the binomial distribution

The binomial distribution answers a question you ask constantly: if I repeat the same yes/no trial n times independently, how many successes do I get? Flip a fair coin 10 times — how many heads? Treat 20 patients with a drug that works 70% of the time — how many recover? Survey 100 voters when 55% support a measure — how many say yes? The count of successes, call it X, is a whole number between 0 and n, and its distribution is X ~ Binomial(n, p).

Its probability mass function is P(X = k) = C(n, k) p^k (1 - p)^(n-k). Read it as a story in three pieces. The p^k says k of the trials succeeded, each with probability p. The (1 - p)^(n-k) says the other n - k failed. And C(n, k), the binomial coefficient 'n choose k', counts how many different orderings of k successes among n trials there are — because we do not care which specific trials succeeded, only how many. Multiply the chance of one such pattern by the number of patterns and you have the total probability. Summing over all k from 0 to n gives exactly 1, which is the binomial theorem in disguise. The mean is E[X] = np (each of n trials contributes p on average) and the variance is Var(X) = np(1 - p).

The model rests on four conditions, sometimes abbreviated as a fixed number of trials, two outcomes each, a constant success probability, and independence. Break any one and the binomial is wrong. The most common slip is using it when you sample without replacement from a small finite pool — there p drifts as you go, and the correct model is the hypergeometric, though the binomial is a fine approximation when the pool is large relative to the sample.