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Binomial Probability

Chance of exactly, at most or at least k successes in n independent trials.

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The formula

P(X = k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ

The binomial distribution counts successes across a fixed number of independent trials with the same probability each time — coin flips, free throws, defective parts. The C(n,k) term counts how many orderings give k successes; the powers give the chance of any one of them.

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Example calculations

Common questions this calculator answers — select one to load its values.

What is the chance of exactly 3 heads in 10 coin flips? P(exactly 3 of 10) 0.117188 What is the chance of exactly 5 heads in 10 flips? P(exactly 5 of 10) 0.246094 What is the chance of rolling exactly two sixes in 12 dice rolls? P(exactly 2 of 12) 0.296094

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Frequently asked questions

When does the binomial distribution apply?+
When there is a fixed number of trials, each has only two outcomes, the probability is the same every time, and the trials do not influence each other. Drawing cards without replacement breaks the last two conditions.
Why is “exactly 5 heads in 10 flips” less than 50%?+
Because there are eleven possible outcomes (0 through 10 heads) and the probability is spread across all of them. Five is the single most likely result at about 24.6%, but it is far from guaranteed.
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