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प्रश्न
A is known to speak truth 3 times out of 5 times. He throws a die and reports that it is one. Find the probability that it is actually one.
उत्तर
Let A, E1 and E2 denote the events that the man reports the appearance of 1 on throwing a die, 1 occurs and 1 does not occur, respectively.
\[\therefore P\left( E_1 \right) = \frac{1}{6} \]
\[ P\left( E_2 \right) = \frac{5}{6}\]
\[\text{ Now } , \]
\[P\left( A/ E_1 \right) = \frac{3}{5}\]
\[P\left( A/ E_2 \right) = \frac{2}{5}\]
\[\text{ Using Bayes' theorem, we get }\]
\[\text{ Required probability } = P\left( E_1 /A \right) = \frac{P\left( E_1 \right)P\left( A/ E_1 \right)}{P\left( E_1 \right)P\left( A/ E_1 \right) + P\left( E_2 \right)P\left( A/ E_2 \right)}\]
\[ = \frac{\frac{1}{6} \times \frac{3}{5}}{\frac{1}{6} \times \frac{3}{5} + \frac{5}{6} \times \frac{2}{5}}\]
\[ = \frac{3}{3 + 10} = \frac{3}{13}\]
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