# Hypergeometric Distribution Calculator

Free hypergeometric distribution calculator. Calculate the probability of drawing exactly k successes from a population without replacement.

## What this calculates

Calculate probabilities for sampling without replacement using the hypergeometric distribution. Find the exact probability of drawing a specific number of success items from a finite population.

## Inputs

- **Population Size (N)** — min 1, max 1000 — Total number of items in the population.
- **Success States (K)** — min 0 — Number of success items in the population.
- **Number of Draws (n)** — min 1 — Number of items drawn (without replacement).
- **Desired Successes (k)** — min 0 — Number of successes you want in the draw.

## Outputs

- **P(X = k)** — Probability of exactly k successes.
- **Expected Value E(X)** — Expected number of successes in the draw.
- **Variance** — Variance of the hypergeometric distribution.
- **P(X <= k)** — Cumulative probability of k or fewer successes.
- **Formula** — formatted as text — The hypergeometric formula applied.

## Details

The hypergeometric distribution models the number of successes in n draws from a finite population of N items containing K successes, without replacement. Unlike the binomial distribution, each draw changes the probability of the next.

Formula: P(X = k) = C(K, k) x C(N-K, n-k) / C(N, n), where C(a, b) is the binomial coefficient. The key properties are: E(X) = nK/N and Var(X) = n(K/N)((N-K)/N)((N-n)/(N-1)). The factor (N-n)/(N-1) is called the finite population correction.

Classic examples: Drawing cards from a deck (what is the probability of getting exactly 3 hearts in a 5-card hand?), quality control (testing a batch of products without replacement), ecology (capture-recapture methods for estimating population sizes), and lottery calculations where order does not matter.

## Frequently Asked Questions

**Q: What is the difference between hypergeometric and binomial distributions?**

A: The binomial distribution assumes sampling with replacement (or from an infinite population), so each trial has the same probability. The hypergeometric distribution models sampling without replacement from a finite population, where each draw changes the composition. As the population gets very large relative to the sample, the hypergeometric approaches the binomial.

**Q: How is this used in card games?**

A: A standard deck has 52 cards with 13 of each suit. To find the probability of drawing exactly 3 hearts in a 5-card hand: set N=52, K=13, n=5, k=3. The result is C(13,3) x C(39,2) / C(52,5) = approximately 8.15%.

**Q: What is the finite population correction factor?**

A: The factor (N-n)/(N-1) in the variance formula. It accounts for the fact that sampling without replacement from a finite population reduces variability compared to sampling with replacement. When N is much larger than n, this factor is close to 1.

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Source: https://vastcalc.com/calculators/statistics/hypergeometric
Category: Statistics
Last updated: 2026-04-21