Problem Statement
An industrial factory has three machines $A, B,$ and $C$ producing bolts. Of their total production, machine $A$ produces $50\%$, machine $B$ produces $30\%$, and machine $C$ produces $20\%$. Of their respective outputs, $2\%, 3\%,$ and $4\%$ are defective bolts. A bolt is drawn at random from the total 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 a bolt is produced by machine $A, B, C$ respectively.
$$P(E_1) = \frac{50}{100} = 0.50, \quad P(E_2) = \frac{30}{100} = 0.30, \quad P(E_3) = \frac{20}{100} = 0.20$$
Let $D$ be the event that the bolt is defective.
Conditional probabilities:
$$P(D|E_1) = 0.02, \quad P(D|E_2) = 0.03, \quad P(D|E_3) = 0.04$$
Compute total probability of defective bolt P(D)
$$P(D) = P(E_1)P(D|E_1) + P(E_2)P(D|E_2) + P(E_3)P(D|E_3)$$
$$P(D) = (0.50 \times 0.02) + (0.30 \times 0.03) + (0.20 \times 0.04)$$
$$P(D) = 0.010 + 0.009 + 0.008 = 0.027$$
Apply Bayes Theorem to find P(E2 | D)
$$P(E_2|D) = \frac{P(E_2)P(D|E_2)}{P(D)} = \frac{0.30 \times 0.03}{0.027} = \frac{0.009}{0.027} = \frac{9}{27} = \frac{1}{3}$$