Factories A and B produce computers Factory A produces 3 tim

Factories A and B produce computers. Factory A produces 3 times as many computers as factory B. The probability that an item produced by factory A is defective is 0.035 and the probability that an item produced by factory B is defective is 0.041.
A computer is selected at random and it is found to be defective. What is the probability it came from factory A?

Solution

As

P(A) + P(B) = 1

and

P(A) = 3P(B)

then

3P(B) + P(B) = 1

P(B) = 1/4

Thus,

P(A) = 3/4

*******************

Let = defective.

Now, note that

P(D) = P(A) P(D|A) + P(B) P(D|B) = (3/4)*0.035 + (1/4)*0.041 = 0.0365

Thus,

P(A|D) = P(A) P(D|A) / P(D)

= (3/4)*0.035/0.0365

= 0.719178082 [ANSWER]