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प्रश्न
Suppose 5 men out of 100 and 25 women out of 1000 are good orators. An orator is chosen at random. Find the probability that a male person is selected. Assume that there are equal number of men and women.
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उत्तर
Let A, E1 and E2 denote the events that the person is a good orator, is a man and is a woman, respectively.
\[\therefore P\left( E_1 \right) = \frac{1}{2} \]
\[ P\left( E_2 \right) = \frac{1}{2}\]
\[\text{ Now } , \]
\[P\left( A/ E_1 \right) = \frac{5}{100}\]
\[P\left( A/ E_2 \right) = \frac{25}{1000}\]
\[\text{ Using Bayes' theorem, we get} \]
\[\text{ Required probability } = P\left( E_1 /A \right) = \frac{P\left( E_1 \right)P\left( A/ E_1 \right)}{P\left( E_1 \right)P\left( A/ E_1 \right) + P\left( E_2 \right)P\left( A/ E_2 \right)}\]
\[ = \frac{\frac{1}{2} \times \frac{5}{100}}{\frac{1}{2} \times \frac{5}{100} + \frac{1}{2} \times \frac{25}{1000}}\]
\[ = \frac{1}{1 + \frac{1}{2}} = \frac{2}{3}\]
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