Question

Customers who purchase an aluminum-structured Audi A8 can order an engine in any of three sizes: 2.0L, 2.4L, and 2.8L. Of all Audi A8 cars sold, 45% have the 2.0L engine, 35% have the 2.4L, and 20% have the 2.8L. Of cars with the 2.0L engine, 10% have been found to fail an emissions test within four years of purchase, while 12% of those with the 2.4L engine and 15% of those with the 2.8L engine fail the emissions test within four years. An Audi A8 record for a failed emissions test taken within four years of purchase is chosen at random.

What is the probability that it is for an Audi A8 car with a 2.4L engine?

Answer 1

The probability that an Audi A8 car with 2.4L engine is selected given that the car failed emissions test taken within four years of purchase is 0.3589.

Explanation:

Let's denote event as follows:

A = an Audi A8 car has a 2.0L engine.

B = an Audi A8 car has a 2.4L engine.

C = an Audi A8 car has a 2.8L engine.

X = an Audi A8 car failed emissions test taken within four years of purchase.

The information provided is:

`P(A)=0.45\\P(B)=0.35\\P(C)=0.20\\P(X|A)=0.10\\P(X|B)=0.12\\P(X|C)=0.15`

The probability that an Audi A8 car with 2.4L engine is selected given that the car failed emissions test taken within four years of purchase is:

`P(B|X)=(P(X|B)P(B))/(P(X))`

Compute the probability of an Audi A*8 car failed emissions test taken within four years of purchase as follows:

`P(X)=P(X|A)P(A)+P(X|B)P(B)+P(X|C)P(C)\\=(0.45*0.10)+(0.35*0.12)+(0.20*0.15)\\=0.117`

Compute the value of P (B|X) as follows:

`P(B|X)=(P(X|B)P(B))/(P(X))=(0.35*0.12)/(0.117) =0.3589`

Thus, the probability that an Audi A8 car with 2.4L engine is selected given that the car failed emissions test taken within four years of purchase is 0.3589.