Question

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

Answer 1

Let A, E₁ and E₂ denote the events that the IQ is more than 150, the selected student is a boy and the selected student is a girl, respectively. 

\[\therefore P\left( E_1 \right) = \frac{60}{100} \]
\[ P\left( E_2 \right) = \frac{40}{100}\]
\[\text{ Now } , \]
\[P\left( A/ E_1 \right) = \frac{5}{100}\]
\[P\left( A/ E_2 \right) = \frac{10}{100}\]
\[\text{ Using Bayes' theorem, we get } \]
\[\text{ Required probability } = P\left( E_1 /A \right) = \frac{P\left( E_1 \right)P\left( A/ E_1 \right)}{P\left( E_1 \right)P\left( A/ E_1 \right) + P\left( E_2 \right)P\left( A/ E_2 \right)}\]
\[ = \frac{\frac{60}{100} \times \frac{5}{100}}{\frac{60}{100} \times \frac{5}{100} + \frac{40}{100} \times \frac{10}{100}}\]
\[ = \frac{300}{300 + 400} = \frac{300}{700} = \frac{3}{7}\]