Question

In a bolt manufacturing factory, machines A, B, and C produce 25%, 30% and 45%
of the total output, respectively. Of their outputs, 7%, 6% and 4% are defective bolts,
respectively. If a bolt drawn at random from the production is found to be defective,
what is the probability that it was manufactured by machine C?

Answer 1

A : Bolt Manufactured by machine A

B : Bolt Manufactured by machine B

C : Bolt Manufactured by machine C

D : Bolt is defective

We need to find the probability that the bolt is manufactured by machine C , if it is defective i.e. $P(C/D)$

$P(A) = 0.25,$ $P(D/A)=0.07,$

$P(B) = 0.30,$ $P(D/B)=0.06$

$P(C) = 0.45,$ $P(D/C)=0.04$

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

$P(C/D) = \frac {(0.04).(0.45)}{(0.25).(0.07)+(0.30).(0.06)+(0.45).(0.04)} = \frac{36}{107}$