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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.
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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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