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 0.037 and the probability that an item produced by factory B is defective is 0.05.
A computer is selected at random and it is found to be defective. What is the probability it came from factory A?
To find the probability that the defective computer came from Factory A, we can use Bayes' Theorem.
Bayes' Theorem states:
P(A | B) = P(B | A) * P(A) / P(B)
In this case:
A: The computer came from Factory A.
B: The computer is defective.
We need to find P(A | B), which is the probability that the computer came from Factory A given that it is defective.
P(B | A) = 0.037 (The probability that the computer is defective given that it came from Factory A).
P(A) = 3/4 (Factory A produces 3 times as many computers as Factory B, so the probability of selecting a computer from Factory A is 3/4).
P(B) = (P(B | A) * P(A)) + (P(B | not A) * P(not A)) (The probability that a computer is defective is the sum of the probabilities of being defective given that it came from Factory A and the probability of being defective given that it did not come from Factory A).
P(B) = (0.037 * 3/4) + (0.05 * 1/4)
P(B) = 0.02775 + 0.0125
P(B) = 0.04025
Now, we can calculate P(A | B) using Bayes' Theorem:
P(A | B) = (P(B | A) * P(A)) / P(B)
P(A | B) = (0.037 * 3/4) / 0.04025
P(A | B) = 0.111 / 0.04025
P(A | B) ≈ 0.276 (rounded to 3 decimal places)
Therefore, the probability that the defective computer came from Factory A is approximately 0.276.