i Customers who purchase a certain make of car can order an engine in any of th

Question

i Customers who purchase a certain make of car can order an engine in any of th or ai, cars sold 4S% have the smallest have the largest Of cars with the smallest engine, 10% fails ynars of purchase, while largest engine engine, 35% have the medium-sized one, and 20% an emissions test within two 12% of those with the medium size and 15% of those with the What is the probability that a randomly chosen car will fail an emissions test within two years (a) (b) A record for a tailed emissions test is chosen at random. What is the probability that it is for a car with a small engine?

Explanation / Answer

Here, we are given that:

P( smallest ) = 0.45, P( medium ) = 0.35 and P( largest ) = 0.2

Also, we are given that:

P( fail | smallest ) = 0.1,
P( fail | medium ) = 0.12
P( fail | largest ) = 0.15

a) Now using the law of total addition, we get the probability as:

P( fail ) = P( fail | smallest )P( smallest ) + P( fail | medium )P( medium ) + P( fail | largest ) P( largest )

P( fail ) = 0.1*0.45 + 0.12*0.35 + 0.15*0.2 = 0.117

Therefore 0.117 is the required probability here that a randomly chosen car will fail an emission test within two years.

b) Given that a record for a failed emission test is chosen at random, then the probability that it is for a car with small engine here is computed as: (Using bayes theorem )

= P( fail | smallest )P( smallest ) / 0.117

= 0.1*0.45 / 0.117

= 0.3846

Therefore 0.3846 is the required probability here.

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