# Statistics Hypergeometric Distribution – lesscss

By | May 18, 2019

# 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.

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