Chapter 12.5, Problem 36HP

To determine

To write a formula to calculate $\text{P}\left(\text{A or B or C}\right)$ .

Answer to Problem 36HP

The formula for the probability $\text{P}\left(\text{A or B or C}\right)=\text{P}\left(\text{A}\right)+\text{P}\left(\text{B}\right)+\text{P}\left(\text{C}\right)-\text{P}\left(\text{A and B and C}\right)$ .

Explanation of Solution

Given information:

The given statement is:

“Suppose there are three events A, B and C that are not mutually exclusive. List all the probabilities you would need to consider in order to calculate $\text{P}\left(\text{A or B or C}\right)$ . Then write the formula you would use to calculate it.

The probabilities needed to be considered are:

  $\begin{array}{l}\text{P}\left(\text{A or B}\right)\\ \text{P}\left(\text{A or C}\right)\\ \text{P}\left(\text{B or C}\right)\\ \text{P}\left(\text{A and B}\right)\\ \text{P}\left(\text{A and C}\right)\\ \text{P}\left(\text{B and C}\right)\\ \text{P}\left(\text{A and B and C}\right)\end{array}$

The formula used is:

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

Hence, the formula for the probability $\text{P}\left(\text{A or B or C}\right)=\text{P}\left(\text{A}\right)+\text{P}\left(\text{B}\right)+\text{P}\left(\text{C}\right)-\text{P}\left(\text{A and B and C}\right)$ .