What is the hypergeometric distribution?

The hypergeometric distribution models the probability of getting exactly k successes in n draws without replacement from a finite population of size N that contains K successes.

When do I use it instead of the binomial?

Use the hypergeometric when sampling without replacement from a finite population. Use the binomial for independent trials with replacement or very large populations.

What inputs do I need?

You need N (population size), K (number of successes in the population), n (sample size), and k (number of successes observed).

Hypergeometric Distribution Calculator (PMF, CDF, Mean, Variance)

1. Input parameters

Population size (N)

Total items in the population

Number of success states (K)

How many of the N are “success”

Sample size (n)

Drawn without replacement

Observed successes (k)

Exact number you want the probability for

2. Results

P(X = k)

—

Cumulative

P(X ≤ k): —

P(X ≥ k): —

Mean

—

Variance

—

3. Distribution table

All valid k from 0 to n (subject to K and N). Handy for homework / QA.

k	P(X = k)	P(X ≤ k)

Hypergeometric distribution: formula

For a population of N items containing K successes, when you draw n items without replacement, the probability of observing exactly k successes is:


                P(X = k) = [C(K, k) · C(N − K, n − k)] / C(N, n)

where C(a, b) is the number of combinations “a choose b”.

Mean and variance


                E[X] = n · (K / N) 

                Var(X) = n · (K / N) · (N - K)/N · (N - n)/(N - 1)

Where it’s used

Quality control (defectives in a batch)
Card games (drawing particular suits)
Sampling without replacement in surveys