# 7.4 - Hypergeometric Distribution

7.4 - Hypergeometric Distribution

## Example 7-7

A crate contains 50 light bulbs of which 5 are defective and 45 are not. A Quality Control Inspector randomly samples 4 bulbs without replacement. Let $$X$$ = the number of defective bulbs selected. Find the probability mass function, $$f(x)$$, of the discrete random variable $$X$$.

This example is an example of a random variable $$X$$ following what is called the hypergeometric distribution. Let's generalize our findings.

Hypergeometric distribution

If we randomly select $$n$$ items without replacement from a set of $$N$$ items of which:

$$m$$ of the items are of one type and $$N-m$$ of the items are of a second type

then the probability mass function of the discrete random variable $$X$$ is called the hypergeometric distribution and is of the form:

$$P(X=x)=f(x)=\dfrac{\dbinom{m}{x} \dbinom{N-m}{n-x}}{\dbinom{N}{n}}$$

where the support $$S$$ is the collection of nonnegative integers x that satisfies the inequalities:

$$x\le n$$ $$x\le m$$ $$n-x\le N-m$$

Note that one of the key features of the hypergeometric distribution is that it is associated with sampling without replacement. We will see later, in Lesson 9, that when the samples are drawn with replacement, the discrete random variable $$X$$ follows what is called the binomial distribution.

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