Hypergeometric Distribution Calculator

About Hypergeometric Distribution Calculator

The Hypergeometric Distribution Calculator computes exact probabilities for sampling without replacement scenarios. Enter your population size (N), the number of success items (K), the number of draws (n), and the desired number of successes (k) to instantly get point and cumulative probabilities with step-by-step combinatorial solutions and interactive visualizations.

What Is the Hypergeometric Distribution?

The hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population of size N containing exactly K success items, drawn without replacement. Unlike the binomial distribution — which assumes each trial is independent — the hypergeometric distribution accounts for the fact that each draw changes the composition of the remaining population.

The Hypergeometric PMF Formula

The probability mass function (PMF) is:

P(X = k) = C(K, k) × C(N − K, n − k) / C(N, n)

Where C(a, b) = a! / (b! × (a − b)!) is the binomial coefficient ("a choose b"). The numerator counts the favorable ways to choose k successes from K and (n − k) failures from (N − K). The denominator counts all possible ways to draw n items from N.

Parameters Explained

• N (Population Size) — Total number of items in the population.
• K (Success States) — Number of items classified as "success" in the population.
• n (Number of Draws) — How many items are drawn without replacement.
• k (Observed Successes) — The specific number of successes you want to find the probability for.

Mean, Variance, and Standard Deviation

For a hypergeometric random variable X:

• Mean: μ = nK / N
• Variance: σ² = n × (K/N) × ((N−K)/N) × ((N−n)/(N−1))
• Standard Deviation: σ = √σ²

The factor (N − n) / (N − 1) is called the finite population correction factor. It reduces the variance compared to the binomial, reflecting that sampling without replacement is less variable than sampling with replacement.

Hypergeometric vs. Binomial Distribution

• Hypergeometric: Sampling without replacement from a finite population. Each draw changes the probability of the next draw.
• Binomial: Sampling with replacement (or from an infinite population). Each trial has the same probability.
• When the population is very large relative to the sample (N ≫ n), the hypergeometric distribution approximates the binomial.

Common Applications

• Quality Control — What is the probability of finding exactly 3 defective items when inspecting 30 units from a batch of 500 containing 20 defectives?
• Card Games — What is the probability of being dealt exactly 2 hearts in a 5-card poker hand from a standard 52-card deck?
• Lottery Analysis — What are the odds of matching a certain number of drawn numbers?
• Ecology (Capture-Recapture) — Estimating wildlife populations by tagging and recapturing animals.
• Statistical Testing — Fisher's exact test uses the hypergeometric distribution to test independence in 2×2 contingency tables.

How to Use This Calculator

1. Enter the population size N (total items).
2. Enter the number of success states K (must be ≤ N).
3. Enter the number of draws n (must be ≤ N).
4. Enter the observed successes k (must be feasible for the given parameters).
5. Click "Calculate Probability" to see exact and cumulative probabilities, step-by-step solutions, a PMF bar chart, and an urn model visualization.

Frequently Asked Questions

What is the hypergeometric distribution used for?

The hypergeometric distribution is used whenever you sample from a finite population without replacement and want to know the probability of drawing a specific number of items with a particular characteristic. Common use cases include quality control inspection, card game probabilities, lottery odds, and ecological capture-recapture studies.

How is the hypergeometric distribution different from the binomial?

The key difference is replacement. The binomial assumes independent trials (with replacement), while the hypergeometric models dependent draws (without replacement). When the population is much larger than the sample, the two distributions converge.

What are the valid ranges for k?

The observed successes k must satisfy: max(0, n − (N − K)) ≤ k ≤ min(n, K). The lower bound ensures there are enough failure items for the remaining draws, and the upper bound ensures you don't exceed available successes or total draws.

Can I use this for Fisher's exact test?

Yes. Fisher's exact test computes probabilities using the hypergeometric distribution. If you have a 2×2 contingency table, you can use this calculator to compute the probability of observing the given cell counts under the null hypothesis of independence.

What is the finite population correction factor?

The factor (N − n) / (N − 1) in the variance formula accounts for sampling without replacement. It always reduces variance compared to the binomial. When n is small relative to N, this factor is close to 1 and the correction is negligible.