Hypergeometric Distribution Calculator (with Graph)
Find probabilities for samples drawn without replacement from a finite population.
When the hypergeometric distribution applies
The hypergeometric distribution counts the successes in a sample drawn without replacement from a finite population that contains a known number of successes. Because each draw changes what is left, the draws are not independent, which is what separates it from the binomial. Use it for cards drawn from a deck, defective parts in a batch checked without putting them back, and lottery matches.
The formula
P(X = k) = C(K, k) × C(N − K, n − k) ÷ C(N, n), where N is the population size, K is the number of successes in the population, n is the sample size and k is the number of successes in the sample. The mean is n × K ÷ N.
Worked examples
| Question | N | K | n | Answer |
|---|---|---|---|---|
| Match exactly 3 of 6 numbers in a 6-from-49 lottery | 49 | 6 | 6 | 0.01765 |
| Match at least 2 of 6 | 49 | 6 | 6 | 0.1510 |
| Exactly 2 aces in a 5-card poker hand | 52 | 4 | 5 | 0.03993 |
| No aces in a 5-card hand | 52 | 4 | 5 | 0.6588 |
The graph under the result highlights the event you chose, and has Download PNG and Download SVG buttons for reports and slides. A table lists P(X = k), P(X ≤ k) and P(X ≥ k) for the values around the mean.
Hypergeometric or binomial?
If the population is much larger than the sample (for example, more than 20 times larger), sampling without replacement hardly changes the probabilities, and the binomial gives almost the same answer. For small populations, use the hypergeometric. The factor (N − n) ÷ (N − 1) in its variance, called the finite population correction, makes the spread smaller than the binomial’s.
How to cite this page
CalcPulse. (2026). Hypergeometric Distribution Calculator. https://calcpulse.com/hypergeometric-distribution-calculator/
Frequently asked questions
What is a hypergeometric distribution?
The distribution of the number of successes in a sample drawn without replacement from a finite population with a known number of successes.
How is it different from the binomial distribution?
The binomial assumes independent trials with a constant probability, as with replacement. The hypergeometric changes after each draw because items are not put back.
How do I calculate a hypergeometric probability?
Use C(K, k) × C(N − K, n − k) ÷ C(N, n). The calculator also adds the probabilities for 'at most', 'at least' and 'between'.
What is the chance of winning a 6/49 lottery?
Matching all 6 numbers has probability 1 in 13,983,816. Enter N = 49, K = 6, n = 6, k = 6.
What is the mean of a hypergeometric distribution?
n × K ÷ N, the sample size times the share of successes in the population.
Last updated