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.

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