Hypergeometric Distribution Calculator

A hypergeometric experiment involves sampling without replacement from a finite population that contains exactly two types of items (successes and failures). The key properties are: (1) you sample n items from a population of N, (2) the population contains K successes and N−K failures, and (3) once an item is drawn it is not returned to the population.

The hypergeometric distribution describes the probability of obtaining exactly x successes in a sample of n items drawn without replacement from a population of N items containing K successes. It is used instead of the binomial distribution when sampling is done without replacement and the sample size is large relative to the population.

The probability P(X = x) is calculated as C(K, x) × C(N−K, n−x) / C(N, n), where C(a, b) is the binomial coefficient ('a choose b'). The numerator counts favorable outcomes (choosing x successes from K and n−x failures from N−K), and the denominator counts all possible samples of size n from N.

Both distributions count the number of successes in a fixed number of trials, but they differ in sampling method. The binomial distribution assumes sampling with replacement (each trial is independent with constant probability p), while the hypergeometric distribution assumes sampling without replacement (each draw changes the composition of the remaining population). Use hypergeometric when the sample is a significant fraction of the population.

A cumulative probability sums the individual probabilities over a range of outcomes. For example, P(X ≤ x) is the probability of getting at most x successes, computed by summing P(X = 0) + P(X = 1) + … + P(X = x). Similarly, P(X ≥ x) = 1 − P(X ≤ x−1) gives the probability of at least x successes.

The mean (expected value) is μ = n × (K / N), representing the average number of successes expected in the sample. The variance is σ² = n × (K/N) × ((N−K)/N) × ((N−n)/(N−1)), and the standard deviation σ is its square root. The factor (N−n)/(N−1) is called the finite population correction factor.

Use this calculator whenever you are sampling a group of items from a known finite population without replacement and you want to find the probability of getting a certain number of 'successes'. Common applications include quality control (defective items in a batch), card games (drawing specific cards from a deck), ecology (capture-recapture studies), and auditing (finding errors in sampled records).

The inputs must satisfy: K ≤ N (successes cannot exceed population size), n ≤ N (sample cannot exceed population size), and x must fall within max(0, n+K−N) ≤ x ≤ min(n, K). If x is outside this support range, P(X = x) = 0. The calculator will alert you if any constraint is violated.