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Hypergeometric Calculator

Input the values, and the calculator will compute individual and cumulative probability distributions, displaying detailed calculations.

Select Function:

Population Size (N)

Successes in population (K)

Sample Size (n)

Successes in Sample (k)

Related

This online Hypergeometric Calculator allows you to compute individual and cumulative hypergeometric probabilities. It can also generate probability tables, charts, and calculate the mean, variance, and standard deviation of a hypergeometric distribution.

What is Hypergeometric Distribution?

A hypergeometric distribution models the probability of a certain number of successes in a sample drawn without replacement from a finite population. For example, drawing 5 cards from a standard deck and counting red cards demonstrates a hypergeometric distribution:

Outcome	Probability	Cumulative Probability
0 red cards	0.025	0.025
1 red card	0.150	0.175
2 red cards	0.325	0.500
3 red cards	0.325	0.825
4 red cards	0.150	0.975
5 red cards	0.025	1.00

Hypergeometric Distribution Formula

The probability of observing k successes in a sample of size n from a population of size N with K successes is:

h(k; N, n, K) = [C(K, k) × C(N-K, n-k)] / C(N, n)

N = Population size
K = Number of successes in the population
n = Sample size
k = Number of successes in the sample
C = Combinations
h = Hypergeometric probability

Using the Hypergeometric Calculator

This calculator can compute both individual probabilities and cumulative distributions, along with key parameters and charts.

1. Probability Distribution

Inputs:

Select "Hypergeometric Probability Distribution".
Enter Population size (N).
Enter Number of successes in population (K).
Enter Sample size (n).
Enter Number of successes in sample (k).

Outputs:

Hypergeometric Probability: P(X = x)
Lower cumulative probability: P(X < x)
Lower or equal cumulative probability: P(X ≤ x)
Upper cumulative probability: P(X > x)
Upper or equal cumulative probability: P(X ≥ x)
Mean, Variance, Standard Deviation
Hypergeometric Distribution Chart

2. Probability Distribution Table & Chart

Inputs:

Select "Hypergeometric Probability Distribution (Chart)".
Choose the function: PMF, Lower Cumulative (P), or Upper Cumulative (Q).
Enter Population size (N), Successes in population (K), Sample size (n).
Enter initial k value, increment step, and number of steps.

Outputs:

Probability table
Mean, Variance, Standard Deviation
Probability chart

FAQs

When should you use a hypergeometric distribution?

Use it for small populations where sampling without replacement affects probabilities. Example: a group of 10 people where only 7 have A+ blood type.

What is a hypergeometric experiment?

Random selections from a finite population without replacement.
Each item is classified as success or failure.

What is the number of successes?

The count of successes in the sample or population. For example, how many red cards are drawn in a sample of 5 cards.

What is hypergeometric probability?

It is the probability of observing a specific number of successes in a hypergeometric experiment.

References

Wikipedia: Hypergeometric distribution
Probability Theory and Mathematical Statistics: Hypergeometric Distribution Example

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