Binomial Distribution Calculator

Answers questions like “what is the chance of exactly 6 successes in 20 tries?”.

P(X = k)

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Share sends a link that reopens these numbers. Copy details pastes the full breakdown as a list.

P(X ≤ k) —

P(X < k) —

P(X ≥ k) —

P(X > k) —

Mean —

Standard deviation —

Reading this —

The formula

P(X = k) = C(n, k) × p^k × (1 − p)^(n − k) mean = np, variance = np(1 − p)

The cumulative figures come from the regularised incomplete beta function, so they are exact to working precision rather than a normal approximation.

“At least” and “at most” are often the question really being asked, rather than “exactly”, so all of them are shown.

What is a worked example?

For 20 trials at 0.3, P(X = 6) is 0.19164 and P(X ≤ 6) is 0.60801.

When is the normal approximation good enough?

A common rule of thumb is when np and n(1 − p) are both at least about 10.