Calculate exact hypergeometric probabilities for sampling without replacement, with PMF, CDF, distribution tables, and binomial approximation comparison.
The hypergeometric distribution calculator computes exact probabilities for sampling without replacement from a finite population. Unlike the binomial distribution, which assumes each draw is independent, the hypergeometric distribution accounts for the changing composition of the pool as items are removed.
This distribution appears in quality control (sampling inspection lots), card games (drawing specific hands), lottery analysis, ecological capture-recapture studies, and Fisher's exact test for independence in 2×2 tables.
Enter the population size N, number of success states K, draw size n, and desired number of successes k to get exact probabilities, cumulative values, a full distribution table with visual bars, and a comparison showing how closely the binomial approximation matches. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case. Use the example pattern when troubleshooting unexpected results. Validate that outputs match your chosen standards. Run at least one manual sanity check before publishing.
This calculator handles exact combinatorial probabilities that are tedious to compute by hand. The distribution table and visual chart make it easy to see the full probability landscape, while the binomial comparison helps you judge when simpler approximations are adequate.
Essential for statistics coursework, quality assurance professionals designing sampling plans, card game strategists, and researchers using Fisher's exact test.
P(X = k) = C(K,k) × C(N−K, n−k) / C(N,n), where C(a,b) = a! / (b!(a−b)!). Mean = nK/N. Variance = nK(N−K)(N−n) / (N²(N−1)).
Result: P(X = 2) ≈ 0.2743 (27.43%)
Drawing 5 cards from a standard 52-card deck, the probability of getting exactly 2 hearts (13 in the deck) is C(13,2)×C(39,3) / C(52,5) ≈ 0.2743.
In manufacturing, a lot of N = 1000 units might have K = 50 defectives. Drawing n = 20 for inspection, the hypergeometric distribution gives the exact probability of finding k defectives. Acceptance sampling plans (AQL/LTPD) rely on these calculations to balance producer and consumer risk.
The basic hypergeometric handles two categories (success/failure). The multivariate extension handles multiple categories simultaneously — like drawing a poker hand and tracking all four suits. The same C(N,n) denominator applies, with the numerator being a product of binomial coefficients for each category.
The Vandermonde identity, ΣC(K,k)C(N−K,n−k) = C(N,n), is essentially stating that hypergeometric PMF values sum to 1. This distribution sits at the intersection of combinatorics, probability theory, and statistical testing.
Use hypergeometric for sampling without replacement from a finite population (card draws, quality inspection). Use binomial when draws are independent (coin flips, each trial identical).
Each item drawn is not returned to the pool, so the composition changes. Drawing 2 hearts makes the remaining deck have 11 hearts in 50 cards, not 13 in 52.
No. The number of successes k must be between max(0, n+K−N) and min(n, K). Values outside this range have probability zero.
Fisher's exact test calculates the probability of observing a 2×2 contingency table (or more extreme) using the hypergeometric distribution. It's the gold standard for small-sample tests of independence.
In quality control, you draw a sample of n items from a lot of N. If more than c defectives are found, the lot is rejected. The hypergeometric distribution gives exact acceptance/rejection probabilities.
The binomial assumes constant probability p = K/N for each draw. Without replacement, the actual probability changes with each draw. The difference shrinks as N grows relative to n.