A hypergeometric random variable is the number of successes that result from a hypergeometric experiment. The probability distribution of a hypergeometric random variable is called a hypergeometric distribution.

Hypergeometric distribution is defined and given by the following probability function:

Formula

h(x;N,n,K)=[C(k,x)][C(N−k,n−x)]C(N,n)h(x;N,n,K)=[C(k,x)][C(N−k,n−x)]C(N,n)

Where −

·        NN = items in the population

·        kk = successes in the population.

·        nn = items in the random sample drawn from that population.

·        xx = successes in the random sample.

Example

Problem Statement:

Suppose we randomly select 5 cards without replacement from an ordinary deck of playing cards. What is the probability of getting exactly 2 red cards (i.e., hearts or diamonds)?

Solution:

This is a hypergeometric experiment in which we know the following:

·        N = 52; since there are 52 cards in a deck.

·        k = 26; since there are 26 red cards in a deck.

·        n = 5; since we randomly select 5 cards from the deck.

·        x = 2; since 2 of the cards we select are red.

We plug these values into the hypergeometric formula as follows:

h(x;N,n,k)=[C(k,x)][C(N−k,n−x)]C(N,n)h(2;52,5,26)=[C(26,2)][C(52−26,5−2)]C(52,5)=[325][2600]2598960=0.32513h(x;N,n,k)=[C(k,x)][C(N−k,n−x)]C(N,n)h(2;52,5,26)=[C(26,2)][C(52−26,5−2)]C(52,5)=[325][2600]2598960=0.32513

Thus, the probability of randomly selecting 2 red cards is 0.32513.