# 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.

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