In a bolt factory, machines A, B and C manufacture respectively 20%, 30% and 50% of the total bolts. Of their output 3%, 4%, and 2% are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found defective, then th

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Solution

Q.
In a bolt factory, machines A, B and C manufacture respectively 20%, 30% and 50% of the total bolts. Of their output 3%, 4%, and 2% are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found defective, then the probability that it is manufactured by the machine C is [2023]

1 $\frac{3}{7}$

2 $\frac{9}{28}$

3 $\frac{5}{14}$

4 $\frac{2}{7}$

Ans.
(3)

$P (A) = \frac{20}{100},$ $P (B) = \frac{30}{100},$ $P (C) = \frac{50}{100}$

$Let D be the event that the product is defective, then$

$P (D / A) = \frac{3}{100},$ $P (D / B) = \frac{4}{100},$ $P (D / C) = \frac{2}{100}$

$Now, by Bayes' theorem, we have$

$P (C / D) = \frac{P (C) \cdot P (D / C)}{P (A) \cdot P (D / A) + P (B) \cdot P (D / B) + P (C) \cdot P (D / C)}$

$= \frac{\frac{50}{100} \times \frac{2}{100}}{\frac{20}{100} \times \frac{3}{100} + \frac{30}{100} \times \frac{4}{100} + \frac{50}{100} \times \frac{2}{100}} = \frac{5}{14}$

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