Problem Statement
A manufacturing firm has three machines $A, B$, and $C$ producing bolts. Machine $A$ produces $30\%$, machine $B$ produces $45\%$, and machine $C$ produces $25\%$ of the total output. Past experience shows that $2\%, 3\%$, and $4\%$ of the bolts produced by machines $A, B$, and $C$ respectively are defective. A bolt is drawn at random from the output and is found to be defective.
Find the probability that it was manufactured by machine $B$.
Verified Solution & Marking Scheme
Define events and prior probabilities
Let $E_1, E_2, E_3$ be the events that the bolt was made by machine $A, B, C$ respectively.
$P(E_1) = 0.30, \quad P(E_2) = 0.45, \quad P(E_3) = 0.25$
Let $D$ be the event that the chosen bolt is defective.
$P(D|E_1) = 0.02, \quad P(D|E_2) = 0.03, \quad P(D|E_3) = 0.04$
Compute Total Probability of Defective Bolt P(D)
$P(D) = P(E_1)P(D|E_1) + P(E_2)P(D|E_2) + P(E_3)P(D|E_3)$
$P(D) = (0.30)(0.02) + (0.45)(0.03) + (0.25)(0.04)$
$P(D) = 0.006 + 0.0135 + 0.010 = 0.0295$
Apply Bayes Theorem for Machine B
$P(E_2|D) = \frac{P(E_2)P(D|E_2)}{P(D)} = \frac{0.0135}{0.0295} = \frac{135}{295} = \frac{27}{59} \approx 0.4576$