Hypergeometric Distribution Calculator

How it works

1. 1Enter the population size N, the number of successes K in the population, the number of draws n, and the observed successes k.
2. 2The calculator applies P(X = k) = C(K, k) · C(N − K, n − k) ÷ C(N, n), counting the ways to draw exactly k successes and n − k failures.
3. 3The result is an exact probability. Parameters that make the draw impossible (e.g. k > K or n > N) return — to indicate the event cannot occur.

Use cases

• Card games: find the probability of being dealt exactly 2 aces in a 5-card hand from a 52-card deck.
• Quality control: estimate the chance of finding exactly k defective items when inspecting n of N units.
• Ecology: model capture–recapture sampling, where K tagged animals are in a population of N and n are re-sampled.

Frequently asked questions

How does it differ from the binomial?

The binomial assumes independent trials with fixed probability — sampling WITH replacement. The hypergeometric models sampling WITHOUT replacement, so each draw changes the remaining population and trials are dependent.

When does it return an impossible (—) result?

When parameters violate the draw constraints: k > K, k > n, n − k > N − K, n > N, or K > N. Each makes the requested outcome impossible.

What is the mean of the hypergeometric distribution?

The expected number of successes is μ = n · K / N — the same as the binomial mean with p = K / N. Its variance includes a finite-population correction factor (N − n) / (N − 1).