A company has two plants to manufacture bicycles. The first plan manufactures $60 %$ of the bicycles and second plant $40 %$ . $80 %$ of the bicycles are rated of the first plant and $90 %$ of standard quality at the second plant. A bicycle is picked up at random and found to be of standard quality. Find the probability that it comes from the second plant.

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Solution

Let $E_{1}$ be the event the cycle is from the first plant
Let $E_{2}$ be the event the cycle is from $2^{n d}$ plant

Let $A$ be the event the cycle is standard

$P (E_{1}) = 0.6$ and $P (E_{2}) = 0.4$

$P (A / E_{1}) = 0.8$ and $P (A / E_{2}) = 0.9$

From Bayer theorem,

$P (E_{2} / A) = \frac{P (E_{2}) P (A / E_{2})}{P (E_{1}) P (A / E_{1}) + P (E_{2}) P (A / E_{2})}$

$= \frac{(0.4) (0.9)}{(0.6) (0.8) + (0.4) (0.9)}$

$= \frac{0.36}{0.84} = \frac{36}{84} = \frac{9}{21}$

$P (E_{2} / A) = \frac{3}{7}$ .

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