A company has two plants A and B to manufacture motorcycles. 60% motorcycles are manufactured at plant A and the remaining are manufactured at plant B. 80% of the motorcycles manufactured at plant A are rated of the standard quality, while 90% of the moto

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Solution

Q.
A company has two plants A and B to manufacture motorcycles. 60% motorcycles are manufactured at plant A and the remaining are manufactured at plant B. 80% of the motorcycles manufactured at plant A are rated of the standard quality, while 90% of the motorcycles manufactured at plant B are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If $p$ is the probability that it was manufactured at plant B, then $126 p$ is [2024]

1 64

2 56

3 66

4 54

Ans.
(4)

Let $E_{1}$ be the event that motorcycle manufactured in plant A.

$E_{2}$ be the event that motorcycle manufactured in plant B, and E be the event that it found to be of standard quality.

$∴$ $P (E_{1}) = \frac{60}{100}, P (E_{2}) = \frac{40}{100}$

$P (\frac{E}{E_{1}}) = \frac{80}{100}$ and $P (\frac{E}{E_{2}}) = \frac{90}{100}$

Now, $p = P (\frac{E_{2}}{E}) = \frac{P (E | E_{2}) \cdot P (E_{2})}{P (E | E_{2}) \cdot P (E_{2}) + P (E | E_{1}) \cdot (E_{1})}$

$= \frac{\frac{90}{100} \cdot \frac{40}{100}}{\frac{60}{100} \cdot \frac{80}{100} + \frac{90}{100} \cdot \frac{40}{100}} = \frac{36}{48 + 36} = \frac{3}{7}$

$∴$ $126 p = 126 \times \frac{3}{7} = 54$

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