Statistics & Probability

Hypergeometric Probability Calculator

Draw n items without replacement from a population containing K successes.

By TheCalcTools Updated Formula checked and tested

Your numbers

Your results

P(X = k)
P(X = k)
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Expected successes
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Hypergeometric Probability Calculator by TheCalcTools

How the Hypergeometric Probability Calculator works

Enter the population, how many are successes, the sample size and the number of successes you want.

The formula counts the favourable samples and divides by all possible samples.

Unlike the binomial distribution, items are not replaced.

The formula

P(X = k) = C(K, k) × C(N − K, n − k) ÷ C(N, n)

Worked example

Using the default values shown in the calculator:

Population size (N)52
Successes in population (K)4
Sample size (n)5
Successes wanted (k)2
  1. Drawing 5 cards from 52, with 4 aces.
  2. P(exactly 2 aces) = C(4,2) × C(48,3) ÷ C(52,5) = 0.0399.
  3. The expected number of aces is 0.3846.

Result:

P(X = k)0.0399
Expected successes0.3846

Frequently asked questions

How is it different from the binomial distribution?

The binomial assumes independent trials with replacement. The hypergeometric applies when drawing from a finite population without replacement.

What are typical uses?

Card games, quality-control sampling and lottery matching probabilities.

Sources and further reading

Binomial Distribution CalculatorCalculate binomial probabilities for n trials with success probability p Permutation and Combination CalculatorCount how many ways you can choose r items from n, with order (permutations) or without… Poisson Distribution CalculatorFind the probability of k events in an interval when events happen at an average rate λ Normal Distribution CalculatorEnter the mean, standard deviation and a value x to find its z-score and probabilities Z-Score CalculatorSee how many standard deviations a value is from the mean Geometric Distribution CalculatorHow many attempts until the first success?

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