About this tool
The Binomial Distribution Calculator finds the probability of getting exactly, at most, or at least k successes in n independent trials that each succeed with probability p. Enter n, p and k (p as a decimal from 0 to 1 or as a percentage), and every result updates instantly in your browser with no data leaving your device.
The exact probability is P(X = k) = C(n, k)·pᵏ·(1 − p)^(n − k), where the binomial coefficient C(n, k) is built with a numerically stable loop, c = c·(n − k + i)/i for i = 1…k, to avoid factorial overflow at large n. The cumulative values are P(X ≤ k) = Σ from i = 0 to k and P(X ≥ k) = Σ from i = k to n. The mean is n·p, the variance is n·p·(1 − p), and the standard deviation is its square root.
Both n and k must be whole numbers with 0 ≤ k ≤ n and n ≥ 1, and p must lie between 0 and 1; anything else is flagged as invalid. The model assumes fixed n, constant p and independent trials — the classic "coin flips" or "defective parts" setting.