7.5 - More Examples

Example 7-8 Section

a school of rainbow trout

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:

 

80𝑥 600-8015-𝑥 60015 # of ways to draw x tagged fish total # of ways of drawing 15 fish from the 600 in the lake # of ways to draw 15-x untagged fish

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

Example 7-9 Section

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 Section

people waiting in line

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.