Hypergeometric Distribution Calculator

Work out hypergeometric distribution instantly with clear inputs, formula shown and shareable results.

About Hypergeometric Distribution Calculator

The hypergeometric distribution describes sampling without replacement: P(X = k) = C(K,k)·C(N−K, n−k)/C(N,n). Unlike the binomial, each draw changes the composition of what remains.

Formulas

Hypergeometric pmf

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

Mean

E = n·K/N

Worked example

With the default values below, Hypergeometric Distribution Calculator returns P(exactly k successes) of 0.0399298181. Change any input above and every figure here is recalculated in your browser.

Inputs used

Population size Naccepts 1–100,000: 52
Successes in the population Kaccepts 0–100,000: 4
Sample size naccepts 1–100,000: 5
Observed successes kaccepts 0–100,000: 2

Results

P(exactly k successes): 0.0399298181
P(at most k successes): 0.9982454520
Expected successes: 0.384615
Variance: 0.32718413

Step by step

1Values used
2Hypergeometric pmf
3Mean
4P(exactly k successes)= 0.0399298181
5P(at most k successes)= 0.9982454520
6Expected successes= 0.384615
7Variance= 0.32718413

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Chance of exactly 2 aces in a 5-card hand?

C(4,2)·C(48,3)/C(52,5) = 6 × 17296 / 2598960 ≈ 0.0399.

When can I use the binomial instead?

When the sample is a small fraction of the population, say under 5%, the difference becomes negligible.