problem 03 Customers who purchase a certain make of car can order an engine in a

Question

problem 03

Customers who purchase a certain make of car can order an engine in any of three sizes. Of all cars sold, 40% have the smallest engine, 35% have the medium-sized one, a have the largest. Of cars with the smallest engine, 12% fails an emissions test within two years of purchase, while 14% of those with the medium size and 16% of those with the 1. nd 25% largest engine. (a) What is the probability that a randomly chosen car will fail an emissions test within two years? (b) A record for a failed emissions test is chosen at random. What is the probability that it is for a car with a small engine?

Explanation / Answer

Given probabilities are:

P(Small Engine) = 0.4; P(Medium Engine) = 0.35 ; P(Large Engine) = 0.25

P(Failure given Small Engine) = P(Failure | Small) = 0.12

Likewise, P(Failure | Medium) = 0.14 and P(Failure | Large) = 0.16.

a) P (Failure) = P(Small)*P(Failure | Small) + P(Medium)*P(Failure | Medium) + P(Large)*P(Failure | Large)

= (0.4*0.12) + (0.35*0.14) + (0.25*0.16) = 0.048 + 0.049 + 0.04 = 0.137

So, P(Failure of emission within 2 years) = 0.137

b) The expected probability is P(Small given it has Failed Emission).

The conditional probability of this is given by:

P(Small | Failure) = [P(Failure | Small) * P (Small)] / P (Failure)

We know P (Failure | Small) = 0.12, P (Small) = 0.4 and from section a) P(Failure) = 0.137.

So, P (Small | Failure) = (0.12*0.4) / 0.137 = 0.048 / 0.137 = 0.35

Probability that the failed emission test is of the small car = 0.35

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