The Hypergeometric Probability Distribution

Statistics • Discrete Random Variables and Their Probability Distributions

Written by STEM Calculators Team Published December 13, 2025 Updated February 24, 2026

The Hypergeometric Probability Distribution

Use this when you sample without replacement from a finite population (so trials are dependent). Enter the population size N, the number of successes in the population r, the sample size n, and the number of observed successes x.

Quick setup N = population size, r = successes in population, N − r = failures in population, n = sample size, x = successes in sample, n − x = failures in sample.

For cumulative probabilities (like “at most” / “at least”), we use the addition rule because the events X = 0, X = 1, … are mutually exclusive.

Population size (N)

Finite population. For performance, keep N reasonably sized (e.g., ≤ 20000).

Successes in population (r)

Must satisfy 0 ≤ r ≤ N.

Sample size (n)

Must satisfy 0 ≤ n ≤ N (sampling without replacement).

Probability query

x (successes in sample)

Valid range depends on N, r, n.

Ready

Enter N, r, n, then click Calculate.

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

The hypergeometric probability distribution

The hypergeometric distribution models the number of successes X in a sample of size n drawn without replacement from a finite population of size N. Because items are not replaced, the trials are dependent and the success probability changes from draw to draw.

Parameters and meanings

N: total number of elements in the population
r: number of successes in the population
N − r: number of failures in the population
n: sample size (number of draws without replacement)
x: number of successes in the sample
n − x: number of failures in the sample

Probability formula

\[ P(X = x)=\frac{\binom{r}{x}\cdot \binom{N-r}{\,n-x\,}}{\binom{N}{n}} \]

The numerator counts the number of samples that contain x successes and n − x failures. The denominator counts the total number of possible samples of size n.

Feasible values of x

Not every integer is possible. The support is:

\[ \max\!\bigl(0,\;n-(N-r)\bigr)\le x \le \min(n,\;r) \]

Cumulative probabilities

For statements like “at most” or “at least,” we add the probabilities of mutually exclusive outcomes: P(X \le k)=\sum_{x}P(X=x) over the appropriate range.

Binomial comparison

If the population is very large and the sample is a small fraction of the population, the hypergeometric model can be close to a binomial model with p \approx r/N. When sampling without replacement from a finite population, the hypergeometric model is the correct default.

Frequently Asked Questions

What is the hypergeometric probability distribution used for?

It models the number of successes X in a sample of size n drawn without replacement from a finite population of size N that contains r successes. It is the standard model when draws are dependent because items are not replaced.

What is the hypergeometric probability formula for P(X = x)?

The formula is P(X = x) = [C(r, x) x C(N - r, n - x)] / C(N, n). It counts samples with x successes and n - x failures divided by all samples of size n.

What values of x are possible in a hypergeometric problem?

Not every integer is feasible. The support is max(0, n - (N - r)) <= x <= min(n, r).

How do you calculate P(X <= x) or P(X >= x) for a hypergeometric distribution?

Cumulative probabilities are sums of exact probabilities because outcomes X = 0, 1, 2, ... are mutually exclusive. For example, P(X <= k) is the sum of P(X = x) over all x up to k.

When is the binomial distribution a reasonable approximation to hypergeometric?

When the population is very large and the sample is a small fraction of the population, the hypergeometric model can be close to binomial with p approximately r/N. For finite populations with sampling without replacement, hypergeometric is the correct default.

The Hypergeometric Probability Distribution

Calculation steps

Rate this calculator

Frequently Asked Questions

Related calculators

Questions related to this topic