Question

Three factories F1, F2 and F3 respectively produce 25%, 35% and 40% of the total number of electrical parts intended for the assembly of a machine. These factories respectively produce 1%, 2% and 3% of defective parts.

We notice : The event A : "the part is produced by the F1 factory"
The event B :  "the part is produced by the F2 factory"
The event C : "the part is produced by the F3 factory"
The event D : "the part is defective".

A) What is the probability that a randomly selected part will be defective?

B) Suppose you take a random part and it's defective. Calculate the probability that this part will be produced by the F2 factory.

Answer 1

P(A): Probability part is produced by the F1 factory = 0.25

P(B): Probability part is produced by the F2 factory = 0.35

P(C): Probability part is produced by the F3 factory = 0.4

P(D|A): Probability that part is defective given that it is from F1 factory = 0.01

P(D|B): Probability that part is defective given that it is from F2 factory = 0.02

P(D|C): Probability that part is defective given that it is from F3 factory = 0.03

a) P(D): Probability part is defective (Using Bayes Thorem) = P(A)*P(D|A) + P(B)*P(D|B) + P(C)*P(D|C)

= 0.25*0.01 + 0.35*0.02 + 0.4*0.03

= 0.0025+0.007+0.012

= 0.0215

b) P(B|D): Probability that part is produced from F2 factory given that it is defective = (P(B)*P(D|B))/P(D)

= (0.35*0.02)/0.0215

= 0.007/0.0215

= 0.326