A consluting firm rents cars from three agencies 20 from age

A consluting firm rents cars from three agencies, 20% from agency A, 30% from agency B, and 50% from agency C. 10% of the cars from A, 12% of the cars from B, and 5% of the cars from C have bad tires. If the car rented has good tires, what is the probability that it was rented from agency C?

Solution

Let

A, B , C = agency where the car came from
G = the car has good tires

Thus,

P(C|G) = P(C) P(G|C) / P(G)

and

P(G) = P(A) P(G|A) + P(B) P(G|B) + P(C) P(G|C) = 0.20*(1-0.20) + 0.30*(1-0.12) + 0.50*(1-0.05) = 0.899

Thus,

P(C|G) = 0.50*(1-0.05)/0.899 = 0.52836485 [ANSWER]