Chapter 13.5, Problem 33E

To determine

To calculate: The probability that a student has brown eyes provided that the student has brown hair.

Answer to Problem 33E

The probability that a student has brown eyes provided that the student has brown hairis $\frac{1}{6}$ .

Explanation of Solution

Given information:

Percentage of students with brown hair is $60\%$ , percentage of students with brown eyes is $30\%$ and percentage of students with both brown hair and brown eyes is $10\%$ .

Formula used:

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

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 thatPercentage of students with brown hair is $60\%$ , percentage of students with brown eyes is $30\%$ and percentage of students with both brown hair and brown eyes is $10\%$ .

Let A be an event that student also has brown eyes and B be an event student has brown hair.

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

Probability that student has brown hair,

  $P\left(B\right)=\frac{60}{100}$

Probability that student has brown eyes also,

  $P\left(A\text{ and B}\right)=\frac{10}{100}$

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 probability that a student has brown eyes provided that the student has brown hair.

  $\begin{array}{c}P\left(A|B\right)=\frac{\frac{10}{100}}{\frac{60}{100}}\\ =\frac{10}{60}\\ =\frac{1}{6}\end{array}$

Thus, the probability that a student has brown eyes provided that the student has brown hairis $\frac{1}{6}$ .