Binomial Distribution Calculator

It models the number of successes in n independent yes-or-no trials where each trial has the same probability p of success. Coin flips, free-throw streaks, and pass-or-fail quality checks are classic examples.

P(X = k) = C(n, k) x p^k x (1 - p)^(n - k), where C(n, k) is the binomial coefficient n! / (k! (n - k)!). For n = 10, p = 0.5, k = 5 this gives 252 x 0.5^10 = 0.2461.

It evaluates the PMF in log space using log-gamma: log P(X = k) = lgamma(n + 1) - lgamma(k + 1) - lgamma(n - k + 1) + k ln(p) + (n - k) ln(1 - p), then exponentiates. That keeps every term finite for n up to 10,000.

Mean = n x p, variance = n x p x (1 - p), and the mode is floor((n + 1) x p). For n = 20, p = 0.3 you get mean 6.0, variance 4.2, std deviation about 2.05, and mode 6.

Use the binomial directly whenever you can. The normal approximation is only a shortcut for hand calculation when n is large and p is not too close to 0 or 1 (a common rule is np and n(1-p) both at least 5). This calculator computes the exact binomial probabilities.