Question 498471: 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 0.021 and the probability that an item produced by factory B is defective is 0.024.
A computer is selected at random and it is found to be defective. What is the probability it came from factory A?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let x be the total number of computers that are built.
since 2 out of 3 come from factory A, then factory A builds (2/3)*x and factory B builds (1/3)*x.
the number of defective computers that come from factory A are .021 * (2/3) * x.
the number of defective computers that come from factory B are .024 * (1/3) * x.
simplifying these expressions gets:
the number of defective computers that come from factory A are .014 * x.
the number of defective computers that come from factory B are .008 * x.
therefore:
the probability that a defective computer comes from factory A is .014.
the probability that a defective computer comes from factory B is .008
the probability of getting a defective computer is .014 + .008 = .022.
as an example:
assume that 3000 computers were built.
2000 came from factory A.
1000 came from factory B.
.021 * 2000 = 42 were defective from factory A.
.024 * 1000 = 24 were defective from factory B.
total defective were 66.
probability of getting a defective computer from factory A is 42/3000 = .014
probability of getting a defective computer from factory B is 24/3000 = .008
probability of getting a defective computer is 66/3000 = .022
the question to be answered is:
A computer is selected at random and it is found to be defective. What is the probability it came from factory A?
This can be translated to:
given that a computer is defective, what is the probability that it came from factory A?
the formula for p(X given Y) is equal to p(X intersect Y) / p(Y)
in this problem we have:
p(X) = probability that a defective computer came from factory A.
p(Y) = probability that a defective computer came from factory A or factory B.
p(X intersect Y) = probability that a defective computer came from factory A and also came from factory A or factory B. That's the same as the probability that it came from factory A.
the formula of p(X intersect Y) / p(Y) beecomes .014/.022 = .6363636363 = .64
in our example we had:
3000 computers were built.
2000 came from factory A
1000 came from factory B
42 were defective from factory A
24 were defective from factory B
66 were defective from factory A or B
given that the computer was defective, then it had to be one of the 66 defective computers in the lot.
since there are 42 out of 66 defective that are from factory A, then the probability that a defective computer came from factory A would be 42/66 = .636363636 = .64 rounded to the nearest hundredth.
i believe this is the answer you are looking for.
let me know if otherwise.
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