Sampling Without Replacement

Unlike the binomial distribution, where trials are independent, the hypergeometric distribution describes scenarios where objects are drawn without replacement. Suppose an urn contains $N$ items of which $K$ are labeled as successes. Drawing $n$ items without replacement leads to dependent outcomes because removing one item affects the probabilities of future draws.

Calculating the PMF

The probability of obtaining exactly $k$ successes in your sample equals

$P\left(k\right)=\frac{\text{C(K,k) C(N-K,n-k)}}{\text{C(N,n)}}$

This expression counts the number of favorable combinations over the total number of ways to draw $n$ items from $N$.

Cumulative Probability

The CDF sums the probabilities for all values up to $k$. It tells you how likely it is to see at most $k$ successes in your sample. This is useful for quality inspection, card games, and any situation involving finite populations.