Binomial Distribution Explorer

The binomial distribution represents the probability of obtaining k successes in n independent Bernoulli trials: P(X=k) = nCr * p^k * (1-p)^{n-k}. As the number of trials n becomes large, the binomial distribution approaches the Normal distribution (de Moivre-Laplace theorem) or the Poisson distribution when the probability p is small and constant, serving as a cornerstone of statistical inference.

Imagine flipping the same biased coin n times and simply counting how many land heads. That count is a binomial random variable: each flip is independent, and every flip shares the same success probability p. The tool lets you slide n and p and watch where the probability piles up. Your best single guess for the number of successes is the mean, np — so 20 flips of a fair coin center on 10 heads.

The "binomial" in binomial distribution is the same binomial as in algebra's binomial theorem. Expanding (p + q)n with q = 1 − p produces exactly the terms nCk · pk · qn−k — the probabilities for every k. Since (p + q) = 1, that expansion equals 1n = 1, which is why the probabilities must sum to one. The coefficients nCk are literally the rows of Pascal's triangle.