How to use
- Pick a calculation: two events, repeated tries or binomial (exactly k).
- Enter each probability as a decimal (0.6), a percentage (60%) or a fraction (1/6).
- Read the main answer, the working with your numbers, and every related result as a decimal, percentage and “1 in N”.
All results assume independent events: one result doesn’t change the chance of another.
Probability formulas
- Both: P(A and B) = P(A) × P(B)
- Either: P(A or B) = P(A) + P(B) − P(A and B)
- Not A: P(not A) = 1 − P(A)
- At least once in n tries: 1 − (1 − p)ⁿ
- Exactly k of n (binomial): C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, where C(n, k) counts the ways to choose k of the n trials
The binomial mean is np and the standard deviation is √(np(1 − p)).
Examples
- P(A) = 0.6 and P(B) = 0.5: both happen with 0.6 × 0.5 = 0.3; at least one with 0.6 + 0.5 − 0.3 = 0.8.
- At least one six in 4 dice rolls: 1 − (5/6)⁴ = 0.517747, just over half.
- Exactly 3 heads in 10 coin flips: C(10, 3) × 0.5³ × 0.5⁷ = 120 × 0.125 × 0.0078125 = 0.117188.
- A 1% risk on each of 50 tries: the chance it happens at least once is 1 − 0.99⁵⁰ ≈ 0.395.
Chance of at least one six
| Rolls | Probability | Percent |
|---|---|---|
| 1 | 0.166667 | 16.67% |
| 2 | 0.305556 | 30.56% |
| 4 | 0.517747 | 51.77% |
| 6 | 0.665102 | 66.51% |
| 10 | 0.838494 | 83.85% |
| 20 | 0.973916 | 97.39% |
Common mistakes
Adding probabilities for “A or B” counts the overlap twice, so 0.6 + 0.5 would give an impossible 1.1.
Past results don’t change independent trials. After five heads, a fair coin is still 50/50; expecting a “due” result is the gambler’s fallacy.
Independent is not the same as mutually exclusive. Mutually exclusive events can’t happen together, so P(A and B) = 0 and this calculator’s formulas don’t apply.
To turn a probability into a percentage or compare two chances, try the percentage calculator.