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 0.03 and the probability that an item produced by factory B is defective is 0.045. A computer is selected at random and it is found to be defective. What is the probability it came from factory A?

P (A∣D) = 0.667

Step-by-step explanation:

We are given;

P (A) = 3P (B)

P (D|A) = 0.03

P (D|B) = 0.045

Now, we want to find P (A∣D) which is the posterior probability that a computer comes from factory A when given that it is defective.

Using Bayes' Rule and Law of Total Probability, we will get;

P (A∣D) = [P (A) * P (D|A) ]/[ (P (A) * P (D|A)) + (P (B) * P (D|B)) ]

Plugging in the relevant values, we have;

P (A∣D) = [3P (B) * 0.03]/[ (3P (B) * 0.03) + (P (B) * 0.045) ]

P (A∣D) = [P (B) / P (B) ] [0.09]/[0.09 + 0.045]

P (B) will cancel out to give;

P (A∣D) = 0.09/0.135

P (A∣D) = 0.667