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प्रश्न
A company has two plants to manufacture bicycles. The first plant manufactures 60% of the bicycles and the second plant 40%. Out of the 80% of the bicycles are rated of standard quality at the first plant and 90% of standard quality at the second plant. A bicycle is picked up at random and found to be standard quality. Find the probability that it comes from the second plant.
उत्तर
Let A, E1 and E2 denote the events that the cycle is of standard quality, plant I is chosen and plant II is chosen, respectively.
\[\therefore P\left( E_1 \right) = \frac{60}{100}\]
\[ P\left( E_2 \right) = \frac{40}{100} \]
\[\text{ Now } , \]
\[P\left( A/ E_1 \right) = \frac{80}{100}\]
\[P\left( A/ E_2 \right) = \frac{90}{100}\]
\[\text{ Using Bayes' theorem, we get} \]
\[\text{ Required probability } = P\left( E_2 /A \right) = \frac{P\left( E_2 \right)P\left( A/ E_2 \right)}{P\left( E_1 \right)P\left( A/ E_1 \right) + P\left( E_2 \right)P\left( A/ E_2 \right) + P\left( E_3 \right)P\left( A/ E_3 \right)}\]
\[ = \frac{\frac{40}{100} \times \frac{90}{100}}{\frac{60}{100} \times \frac{80}{100} + \frac{40}{100} \times \frac{90}{100}}\]
\[ = \frac{36}{48 + 36} = \frac{36}{84} = \frac{3}{7}\]
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