Chapter 3, Problem 24CT

To determine

To calculate: The number of runners that run only during the week.

Answer to Problem 24CT

The number of runners that run only during the week is 10.

Explanation of Solution

Given information:

Of $145$ runners, $72$ run only on weekends and $63$ run on both weekends and during the week.

Formula used:

If A and B are two mutually exclusive events then $\text{P}\left(\text{A}\cup \text{B}\right)=\text{P}\left(\text{A}\right)+\text{P}\left(\text{B}\right)$ .

Calculation:

Consider the provided information that of $145$ runners, $72$ run only on weekends and

  $63$ run on both weekends and during the week..

Recall that if A and B are two sets then $\text{P}\left(\text{A}\cup \text{B}\right)=\text{P}\left(\text{A}\right)+\text{P}\left(\text{B}\right)$ .

Let A represent the set of runners that run on weekends and B represents the set of runners that run during the week.

Here, $\text{P}\left(\text{A}\cup \text{B}\right)=145$ , and $\begin{array}{c}\text{P}\left(\text{A}\right)=63+72\\ =135\end{array}$ .

Substitute the values in the formula and solve it as,

  $\begin{array}{c}\text{P}\left(\text{A}\cup \text{B}\right)=\text{P}\left(\text{A}\right)+\text{P}\left(\text{B}\right)\\ 145=135+\text{P}\left(\text{B}\right)\\ 145-135=\text{P}\left(\text{B}\right)\\ \text{P}\left(\text{B}\right)=10\end{array}$

Thus, number of runners that run only during the week is 10.