Prob.4 Customers who purchase a certain car can order an engine in any of three

Question

Prob.4 Customers who purchase a certain car can order an engine in any of three sizes. Of all cars sold, 45% have small-size engines, 35% have medium-size, and 20% have large-size engines. Of the cars with small size engines, 10% fail emission test within two years of purchase, while 12% of those with medium size and 15% of large size engine fail. (a) What is the probability that a randomly chosen car will fail emission test within two years? (b) A record for failed emission test is chosen at random. What is the probability that it is for a car with large size engine?

Explanation / Answer

Let M shows the car sold has medium size engine, S shows the car sold has small size engine and L shows the car sold has large size engine.

So we have

P(M) = 0.35, P(S) = 0.45, P(L) = 0.20

Let F shows the event car fail emission test. So

P(F|M) = 0.12, P(F|S) = 0.10, P(F|L) = 0.15

(a)

By the law of total probability, the probability that car fail emission is

P(F) = P(F|M)P(M)+P(F|S)P(S)+P(F|L)P(L) = 0.12 * 0.35 + 0.10 * 045 + 0.15* 0.20 =0.117

(b)

P(L|F) = [ P(F|L)P(L) ] / P(F) = [0.15*0.20] / 0.117 = 0.2564

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