Question

In a bolt manufacturing factory, machines A, B, and C produce 25%, 30% and 45% of the total outputs, respectively. Of their outputs, 7%, 6% and 4% are defective bolt, respectively.

(i) What is the probability that a bolt drawn at random from production will be defective?

(ii) If a bolt drawn at random from production is found to be defective, what is the probability that it was manufactured by machine C?

Answer 1

D = event that bolt is defective

P(A)=0.25

P(B)=0.35

P(C)=0.40

P(D|A) = 0.07

P(D|B)=0.06

P(D|C)=0.04

(i) $P(D) = P(A) \times P(D|A) + P(B) \times P(D|B) + P(C) \times P(D|C)$

$= 0.25 \times 0.07 + 0.35 \times 0.06 + 0.40 \times 0.04 \\ = 0.0175+0.021+0.016 \\ = 0.0545$

(ii)

$P(C|D) = \frac{P(D|C) \times P(C)}{P(D)} \\ = \frac{0.04 \times 0.40}{0.0545} \\ = 0.2935$