Hypergeometric Distribution Calculator

About This Test

What Is the Hypergeometric Distribution?

The hypergeometric distribution gives the probability of a specific number of successes when drawing a sample without replacement from a finite population. It's closely related to the binomial distribution, but the binomial assumes each draw doesn't change the population — the hypergeometric accounts for the fact that once you draw something, it's gone.

When to Use It

Drawing cards from a deck, or balls from an urn, without putting them back
Quality control: sampling items from a finite batch to check for defects
Any fixed, finite population where sampling is done without replacement

The Formula

\[ P(X = x) = \dfrac{\binom{K}{x}\binom{N-K}{n-x}}{\binom{N}{n}} \]

Worked Example

A standard deck has 52 cards, 13 of them hearts. If you draw 5 cards, what's the probability of getting exactly 2 hearts?

\( N = 52, \ K = 13, \ n = 5, \ x = 2 \)
\( P(X = 2) \approx 0.2743 \), about 27.4%

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