Hypergeometric Distribution Calculator

The hypergeometric distribution describes the number of successes in a fixed-size sample drawn without replacement from a finite population. The population contains N objects, exactly K of which have the feature called success. A sample of n objects is selected, and X counts how many of those sampled objects are successes. Typical applications include drawing cards from a deck, auditing a shipment, selecting committee members, and testing products from a finite batch. Exact probability comes from counting combinations. The number of samples containing exactly k successes is the combination of K choose k multiplied by the combination of N minus K choose n minus k. Dividing that product by N choose n, the number of all possible samples of size n, gives P(X = k). Since order does not matter, combinations are the natural counting method. The calculator evaluates these factors without needing a list of individual objects. Feasible inputs obey several constraints. Population size must be positive. The population success count and sample size cannot exceed the population. The requested sample successes cannot exceed either the successes available or the sample size. It also cannot be so small that the sample would require more failures than exist in the population. Validating all these limits prevents a mathematically impossible event from being confused with an ordinary zero-probability result. The expected number of sample successes is n times K divided by N. Variance equals n times the population success proportion times its complement, multiplied by the finite population correction (N - n) divided by (N - 1). That correction captures dependence caused by sampling without replacement. As more of the population is sampled, uncertainty decreases; when the entire population is selected, the number of successes is known and variance is zero. The hypergeometric and binomial distributions are related but not interchangeable. A binomial model assumes independent trials with a constant success probability, which fits replacement sampling or a population effectively much larger than the sample. Hypergeometric draws are dependent because each selection changes the composition remaining. When the sample is a small fraction of a large population, the binomial distribution can approximate the result. For finite batches or substantial sampling fractions, the exact hypergeometric calculation is preferable. Use this tool for transparent probabilities in quality control, genetics, surveys, games, and any selection process where sampled items are not returned.