# Statistics - Hypergeometric Distribution

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

${h(x;N,n,K) = \frac{[C(k,x)][C(N-k,n-x)]}{C(N,n)}}$

Where −

• ${N}$ = items in the population

• ${k}$ = successes in the population.

• ${n}$ = items in the random sample drawn from that population.

• ${x}$ = 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:

${h(x;N,n,k) = \frac{[C(k,x)][C(N-k,n-x)]}{C(N,n)} \\[7pt] h(2; 52, 5, 26) = \frac{[C(26,2)][C(52-26,5-2)]}{C(52,5)} \\[7pt] = \frac{}{2598960} \\[7pt] = 0.32513 }$

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