Binomial Distribution Calculator

About this calculator

The binomial distribution models the number of successes in n independent trials, each with the same success probability p. The exact probability of getting precisely k successes is P(X=k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ, where C(n,k) is the number of ways to choose which k trials succeed; this calculator also sums that formula to give the cumulative probabilities P(X≤k) and P(X≥k).

Statisticians, quality-control engineers, and data scientists use the binomial distribution whenever an outcome is binary and trials are independent — defect rates in manufacturing batches, coin-flip and dice problems in intro statistics, and conversion-rate comparisons in A/B testing all reduce to this model. Clinical researchers use it too, for example to compute the probability of a given number of patients responding to treatment out of a fixed trial size.