Table of Contents
What Is the Hypergeometric Distribution?
The hypergeometric distribution models the number of successes in a fixed number of draws from a finite population without replacement. Unlike the binomial distribution, the probability changes with each draw because items are not returned to the population.
Common applications include quality control sampling (how many defective items in a batch?), card games (what is the probability of drawing a certain hand?), and ecological capture-recapture studies.
Formula
Where C(a,b) is the binomial coefficient "a choose b", N is population size, K is total successes in population, n is number of draws, and k is the number of observed successes.
Examples
| Scenario | N | K | n | k | P(X=k) |
|---|---|---|---|---|---|
| Cards: 2 aces in 5 draws | 52 | 4 | 5 | 2 | 0.0399 |
| QC: 1 defect in 10 samples | 100 | 5 | 10 | 1 | 0.3394 |
| Lottery: 3 matches from 6 | 49 | 6 | 6 | 3 | 0.0177 |
Frequently Asked Questions
When should I use hypergeometric vs binomial?
Use hypergeometric when sampling without replacement from a finite population. Use binomial when sampling with replacement or when the population is so large that removing items has negligible effect (rule of thumb: sample is less than 5% of population).
Can the hypergeometric distribution be approximated?
Yes, when the population is large relative to the sample, it can be approximated by the binomial distribution. When both N and n are large with K/N close to a value p, the normal approximation may also be used.
What are the constraints on the parameters?
The parameters must satisfy: 0 ≤ K ≤ N, 0 ≤ n ≤ N, and max(0, n+K-N) ≤ k ≤ min(n, K). The number of desired successes cannot exceed the draws or the available successes.