Hypergeometric Distribution¶

Definition¶

Let a population contain $N$ objects, of which $K$ are successes, and suppose $n$ objects are drawn uniformly without replacement, so every subset of size $n$ is equally likely, where $N,K,n$ are integers with $N\ge0$ and $0\le K,n\le N$ $N$ . A random variable $X$ has a hypergeometric distribution if $X$ counts the number of successes in the draw, with

\begin{gathered} \mathbb{P}(X=k)=\frac{\binom Kk\binom{N-K}{n-k}}{\binom Nn},\\ \max(0,\,n-N+K)\le k\le\min(n,K), \end{gathered}

and $\mathbb{P}(X=k)=0$ for all other $k$ .

Interpretation¶

A hypergeometric distribution counts successes when sampling without replacement. The draws change what remains, so their success indicators are generally dependent; deterministic cases such as $K=0$ are exceptions.

Probability bars for the number of successes among 6 objects drawn without replacement from 20 objects of which 8 are successes, peaking at 2 with mean 2.4.

The diagram counts successes in a sample of $n=6$ objects drawn without replacement from $N=20$ objects, of which $K=8$ are successes. The PMF is $\mathbb{P}(X=k)=\binom8k\binom{12}{6-k}/\binom{20}6$ for $0\le k\le6$ $6$ . Its mean is $6\cdot\tfrac{8}{20}=2.4$ $2.4$ , and its variance $1.44\cdot\tfrac{14}{19}\approx1.061$ is smaller than the variance $6(0.4)(0.6)=1.44$ of $\operatorname{Bin}(6,0.4)$ , the corresponding model with replacement.

A combination counts an unordered selection without replacement. Counting the chosen successes and failures separately and dividing by all samples gives the hypergeometric probabilities.

Key Results¶

The hypergeometric distribution models sampling without replacement.
Unlike the independent trials of the binomial distribution, success indicators are generally dependent.
For $N>0$ $0$ , its mean is $nK/N$ . If $N=0$ $0$ , necessarily $K=n=0$ and $X=0$ $0$ .
For $N>1$ $1$ , its variance is

$\operatorname{Var}(X)=n\frac{K}{N}\left(1-\frac{K}{N}\right)\frac{N-n}{N-1}.$

For $N=0$ or $N=1$ $1$ , the admissible sample count is deterministic and its variance is $0$ .

This is the binomial variance with $p=K/N$ times the finite population correction $\frac{N-n}{N-1}$ , which is below $1$ when $n\ge2$ and close to $1$ when the population is much larger than the sample.

Examples¶

The number of aces in a five-card hand has a hypergeometric distribution.
The number of defective items in a sample drawn without replacement from a shipment is hypergeometric.
The number of red balls drawn from an urn without replacement is hypergeometric.

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 3

Keywords: hypergeometric random variable, sampling without replacement

Last updated: 27 October 2026 11:52