, a fraction and “1 in N”.
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