Problem 11 A company is buying certain machine pieces from t

Problem 11 A company is buying certain machine pieces from three different vendors. Vendor A^?s machine you get are broken 15% of times vendor B^?s 20% and vendor C 30% of time. Overall the company receives broken machines 20% of time. An inspector randomly chooses one of the broken machines. What is a probability that the machine is from vendor A?

Solution

Note that

P(A and broken) = P(A) P(broken|A)

If all vendors have the same proportion of parts, then

P(A) = P(B) = P(C) = 1/3

Thus,

P(A and broken) = (1/3)(0.15) = 0.05

Now,

P(A|broken) = P(A and broken) / P(broken)

as

P(broken) = 0.20,

P(A|broken) = P(A and broken) / P(broken) = 0.05/0.20

= 0.25 [ANSWER]