Chapter 13.5, Problem 41E

To determine

To calculate: The probability that she studied for the test.

Answer to Problem 41E

The probability that she studied for the test is $\frac{5}{6}$ .

Explanation of Solution

Given information:

The probability that she passes the test if she studied for it is $\frac{4}{5}$ . The probability that classes passes the test and studied for it is $\frac{2}{3}$ .

Formula used:

Probability of an event E is $P\left(\text{E}\right)=\frac{\text{number of outcome}}{\text{total number of outcome}}$ .

Given an event Q, the conditional probability of event A is $P\left(A|Q\right)=\frac{P\left(A\text{ and Q}\right)}{P\left(Q\right)}$ where $P\left(Q\right)\ne 0$ .

Calculation:

Consider the provided information probability that she passes the test if she studied for it is $\frac{4}{5}$ . The probability that classes passes the test and studied for it is $\frac{2}{3}$ .

Let A be the event she passed the test and B be the event that she studied.

Recall that probability of an event E is $P\left(\text{E}\right)=\frac{\text{number of outcome}}{\text{total number of outcome}}$ .

The probability that classes passes the test and studied for it is $\frac{2}{3}$ .

  $P\left(A\text{ and B}\right)=\frac{2}{3}$

Recall that given an event Q, the conditional probability of event A is $P\left(A|Q\right)=\frac{P\left(A\text{ and Q}\right)}{P\left(Q\right)}$ where $P\left(Q\right)\ne 0$ .

The she passes the test if she studied for it is $\frac{4}{5}$ .

  $\begin{array}{c}P\left(A|B\right)=\frac{\frac{2}{3}}{P\left(B\right)}\\ \frac{4}{5}=\frac{\frac{2}{3}}{P\left(B\right)}\\ P\left(B\right)=\frac{\frac{2}{3}}{\frac{4}{5}}\\ P\left(B\right)=\frac{5}{6}\end{array}$

Thus, the probability that she studied for the test is $\frac{5}{6}$ .