Hypergeometric Distribution Calculator

How it is calculated

Draws are made without replacement from a population of N items, of which K are successes. The probability of getting exactly k successes in n draws is P(X = k) = C(K, k) × C(N − K, n − k) ÷ C(N, n), where C(a, b) is the number of ways to choose b items from a. The numerator counts the samples with k successes and n − k failures, the denominator counts all possible samples. The second result is the mean, n × K ÷ N. Only the probability of exactly k is given, not 'k or more' or 'k or fewer'; add up the exact probabilities yourself for those. All four inputs must be whole numbers, with K ≤ N, n ≤ N, k ≤ n and k ≤ K, and the population must hold at least n − k failures, otherwise no result is shown. Population sizes in the thousands or millions work (up to 1 billion); the probability is shown to four decimals.

Common mistakes

• Using it when items are put back after each draw: with replacement the draws are independent and the binomial distribution applies, not this one.
• Entering K as a percentage or a probability instead of a count of success items in the population; all four inputs must be whole numbers, so a k of 2.5 gives no result.
• Reading the result as 'at least k': the tool gives exactly k, so for 2 or more hearts you must add the probabilities for 2, 3, 4 and 5.
• Swapping n and k, so that the sample size is the number of successes wanted; n is how many you draw, k how many of those should be successes.
• Entering a sample the population cannot supply, such as k larger than n or K, or more failures than the population contains (N = 52, K = 13, n = 45, k = 2); such combinations give no result.