Binomial Distribution & Probability Calculator

Calculate exact and cumulative probabilities for independent trials with a fixed chance of success.

Result and method

When the binomial model applies

There must be a fixed number of independent trials, two outcomes per trial, and the same success probability each time. Sampling without replacement from a small population generally requires a different model.

Formula and example

P(X = k) = C(n,k) p^k(1−p)^n−k. For ten fair coin flips, exactly five heads has probability 252/1024 = 0.24609375.

Exact versus cumulative

Exactly k is one outcome count. At most k sums counts 0 through k. At least k sums counts k through n. The calculator also returns strict inequalities to avoid boundary confusion.

Precision and limits

Supports 1–1,000 trials. Terms are evaluated with logarithms and normalized to limit numerical error. Extremely small probabilities can round or underflow to zero; this tool does not provide arbitrary precision.

Reference: NIST: Binomial Distribution.