What is the hypergeometric distribution?

The hypergeometric distribution models the probability of k successes in n draws without replacement from a population of N items containing exactly K successes. Unlike the binomial distribution, it does not assume replacement, so successive draws are not independent.

What is the PMF formula?

P(X = k) = C(K, k) × C(N − K, n − k) / C(N, n), where C(a, b) is the binomial coefficient "a choose b". The valid range for k is max(0, n + K − N) ≤ k ≤ min(n, K).

How does it differ from the binomial distribution?

The binomial distribution assumes each draw is independent (sampling with replacement), so the success probability stays constant. The hypergeometric distribution accounts for a shrinking population after each draw — sampling without replacement — making the probability change with each pick.

What are the mean and variance?

Mean = n × K / N. Variance = n × (K/N) × (1 − K/N) × (N − n)/(N − 1). The factor (N − n)/(N − 1) is the finite population correction, which is always ≤ 1 and reduces variance compared to the binomial.

What constraints must the inputs satisfy?

N (population) must be a positive integer. K (successes in population) must satisfy 0 ≤ K ≤ N. n (sample size) must satisfy 1 ≤ n ≤ N. k (observed successes) must satisfy max(0, n + K − N) ≤ k ≤ min(n, K).

When does the hypergeometric distribution approximate the binomial?

When the sample size n is small relative to the population N (typically n/N < 0.05), the hypergeometric distribution is well approximated by a binomial distribution with p = K/N, because the change in population after each draw becomes negligible.

Hypergeometric Distribution Calculator

Name: Hypergeometric Distribution Calculator
Availability: InStock
Author: Nham Vu

Enter population size, number of success states, and sample size to compute hypergeometric PMF, CDF, mean, and variance instantly.

Parameters

Population size N

Total number of items in the population.

Successes in population K

Number of success states in the population (0 ≤ K ≤ N).

Sample size n

Number of draws without replacement (1 ≤ n ≤ N).

Observed successes k

Number of successes observed in the sample.

Distribution Statistics

Mean (Expected value): —
Variance: —
Standard Deviation: —
Valid k range: —

Results for k = 2

PMF — P(X = k)

—

Probability of exactly k successes

CDF — P(X ≤ k)

—

Probability of at most k successes

P(X ≥ k)

—

Probability of at least k successes

p = K / N

—

Population success proportion

Probability Table

All valid k values

k	P(X = k)	P(X ≤ k)

Summary

Enter population size, number of success states, and sample size to compute hypergeometric PMF, CDF, mean, and variance instantly.

How it works

Enter the population size N (total items in the group).
Enter K, the number of success states in the population.
Enter the sample size n (number of items drawn without replacement).
Enter k, the number of observed successes you want to evaluate.
The calculator instantly shows P(X = k), P(X ≤ k), P(X ≥ k), and distribution statistics.
The probability table lists all possible values of k with their PMF and CDF.

Use cases

Card games: probability of drawing exactly k aces in a 5-card hand from a standard deck.
Quality control: chance of finding exactly k defects in a batch sample.
Election auditing: probability that a sample of ballots contains a given number of errors.
Medical trials: likelihood of k responders in a group selected from a clinical pool.
Lottery analysis: probability of matching exactly k numbers when drawing from a pool.
Ecology: estimating species counts in a sampled area from a known total.
Market research: probability that k out of n surveyed belong to a target subgroup.
Genetics: probability of inheriting k copies of a trait from a finite gene pool.

Frequently Asked Questions

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Last updated: 2026-05-23 · Reviewed by Nham Vu