Chapter 12.5, Problem 17PPS

To determine

To state whether the events are mutually exclusive or not mutually exclusive and find the probability.

Answer to Problem 17PPS

The events are not mutually exclusive and the probability that a randomly selected essay will have the required length and the required grade-point average is $\frac{2956}{3000}\text{ or 98}\text{.5\%}$ .

Explanation of Solution

Given information:

The given statement is:

“ $3000$ essays were received for a $\$5000$ college scholarship. $2865$ essays were the required length, $2577$ of the applicants had the minimum required grade-point average, and $2486$ had the required length and minimum grade-point average. What is the probability that an essay selected at random will have the required length or the required grade-point average?"

Formula used:

The following formula is used:

  $\text{Probability}=\frac{\text{Number of possible outcomes}}{\text{Total number of outcomes}}$

Calculation :

The events are not mutually exclusive because some essays had both the required length and the minimum grade-point average.

The probability can be calculated by;

  $\begin{array}{l}\text{P}\left(\text{minimum length}\right)=\frac{2865}{3000}\\ \text{P}\left(\text{minimum required grade point average}\right)=\frac{2577}{3000}\\ \text{P}\left(\text{minimum length and minimum required grade point average}\right)=\frac{2486}{3000}\\ \text{P}\left(\text{minimum length or minimum required grade point average}\right)=\frac{2865}{3000}+\frac{2577}{3000}-\frac{2486}{3000}\\ \text{P}\left(\text{minimum length or minimum required grade point average}\right)=\frac{2956}{3000}\text{ or 98}\text{.5\%}\end{array}$

Hence, the events are not mutually exclusive and the probability that a randomly selected essay will have the required length and the required grade-point average is $\frac{2956}{3000}\text{ or 98}\text{.5\%}$ .