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Question
In a certain college, 4% of boys and 1% of girls are taller than 1.75 metres. Further more, 60% of the students in the colleges are girls. A student selected at random from the college is found to be taller than 1.75 metres. Find the probability that the selected students is girl.
Solution
Let A, E1 and E2 denote the events that the height of the student is more than 1.75 m, selected student is a girl and selected student is a boy, respectively.
\[\therefore P\left( E_1 \right) = \frac{60}{100} \]
\[ P\left( E_2 \right) = \frac{40}{100}\]
\[\text{Now }, \]
\[P\left( A/ E_1 \right) = \frac{1}{100}\]
\[P\left( A/ E_2 \right) = \frac{4}{100}\]
\[\text{ Using Bayes' theorem, we get } \]
\[\text{ Required probability } = P\left( E_1 /A \right) = \frac{P\left( E_1 \right)P\left( A/ E_1 \right)}{P\left( E_1 \right)P\left( A/ E_1 \right) + P\left( E_2 \right)P\left( A/ E_2 \right)}\]
\[ = \frac{\frac{60}{100} \times \frac{1}{100}}{\frac{60}{100} \times \frac{1}{100} + \frac{40}{100} \times \frac{4}{100}}\]
\[ = \frac{6}{6 + 16} = \frac{6}{22} = \frac{3}{11}\]
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