Hypergeometric Distribution Calculator

The hypergeometric distribution models drawing a fixed-size sample of n items, without replacement, from a finite population of N items that contains exactly K "successes" — the classic example is dealing a 5-card hand from a 52-card deck and asking how many of the 13 hearts it contains. The exact probability of getting precisely k successes is the combinatorial ratio P(X = k) = C(K, k) × C(N − K, n − k) ÷ C(N, n): choose k successes from the K available, choose the rest of the sample from the N − K non-successes, divided by the number of ways to draw any sample of size n from the whole population.

This is distinct from the site's binomial-probability calculator, whose success probability p stays fixed on every trial because items are effectively replaced (or the population is treated as infinite). Here, every item drawn changes what's left to draw from, so the chance of success shifts as the sample is taken — which is exactly why card hands, lottery draws, and quality-control sampling from a finite lot all need the hypergeometric model rather than the binomial one. The mean number of successes is n × K ÷ N, and the variance, n × (K/N) × ((N − K)/N) × ((N − n)/(N − 1)), includes that extra (N − n)/(N − 1) "finite population correction" factor the binomial distribution doesn't need.