Hypergeometric distribution

The hypergeometric distribution gives the probability of a set number of successes when sampling without replacement from a finite population, such as defects in a batch.

ByHoang TruongUpdated July 11, 2026

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A batch of 20 components contains 4 defective units and 16 good ones. Drawing a sample of 3 without replacement, the probability that exactly 1 of the 3 is defective is (4 choose 1) times (16 choose 2), divided by (20 choose 3), which works out to 8⁄19, about 42.1%. Each draw changes the odds for the next, unlike the binomial.

Where it fits

TopicProbability & DistributionsAdvancedSubjectData Analysis & StatisticsAdvanced

The formula

LaTeX

P(X=k) = \dfrac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}

Variables

$N$: Size of the whole population
$K$: Number of successes in the population
$n$: Size of the sample drawn
$k$: Number of successes in the sample

Gives the probability of drawing exactly k successes in a sample of n taken without replacement from a population of N containing K successes.

LaTeX

E[X] = n\frac{K}{N}

Variables

$n$: Size of the sample drawn
$K$: Number of successes in the population
$N$: Size of the whole population

Gives the expected number of successes in a hypergeometric sample.