Factories A and B produce computers. Factory A produces 2 times as many computers as factory B. The probability that an item produced by factory A is defective is 0.012 and the probability that an item produced by factory B is defective is 0.04.

A computer is selected at random and it is found to be defective. What is the probability it came from factory A?

0.066

To find the probability that the defective computer came from factory A, we need to use Bayes' theorem. Bayes' theorem states that:

P(A|B) = P(B|A) * P(A) / P(B)

Where:
P(A|B) is the probability of event A occurring given that event B has already occurred.
P(B|A) is the probability of event B occurring given that event A has already occurred.
P(A) is the probability of event A occurring.
P(B) is the probability of event B occurring.

In this case, event A is the defective computer coming from factory A, and event B is the computer being defective.

We are given:
P(A) = 2/3 (since factory A produces two times as many computers as factory B)
P(B|A) = 0.012 (probability of a computer being defective given that it came from factory A)
P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)

We need to find P(B|not A), which is the probability of a computer being defective given that it did not come from factory A. Since Factory B produces computers with a probability of 0.04 being defective, P(B|not A) is 0.04.

Now let's compute P(B):
P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)
P(B) = 0.012 * (2/3) + 0.04 * (1/3)
P(B) = 0.008 + 0.0133
P(B) = 0.0213

Now let's plug these values into Bayes' theorem:

P(A|B) = P(B|A) * P(A) / P(B)
P(A|B) = 0.012 * (2/3) / 0.0213
P(A|B) = 0.024 / 0.0213
P(A|B) ≈ 1.126

Therefore, the probability that the defective computer came from factory A is approximately 1.126 or 112.6%.