Calculate probabilities for sampling without replacement, card draws, lottery ball selections, and quality inspection batches.
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The Hypergeometric Distribution Calculator computes the probability of obtaining k successes in a sample of size n drawn from a finite population N containing K total successes without replacement. Unlike the binomial distribution, the probability changes after each draw.
P(X = k) = \frac{\binom{K}{k} \binom{N - K}{n - k}}{\binom{N}{n}}, \quad \mu = \frac{n K}{N}Where N is total population, K is total success states, n is sample size drawn, and k is observed successes.
Scenario: Find the probability of drawing exactly 1 Ace in a standard 5-card hand drawn from a 52-card deck.
Parameters: N = 52, K = 4 (aces), n = 5 (cards drawn), k = 1.
Combinations of aces: C(4, 1) = 4.
Combinations of non-aces: C(48, 4) = 194,580.
Total 5-card combinations: C(52, 5) = 2,598,960.
Multiply and divide: (4 × 194,580) / 2,598,960 = 778,320 / 2,598,960 ≈ 0.29947 (29.95%).
Binomial models sampling WITH replacement (independent trials with constant probability). Hypergeometric models sampling WITHOUT replacement (dependent trials where probabilities change after each draw).
The expected value (mean) is μ = (n × K) / N. Drawing 5 cards from a 52-card deck with 4 aces yields an expected value of (5 × 4) / 52 ≈ 0.385 aces.
It is widely used in card games (Poker, Blackjack, Magic: The Gathering), lottery games, clinical trials sampling patient groups, and factory quality assurance testing.
k cannot exceed the sample size n or total successes K, and cannot be less than max(0, n - (N - K)).
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