In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of ${\displaystyle k}$ successes (random draws for which the object drawn has a specified feature) in ${\displaystyle n}$ draws, without replacement, from a finite population of size ${\displaystyle N}$ that contains exactly ${\displaystyle K}$ objects with that feature, where in each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of ${\displaystyle k}$ successes in ${\displaystyle n}$ draws with replacement.

The following conditions characterize the hypergeometric distribution:

A random variable ${\displaystyle X}$ follows the hypergeometric distribution if its probability mass function (pmf) is given by

where

The pmf is positive when ${\displaystyle \max(0,n+K-N)\leq k\leq \min(K,n)}$.

A random variable distributed hypergeometrically with parameters ${\displaystyle N}$, ${\displaystyle K}$ and ${\displaystyle n}$ is written ${\textstyle X\sim \operatorname {Hypergeometric} (N,K,n)}$ and has probability mass function ${\textstyle p_{X}(k)}$ above.

As required, we have

which essentially follows from Vandermonde's identity from combinatorics.