How to use
- Pick a calculation at the top.
- Enter probabilities as 0.25, 25% or 1/4 — whichever is easiest. For a die,
1/6is exact. - The formula and the numbers plugged into it appear under the result.
Formulas
Two events A and B
| Probability | Independent | Mutually exclusive |
|---|---|---|
| A and B | P(A) × P(B) | 0 |
| A or B | P(A) + P(B) − P(A)P(B) | P(A) + P(B) |
| Neither | (1 − P(A))(1 − P(B)) | 1 − P(A) − P(B) |
| Not A (complement) | 1 − P(A) | 1 − P(A) |
- Independent: one event does not change the other’s probability (tossing a coin and rolling a die).
- Mutually exclusive: the events cannot happen together (rolling a 1 and a 6 on a single roll).
Binomial distribution — the number of successes X in n independent trials, each with success probability p
- P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, where C(n, k) = n! ÷ (k!(n − k)!)
- Mean np, variance np(1 − p)
At least once — P = 1 − (1 − p)ⁿ. The tries needed for a target is the smallest whole n with n ≥ ln(1 − target) ÷ ln(1 − p).
Examples
- Exactly 3 heads in 10 coin tosses: C(10, 3) × 0.5¹⁰ = 120 ÷ 1,024 = 0.1172 (11.72%); at most 3 is 176 ÷ 1,024 = 17.19%.
- At least one six in 4 rolls of a die: 1 − (5/6)⁴ = 51.77% — the problem famously posed by the gambler Chevalier de Méré in the 17th century.
- A 1% drop over 100 tries appears at least once with probability 63.40%. Passing 50% takes 69 tries, 90% takes 230 and 99% takes 459.
Pitfalls
- Only multiply independent probabilities. If an earlier outcome changes the next one — drawing cards without putting them back, for instance — the independent formulas no longer apply.
- “100 tries at 1% guarantees a hit” is false. There is still a 36.6% chance of getting nothing.
- Very small probabilities are shown to six significant digits.
FAQ
How do I turn a probability into a percentage?
Multiply by 100: 0.125 is 12.5%. For general percentage math, see the percentage calculator.
When does the binomial distribution apply?
When each trial has two outcomes, the success probability stays the same every time, and trials do not affect each other — for example, the chance that at most 2 of 50 parts are defective when the defect rate is 2%.
Can I get a margin of error for a survey?
Yes — use the confidence interval calculator for proportion intervals and margins of error.