Chapter 11.8, Problem 9E

To determine

To determine whether the events are disjoint or overlapping and find $\text{P}\left(\text{A or B}\right)$ .

Answer to Problem 9E

The events A and B are overlapping.

The probability $\text{P}\left(\text{A or B}\right)=\frac{2}{3}$ .

Explanation of Solution

Given information:

The following spinner is divided into equal parts:

  

Event A: Stops on an odd number.

Event B: Stops on a multiple of $3$ .

Formula used:

The following formula is used:

  $\text{P}\left(\text{A or B}\right)=\text{P}\left(\text{A}\right)+\text{P}\left(\text{B}\right)-\text{P}\left(\text{A and B}\right)$

Calculation:

The events A and B are overlapping because $3$ is an oddnumber and is a multiple of 3.

The probability $\text{P}\left(\text{A or B}\right)$ can be calculated by:

  $\begin{array}{l}\text{Probability of event A, P}\left(\text{A}\right)=\frac{3}{6}\\ \text{Probability of event B, P}\left(\text{B}\right)=\frac{2}{6}\\ \text{Probability of }\left(\text{A and B}\right)\text{, P}\left(\text{A and B}\right)=\frac{1}{6}\\ \text{P}\left(\text{A or B}\right)=\text{P}\left(\text{A}\right)+\text{P}\left(\text{B}\right)-\text{P}\left(\text{A and B}\right)\left[\text{Probability of overlapping events}\right]\\ \text{P}\left(\text{A or B}\right)=\frac{3}{6}+\frac{2}{6}-\frac{1}{6}\left[\text{Substitute the values}\right]\\ \text{P}\left(\text{A or B}\right)=\frac{4}{6}=\frac{2}{3}\left[\text{Simplify}\right]\end{array}$

Hence, the probability $\text{P}\left(\text{A or B}\right)=\frac{2}{3}$ .