Calculate probabilities for sampling without replacement from a finite population. Essential for quality control and lottery-type scenarios.
Last updated: March 2026
Total items in population
Items with desired property
Sample size (without replacement)
Number of successes in sample
The hypergeometric distribution describes the probability of obtaining exactly k successes when drawing n items from a population of N items containing K successes, without replacement. This is distinct from the binomial distribution, which assumes replacement (infinite population).
This distribution is crucial in quality control, auditing, lottery analysis, and ecological studies. For example: drawing defective items from a lot, selecting cards from a deck, or sampling tagged animals from a population. The key feature is that probabilities change with each draw because the population shrinks and composition changes.
As the population size N increases while keeping K/N constant, the hypergeometric distribution approaches the binomial distribution, because sampling without replacement becomes equivalent to sampling with replacement.
Quality Control: Defective Items in a Batch
Use hypergeometric when sampling WITHOUT replacement from a small finite population. Use binomial when sampling WITH replacement or from a very large population where replacement is negligible.
The hypergeometric formula uses combinations (binomial coefficients). C(n,k) counts the ways to choose k items from n items. The formula divides favorable outcomes by total possible outcomes.
K represents items with a property in a population of N total items. You can't have more 'success' items than total items. If K > N, the scenario is mathematically impossible.
If n = N, there's no randomness. You always get exactly K successes with probability 1. This represents a census, not a sample.
Smaller N increases variance in outcomes. Large N makes hypergeometric approach binomial. For N > 1000, you can often use binomial as an approximation with p = K/N.
Mean = n × K/N (sample size × proportion of successes in population). This follows the intuition: expected value equals sample size times success rate.
Yes, if k is forced to a specific value by the constraints. For example, if n = K and N - K ≥ 0, you must always get exactly k = K successes with zero variance.
Auditors use hypergeometric to determine error detection probability. For a population of N transactions with K known errors, sampling n items gives the probability of finding 0, 1, 2, etc. errors.
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