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

Let

$A=$ the event "bolt is manufactured by machine A",

$B=$ the event "bolt is manufactured by machine B",

$C=$ the event "bolt is manufactured by machine C", and

$D=$ the event "bolt is defective".

Given

$P(A)=0.25, P(B)=0.3, P(C)=0.45,$

$P(D|A)=0.07, P(D|B)=0.06,P(D|C)=0.04$

Then by Bayes' Theorem

$$P(C|D)={P(D|C)P(C)\over P(D|A)P(A)+P(D|B)P(B)+P(D|C)P(C)}=$$

$$={0.04(0.45)\over 0.07(0.25)+0.06(0.3)+0.04(0.45)}={0.018\over 0.0535}\approx$$

$$\approx0.3364$$