Chapter 13.5, Problem 34E

To determine

To calculate: The probability that a student does not have brown hair provided that the student has brown eyes.

Answer to Problem 34E

The probability that a student does not have brown hair provided that the student has brown eyesis $\frac{2}{3}$ .

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 does not have brown hair and B be an event student has brown eyes.

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 eyes,

  $P\left(B\right)=\frac{30}{100}$

Now,

  $\begin{array}{c}P\left(A\text{'}\cap B\text{'}\right)=1-P\left(AUB\right)\\ =1-\left[P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)\right]\\ =1-\left[0.6+0.3-0.1\right]\\ =0.2\end{array}$

Probability that student does not have brown hair,

  $P\left(A\text{ and B}\right)=0.2$

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.

  $\begin{array}{c}P\left(A|B\right)=\frac{0.2}{\frac{30}{100}}\\ =\frac{0.2}{0.3}\\ =\frac{2}{3}\end{array}$

Thus, the probability that a student does not have brown hair provided that the student has brown eyesis $\frac{2}{3}$ .