Hypergeometric Distribution Calculator

What Is the Hypergeometric Distribution?

The hypergeometric distribution describes the probability of drawing exactly $k$ success items in a sample of size $n$ from a finite population of size $N$ containing $K$ success items, without replacement. It applies to card draws, quality control sampling, and Fisher's exact test. Compare with the Binomial Probability Distribution Calculator for sampling with replacement.

Key Formulas

The probability mass function is:

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

$$\mu = \frac{nK}{N}, \quad \sigma^2 = \frac{nK(N-K)(N-n)}{N^2(N-1)}$$

Example: Chocolate Bar Sample

A bag has 12 dark and 36 white chocolate bars ($N = 48$, $K = 12$). You draw 10 bars without replacement and want exactly 4 dark bars ($n = 10$, $k = 4$). The probability is $P(X = 4) \approx 0.1474$, or about 14.74%.

Frequently Asked Questions

When should I use hypergeometric instead of binomial?

Use hypergeometric when sampling without replacement from a finite population where each draw changes the remaining composition. Use binomial when trials are independent with constant success probability.

What is the valid range for k?

$k$ must satisfy $\max(0, n - (N-K)) \leq k \leq \min(n, K)$. This ensures you cannot draw more successes than exist in the population or sample.

What does the mean represent?

The mean $nK/N$ is the expected number of success items in a random sample of size $n$ drawn without replacement.

Can I compute cumulative probabilities?

Yes. Select P(X ≤ k), P(X < k), P(X ≥ k), or P(X > k) from the compute dropdown to get cumulative hypergeometric probabilities.

How is this related to Fisher's exact test?

Fisher's exact test uses hypergeometric probabilities to test independence in 2×2 contingency tables. See our Fisher's Exact Test Calculator.

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