Factories A B and C produce computers Factory A produces 4 t

Factories A, B and C produce computers. Factory A produces 4 times as many computers as factory C, and factory B produces 6 times as many computers as factory C. The probability that a computer produced by factory A is defective is 0.037, the probability that a computer produced by factory B is defective is 0.033, and the probability that a computer produced by factory C is defective is 0.042.
A computer is selected at random and it is found to be defective. What is the probability it came from factory B?

Solution

Let D = defective

A, B, C = that they came from those respective factories

Here,

P(A) + P(B) + P(C) = 1

As P(A) = 4P(C), P(B) = 6P(C),

4P(C) + 6P(C) + P(C) = 1

P(C) = 1/11
P(A) = 4/11
P(B) = 6/11

Now,

P(D) = P(A) P(D|A) + P(B) P(D|B) +P(C) P(D|C) = (4/11)*(0.037) + (6/11)*(0.033) + (1/11)*(0.042) =

P(D) = 0.035272727

Also,

P(B and D) = P(B) P(D|B) = (6/11)*(0.033) = 0.018

Thus,

P(B|D) = P(B and D) / P(D) = 0.018/0.035272727

=0.510309278 [ANSWER]