Question

In a class 5% of boys and 10% of girls have an I.Q of more than 150.In this class 60% of students are boys. If a student is selected at random and found to have an I.Q. of more than 150. Find the probability that the student is a boy.

• A. $\frac{3}{7}$
• B. $\frac{23}{7}$
• C. $\frac{3}{5}$
• D. None of these

Answer 1

The correct option is A $\frac{3}{7}$

Let us consider the problem

$E_1$ : Event that boys are selected

$E_2$ : Event that girls are selected

$A$ : event that have IQ $150$

Implies that,

$$\begin{array}{c}P\left(A/E_1\right)=\frac{5}{100}\\ P\left(A/E_2\right)=\frac{10}{100}\\ P\left(E_1\right)=\frac{60}{100},P\left(E_2\right)=\frac{40}{100}\\ P\left(A|E_1\right)=\frac{P\left(E_1\right)P\left(A|E_1\right)}{P\left(E_1\right)P\left(A|E_1\right)+P\left(E_2\right)P\left(A|E_2\right)}\\ P\left(E_1|A\right)=\frac{\frac{60}{100}\times \frac{5}{100}}{\frac{60}{100}\times \frac{5}{100}+\frac{40}{100}\times \frac{10}{100}}\\ P\left(E_1|A\right)=\frac{60\times 5}{60\times 5+40\times 10}\\ P\left(E_1|A\right)=\frac{300}{300+400}\\ P\left(E_1|A\right)=\frac{3}{7}\end{array}$$

Hence, the probability is $\frac{3}{7}$