This situation involves N items. a of the items are assigned the label 1 and N – a are assigned the label 0. Select at random a sample of n items from the N available without replacement. The number of items in the sample labeled 1 is the random variable of interest.

The values that result in nonzero probabilities are enumerated by the statement following the probability statement. All other values have 0 probability.

This distribution is used when samples are drawn from a larger, but finite set.

Example: : You are dealt a hand of 5 cards from a standard bridge deck of 52 cards. Before looking at the hand you wonder about the number of aces among the five cards. This is a case for the hypergeometric distribution.

For the example, there is a fixed number from which to draw (N = 52 cards), the number of aces in the deck (a = 4 aces) and the sample is 5 cards (n = 5). The probability of drawing four aces is very low. Of course the probability of drawing five aces is 0 because there are only four aces in the deck.