Question

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.

Answer 1

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.

\[\therefore P\left( E_1 \right) = \frac{60}{100}\]
\[ P\left( E_2 \right) = \frac{40}{100} \]
\[\text{ Now } , \]
\[P\left( A/ E_1 \right) = \frac{80}{100}\]
\[P\left( A/ E_2 \right) = \frac{90}{100}\]
\[\text{ Using Bayes' theorem, we get} \]
\[\text{ Required probability }  = P\left( E_2 /A \right) = \frac{P\left( E_2 \right)P\left( A/ E_2 \right)}{P\left( E_1 \right)P\left( A/ E_1 \right) + P\left( E_2 \right)P\left( A/ E_2 \right) + P\left( E_3 \right)P\left( A/ E_3 \right)}\]
\[ = \frac{\frac{40}{100} \times \frac{90}{100}}{\frac{60}{100} \times \frac{80}{100} + \frac{40}{100} \times \frac{90}{100}}\]
\[ = \frac{36}{48 + 36} = \frac{36}{84} = \frac{3}{7}\]