Factories A, B and C produce computers. Factory A produces 3 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 0.02, the probability that a computer produced by factory B is defective is 0.025, and the probability that a computer produced by factory C is defective is 0.038.
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 relates the conditional probability of an event A given that event B has occurred, with the conditional probability of event B given that event A has occurred:
P(A|B) = (P(B|A) * P(A)) / P(B)
In this case, event A represents the computer coming from factory A, event B represents the computer being defective, and we want to find P(A|B), the probability that the computer came from factory A given that it is defective.
We are given the following probabilities:
P(A) = probability that a computer is from factory A = ?
P(B) = probability that a computer is defective = ?
P(B|A) = probability that a computer is defective given that it is from factory A = 0.02
We need to find the values of P(A) and P(B).
To find P(A), we need to find the probability that a randomly selected computer comes from factory A. From the given information, we know that factory A produces 3 times as many computers as factory C, and factory B produces 6 times as many computers as factory C.
Let's assume that factory C produces x computers. Then factory A produces 3x computers and factory B produces 6x computers.
The total number of computers produced is given by the sum of computers produced by all factories:
Total = x + 3x + 6x = 10x
The probability of selecting a computer from factory A is given by the ratio of computers produced by factory A to the total number of computers:
P(A) = (3x) / (10x) = 3/10 = 0.3 (or 30%)
To find P(B), we need to find the probability that a randomly selected computer is defective.
The total number of defective computers is the sum of defective computers produced by all factories:
Total defective = (3x * 0.02) + (6x * 0.025) + (x * 0.038)
The probability of selecting a defective computer is given by the ratio of the total number of defective computers to the total number of computers:
P(B) = Total defective / Total
Now we have P(A), P(B), and P(B|A), and we can calculate P(A|B) using Bayes' theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)