# 7.5 - More Examples

7.5 - More Examples## Example 7-8

A lake contains 600 fish, eighty (80) of which have been tagged by scientists. A researcher randomly catches 15 fish from the lake. Find a formula for the probability mass function of \(X\), the number of fish in the researcher's sample which are tagged.

#### Solution

This problem is very similar to the example on the previous page in which we were interested in finding the p.m.f. of \(X\), the number of defective bulbs selected in a sample of 4 bulbs. Here, we are interested in finding \(X\), the number of tagged fish selected in a sample of 15 fish. That is, \(X\) is a hypergeometric random variable with \(m = 80\), \(N = 600\), and \(n = 15\). Therefore, the p.m.f. of \(X\) is:

for the support \(x=0, 1, 2, \ldots, 15\).

## Example 7-9

Let the random variable \(X\) denote the number of aces in a five-card hand dealt from a standard 52-card deck. Find a formula for the probability mass function of \(X\).

#### Solution

The random variable \(X\) here also follows the hypergeometric distribution. Here, there are \(N=52\) total cards, \(n=5\) cards sampled, and \(m=4\) aces. Therefore, the p.m.f. of \(X\) is:

\(f(x)=\dfrac{\dbinom{4}{x} \dbinom{48}{5-x}}{\dbinom{52}{5}}\)

for the support \(x=0, 1, 2, 3, 4\).

## Example 7-10

Suppose that 5 people, including you and a friend, line up at random. Let the random variable \(X\) denote the number of people standing between you and a friend. Determine the probability mass function of \(X\)* *in tabular form. Also, verify that the p.m.f. is a valid p.m.f.