Free Hypergeometric Distribution Calculator

Introduction to the Hypergeometric Distribution Calculator

The hypergeometric distribution calculator is a dedicated tool for calculating probabilities associated with the hypergeometric probability distribution, a discrete probability distribution that models the number of successes in a fixed-size sample drawn without replacement from a finite population. This hypergeometric probability calculator computes both the probability mass function (PMF) and the cumulative distribution function (CDF) for given parameters, making it an indispensable resource for statistics, quality control, and games of chance.

What Is the Hypergeometric Distribution?

The hypergeometric distribution describes the probability of obtaining exactly $k$ successes (items with a particular attribute) in a sample of size $n$ drawn from a population of size $N$ that contains exactly $K$ successes. Critically, each item can be selected only once—sampling without replacement—so the probability of success changes as items are removed. This contrasts with binomial sampling, where replacement keeps probabilities constant.

The distribution appears naturally in many real-world contexts: drawing cards from a deck, selecting defective products from a batch, or in Fisher’s exact test for small contingency tables. The hypergeometric distribution probability calculation is essential whenever the population is finite and sampling is without replacement.

Parameters of the Hypergeometric Distribution

Four values define a specific hypergeometric scenario:

• $N$ – total number of items in the population.
• $K$ – number of “success” items in the population.
• $n$ – number of items drawn (sample size).
• $k$ – number of successes observed in the sample (used as input for the PMF).

A random variable $X$ following the hypergeometric distribution is denoted as $X \sim \text{Hypergeometric}(N, K, n)$.

Key Formulas

The probability of obtaining exactly $k$ successes is given by:

$$P(X = k) = \frac{\binom{K}{k} \binom{N - K}{n - k}}{\binom{N}{n}}$$

where $\binom{a}{b}$ is the binomial coefficient.

Equivalent factorial form:

$$P(X = k) = \frac{K! \ (N-K)! \ n! \ (N-n)!}{N! \ k! \ (K-k)! \ (n-k)! \ (N-K-n+k)!}$$

The mean (expected value) of the hypergeometric distribution is:

$$\mu = n \cdot \frac{K}{N}$$

The variance is:

$$\sigma^2 = \frac{n \cdot K \cdot (N - K) \cdot (N - n)}{N^{2} \cdot (N - 1)}$$

The standard deviation is the square root of variance:

$$\sigma = \sqrt{\sigma^{2}}$$

How to Use the Hypergeometric PMF and CDF Calculator

Operating this hypergeometric distribution calculator is straightforward:

1. Enter the population size $N$, number of successes in population $K$, and sample size $n$.
2. Specify the number of observed successes $k$.
3. Select the probability type: PMF for $P(X = k)$, or CDF for cumulative probabilities like $P(X > k)$, $P(X \geq k)$, $P(X < k)$, or $P(X \leq k)$.
4. The calculator instantly returns the requested probability. Below the result, it also displays the mean, variance, and standard deviation of the distribution for your parameters.

This tool eliminates tedious factorial computations, giving you accurate hypergeometric distribution probability results in seconds.

Worked Example

Consider a box containing 12 dark chocolate bars and 36 white chocolate bars (total $N = 48$, $K = 12$). You draw 10 bars at random without replacement ($n = 10$). What is the probability of drawing exactly 4 dark bars?

$$P(X = 4) = \frac{\binom{12}{4} \binom{36}{6}}{\binom{48}{10}} \approx 0.1474$$

Using the calculator, you can also find:

• $P(X \geq 4) \approx 0.2023$
• Mean: $\mu = 10 \times 12/48 = 2.5$
• Variance: $\sigma^2 \approx 1.5160$
• Standard deviation: $\sigma \approx 1.2313$

The hypergeometric CDF calculator can quickly sum the individual probabilities for you.

Hypergeometric vs. Binomial Distribution

Both distributions count successes in $n$ trials, but they differ in sampling method:

Feature	Hypergeometric	Binomial
Sampling	Without replacement	With replacement
Trial dependence	Dependent (population changes)	Independent (constant probability)
Population size	Finite, known	Infinite or very large
Success probability	Changes each draw	Constant $p$

When the population size $N$ is large relative to the sample size $n$, the hypergeometric distribution approximates the binomial distribution, but they remain distinct.

Use the hypergeometric distribution when:

• Sampling without replacement.
• The population is finite and not extremely large compared to the sample.

Use the binomial distribution when:

• Sampling with replacement.
• Each trial is independent with constant probability.

Conclusion

The hypergeometric distribution probability calculator offers a fast and reliable way to compute probabilities, means, and variances for sampling without replacement problems. Whether you are analyzing quality control samples, card games, or performing Fisher’s exact test, this hypergeometric probability calculator and sampling without replacement calculator will handle the heavy math.