Hypergeometric Distribution

Consider a statistical experiment where a sample of n observations are to be taken from a population of size N. The population contains k items that are labeled 'success' and N - k items that are labeled 'failure'. If a random variable X assumes the value equal to the number of successes in the sample of size n, then X has a hypergeometric distribution with parameters N, n and k. The random variable X is said to be hypergeometrically distributed with parameter N, n and k, and has the following probability mass function:

The mean and variance of a random variable X that is hypergeometrically distributed with parameter N, n and k are computed as follows:

EX. In a certain game show, a contestant is asked to choose a number of boxes from a set of 10 boxes. One of these boxes has a prize, while the rest are empty. The number of boxes that the contestant may choose ranges from 1 to 3.

The random variable X, the number of boxes chosen by the contestant that contain a prize, will have a value that is either 0 or 1. It is hypergeometrically distributed with parameters N = 10, n and k = 1.

If the contestant can only choose one box, n = 1.

If he or she can choose two boxes, n = 2.

If three boxes can be chosen, then n = 3.

As the number of boxes picked increases, the chances of choosing the box with the prize increases.