Answer to Question #141441 in Statistics and Probability for Soong

Question

A consulting firm rent cars from two agencies, 45% from agency P and 55% from agency
Q. It is known that 8% of the cars from agency P and 6% of the cars from agency Q have
bad tyres.

iii) If a car rented by firms has good tyres, find the probability that it came from
agency Q.

Answer 1

Probability of having a car rented from different agencies denoted by P, Q:

P(P) = 0.45

P(Q) = 0.55

Probability of having a car with bad tyres denoted by bad given corresponding agency:

P(bad|P) = 0.08

P(bad|Q) = 0.06

Hence:

P(good|P) = 1 – 0.08 = 0.92

P(good|Q) = 1 – 0.06 = 0.94

Probability of having a car with a good tyres coming from agency Q is denoted P(Q|good).

$P(Q|good)=\frac{P(Q\cap good)}{P(good)}=\frac{0.55\cdot0.94}{0.55\cdot0.94+0.45\cdot0.92}=0.56$