Hypergeometric Distribution Calculator

This hypergeometric distribution calculator gives the probability of drawing a given number of successes when you sample without replacement from a finite population. Enter the population size, how many successes are in the population, the sample size and the number of successes you want, and the calculator returns the probability. Unlike the binomial distribution, the hypergeometric accounts for the population shrinking as you draw, so it is the right model for things like quality control batches, card hands and capture-recapture. The formula, a worked example and the assumptions are below.

Population N

Successes in population K

Sample size n

Observed successes k

Expected successes

Observed k

The formula

The hypergeometric probability is P(X = k) = C(K, k) times C(N minus K, n minus k), all divided by C(N, n), where N is the population, K the successes in it, n the sample size and k the observed successes. The expected number of successes is n times K over N.

Worked example

From a population of 10 with 5 successes, drawing a sample of 4, the chance of exactly 2 successes is C(5,2) times C(5,2) over C(10,4) = 100/210, about 0.476. Enter N=10, K=5, n=4, k=2 to confirm.

Frequently asked questions

When do I use the hypergeometric distribution?

For sampling without replacement from a finite population, where each draw changes the remaining proportions.

How is it different from the binomial?

The binomial assumes a constant success probability (sampling with replacement). The hypergeometric corrects for the shrinking population.

What is the expected number of successes?

n times K over N, the sample size times the proportion of successes in the population.

Who this calculator is for

This calculator is for statistics students, researchers and analysts.

What this calculator assumes

You enter valid numbers.
The standard formula is applied.
Results are rounded for display.

Formula and sources

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