Hypergeometric Distribution Calculator
Calculate probabilities for sampling without replacement, card draws, lottery ball selections, and quality inspection batches.
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What Is a Hypergeometric Distribution Calculator?
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.
How to Use This Calculator
- Enter the Total Population Size (N) (e.g. 52 cards in a deck).
- Enter the Number of Success States in Population (K) (e.g. 4 aces in deck).
- Enter the Sample Size Drawn (n) (e.g. 5-card hand).
- Enter the Target Successes (k) (e.g. 1 ace).
- Review exact P(X = k) and cumulative probabilities.
Hypergeometric PMF Formula
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.
Worked Example
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%).
Tips & Key Notes
- If the population size N is very large compared to sample size n (e.g. N > 20n), sampling without replacement closely approximates the binomial distribution.
- The hypergeometric distribution is essential in industrial quality control to accept or reject batches based on defective part sampling.
Frequently Asked Questions
What is the main difference between Binomial and Hypergeometric distributions?
Binomial models sampling WITH replacement (independent trials with constant probability). Hypergeometric models sampling WITHOUT replacement (dependent trials where probabilities change after each draw).
What is the expected value of a hypergeometric distribution?
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.
Where is the hypergeometric distribution used in real life?
It is widely used in card games (Poker, Blackjack, Magic: The Gathering), lottery games, clinical trials sampling patient groups, and factory quality assurance testing.
What are the bounds for valid values of k?
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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