Hypergeometric Distribution Calculator

The hypergeometric distribution models successes when drawing without replacement.

How the Math Works

The hypergeometric distribution calculates the probability of observing exactly k successes in n draws from a finite population of size N, where there are K total successes in the population. The formula P = C(K,k)·C(N−K,n−k) ÷ C(N,n) works by multiplying the combinations of ways to choose k successes from K available (C(K,k)) with the ways to choose the remaining n−k failures from the N−K non-successes (C(N−K,n−k)), then dividing by the total combinations of any n items from the population (C(N,n)). This accounts for sampling without replacement, where each draw affects subsequent probabilities, unlike the binomial distribution which assumes independence.

Practical Applications

This calculation is essential in quality control processes, where a batch of N items contains K defective units. By sampling n items and finding the probability of k defects, manufacturers can assess risk without needing to test every product. It's also widely used in medical research to evaluate treatment efficacy in small, closed populations, such as calculating the likelihood of a certain number of patients responding to a drug when only K out of N patients are known to respond. Environmental scientists use it to estimate contamination probabilities in water or soil samples drawn from a known total volume.

Day-to-Day Use

In everyday scenarios, this helps make informed decisions under uncertainty. For example, a teacher might use it to estimate the probability of selecting specific students for a project when only K out of N students have prior experience. Athletes or coaches could apply it to predict the chance of drawing certain training equipment from a limited supply. Even in hobbies like card games, it helps calculate the likelihood of drawing key cards from a known deck composition, enhancing strategic decisions without requiring complex probability calculations each time.

Worked example

Drawing 5 cards, P(exactly 1 ace) ≈ 0.299.

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