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Probability Distribution Tool

Hypergeometric Distribution Calculator

Calculate hypergeometric probabilities for random sampling without replacement. Enter the population size, number of successes in the population, sample size, and observed successes to find exact, cumulative, and tail probabilities instantly.

Exact P(X = x) Cumulative Probability Mean & Variance Instant Results

Enter Distribution Values

All inputs should be whole numbers.

Total number of items in the population.
Total population items classified as successes.
Number of items selected without replacement.
Number of successes observed in the sample.
Hypergeometric probability applies when sampling is performed without replacement from a finite population containing two categories, such as success and failure.
Probability Guide

What Is the Hypergeometric Distribution?

The hypergeometric distribution is a discrete probability distribution used when a fixed number of items is selected from a finite population without replacement. The population contains a known number of items considered successes and the remaining items are failures.

The distribution calculates the probability of obtaining a particular number of successes in the sample. Because sampling is without replacement, the probability of success can change after each item is selected.

What Does “Without Replacement” Mean?

Without replacement means that once an item is selected, it is not returned to the population before the next selection. This creates dependence between successive selections and is one of the main reasons the hypergeometric distribution differs from the binomial distribution.

How It Works

How to Use the Hypergeometric Calculator

01

Enter Population Data

Enter the total population size N and the number of successes K contained in that population.

02

Enter Sample Information

Provide the number of items sampled n and the number of successes x you want to evaluate.

03

Calculate Probabilities

The calculator returns the exact probability plus left-tail and right-tail cumulative probabilities.

Example: Suppose a box contains 50 items, including 15 defective items. If 10 items are selected without replacement, this calculator can find the probability that exactly 4 of the selected items are defective.
Probability Formula

Hypergeometric Distribution Formula

The probability of observing exactly x successes is calculated using: P(X=x) = [C(K,x) × C(N−K,n−x)] / C(N,n)

In this formula, N is the total population size, K is the total number of successes in the population, n is the sample size, and x is the number of successes observed in the sample.

Possible Values of X

Not every integer from zero to n is always possible. The smallest possible value is max(0, n − (N − K)), while the largest possible value is min(n, K).

Distribution Statistics

Mean and Variance of a Hypergeometric Distribution

The expected number of successes is: Mean = n(K/N) . This tells you the average number of successes expected across many samples of the same size drawn under identical conditions.

The variance is: n(K/N)(1−K/N)((N−n)/(N−1)) . The final factor is known as the finite population correction and reflects the fact that items are sampled without replacement.

Comparison

Hypergeometric vs Binomial Distribution

Feature Hypergeometric Binomial
Replacement Without replacement Independent trials / constant probability
Population Finite population Often treated as effectively unlimited
Success Probability May change after each draw Remains constant
Typical Application Sampling objects from a fixed collection Repeated independent success/failure trials
If you are selecting objects from a finite collection and do not put each selected object back before the next draw, the hypergeometric model is often the appropriate distribution.
Common Questions

Hypergeometric Distribution Calculator FAQs

Use it when you are taking a fixed-size sample from a finite population without replacement and each population item can be classified into one of two categories, such as success or failure.
N is the total number of items in the population before any items are selected.
K is the total number of items in the population that are classified as successes.
P(X ≤ x) is the probability of observing x or fewer successes in the sample.
P(X ≥ x) is the probability of observing at least x successes in the sample.
No. The number of successes observed in a sample cannot exceed the total number of observations in that sample.
No. When sampling without replacement, the number of sampled items cannot exceed the total population size.
The probability calculations in this page run directly in the browser using JavaScript. The calculator does not need to send your entered values to a calculation server.