Hypergeometric Distribution Calculator
Enter the population size, number of successes in the population, sample size, and number of successes you want to find the exact and cumulative probability.
Last updated: September 5, 2026
Frequently asked questions
What is the hypergeometric distribution used for?
Calculating probabilities when sampling without replacement from a finite population - like drawing cards from a deck, or defective items from a batch, where each draw changes the odds for the next one (unlike the binomial distribution, which assumes replacement).
What's the probability of drawing exactly 2 hearts in a 5-card hand?
About 27.4% - with N=52 (deck), K=13 (hearts), n=5 (hand size), k=2: P(X=2) = C(13,2) × C(39,3) / C(52,5) ≈ 0.2743.
How is this different from the binomial distribution?
Binomial assumes each trial is independent with a fixed probability (sampling with replacement, or an infinite population). Hypergeometric accounts for the population shrinking as you sample (without replacement), which changes the odds on each draw - this matters most when the sample size is a meaningful fraction of the population.
What does the cumulative probability (P(X ≤ k)) tell me?
The probability of getting k or fewer successes, not exactly k - useful for questions like "what's the probability of drawing at most 2 hearts," which sums the individual probabilities for 0, 1, and 2 hearts.
What do N, K, n, and k each represent in plain terms?
N is the total population (e.g., a full deck of 52 cards), K is how many 'success' items are in that population (e.g., 13 hearts), n is how many items you draw (your sample size), and k is how many successes you want among those drawn.
What happens if k is larger than either n or K?
The probability is 0 - you can't draw more successes than you actually sampled (k > n) or more successes than exist in the population (k > K), so those combinations are mathematically impossible.
How is the mean of a hypergeometric distribution calculated?
Mean = n × (K/N) - for the 5-card hand from a 52-card deck with 13 hearts, the expected number of hearts is 5 × (13/52) = 1.25, even though you can't actually draw a fractional number of hearts in any single hand.