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

Answer:

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.

Step-by-step 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:

$P(B|X)=\frac{P(X|B)P(B)}{P(X)}$

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

$P(X)=P(X|A)P(A)+P(X|B)P(B)+P(X|C)P(C)\\=(0.45\times0.10)+(0.35\times0.12)+(0.20\times0.15)\\=0.117$

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

$P(B|X)=\frac{P(X|B)P(B)}{P(X)}=\frac{0.35\times0.12}{0.117} =0.3589$

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.