📊 Hypergeometric Distribution Calculator

The hypergeometric distribution models the probability of obtaining exactly k successes in n draws from a finite population of size N containing exactly K successes, without replacement. Unlike the binomial distribution (which assumes independent trials with replacement), the hypergeometric distribution accounts for the changing probabilities as items are removed from the population. It's widely used in quality control, ecological sampling, card games, and audit testing.

Use the hypergeometric distribution when sampling without replacement from a finite population where the sample size is a significant fraction of the population (typically n/N > 5%). Use the binomial distribution when sampling with replacement or when the population is very large relative to the sample size (n/N < 5%), as the hypergeometric distribution converges to the binomial in that case. The key difference: hypergeometric trials are dependent; binomial trials are independent.

• N — Total population size (e.g., 1000 manufactured parts).
• K — Number of "success" items in the population (e.g., 50 defective parts).
• n — Sample size drawn without replacement (e.g., 30 parts inspected).
• k — Number of successes observed in the sample (e.g., 2 defectives found).

The valid range for k is: max(0, n+K−N) ≤ k ≤ min(n, K).

The probability mass function (PMF) is:
P(X = k) = [C(K, k) × C(N−K, n−k)] / C(N, n)
where C(a, b) is the binomial coefficient ("a choose b"). This formula counts the number of ways to choose k successes from K available, multiplied by the ways to choose n−k failures from N−K available, divided by the total ways to choose n items from N. Our calculator uses numerically stable recurrence relations to compute these values accurately even for large populations.

• Quality Control: Accept/reject a batch based on defect count in a sample.
• Ecology: Estimate animal populations via mark-recapture studies.
• Card Games: Probability of drawing specific hands (e.g., Poker, Bridge).
• Auditing: Probability of detecting fraud in a sample of transactions.
• Lottery Analysis: Odds of matching numbers in lottery draws.
• Genetics: Probability of inheriting specific allele combinations.

The Finite Population Correction factor is √[(N−n)/(N−1)]. It adjusts the variance when sampling without replacement from a finite population. The hypergeometric variance is: Var(X) = n × (K/N) × (1−K/N) × [(N−n)/(N−1)]. The FPC reduces the variance compared to the binomial distribution (which would have variance n×p×(1−p)). When n is small relative to N, the FPC approaches 1, and the hypergeometric approximates the binomial.

Yes! When the population is large and the sample fraction is small (typically n/N < 5%), the hypergeometric distribution can be well approximated by the binomial distribution with p = K/N. For very large N and moderate K/N, a normal approximation may also be used with mean μ = n·K/N and variance σ² = n·(K/N)·(1−K/N)·[(N−n)/(N−1)]. This calculator always computes exact hypergeometric probabilities for maximum accuracy.