Answer to Question #176958 in Statistics and Probability for Sean

Question

car dealer has established that 40% of his potential customers prefer single cab cars while 

60% prefer double cab cars. From a recent survey among his existing clients, he obtained 

additional information which indicates that 15% of clients who bought single cab cars prefer 

air conditioning while 65% of clients who bought double cars prefer air conditioning.

a) What is the probability that a client who bought a single cab does not prefer air 

conditioning? (5)

b) What is the probability that a client prefers a double cab with air conditioning? (10)

Answer 1

Solution:

Notations:

S = customers prefer single cab cars

D = customers prefer double cab cars

A = customers prefer cars with air conditioning

Given-

$P(S)=0.4,P(D)=0.6 \\ P(A|S)=0.15,P(A|D)=0.65 \\ \Rightarrow P(A'|S)=1-0.15=0.85, P(A'|D)=1-0.65=0.35$

(a):

$P(S|A')=\dfrac{P(S)P(A'|S)}{P(S)P(A'|S)+P(D)P(A'|D)} \\= \dfrac{0.4\times 0.85}{0.4\times 0.85+0.6\times0.35}=\dfrac{34}{55}$

(b):

$P(D|A)=\dfrac{P(D)P(A|D)}{P(D)P(A|D)+P(S)P(A|S)} \\= \dfrac{0.6\times 0.65}{0.6\times 0.65+0.4\times0.15}=\dfrac{13}{15}$