Question

3. A certain auto mall sells only Fords, Hondas, and Toyotas. Each customer buys only 1 car. It is known that 45% of all customers buy a Toyota and 35% buy a Ford. It is also known that 40% of the Toyota customers buy a white car whereas only 30% of the Honda customers buy a white car. None of the Ford customers buy a white car. a. Express all numbers given in the problem statement as probabilities of events. (2 pts) b. What is the probability that the customer bought a white car? (2 pts) c. If the customer bought a white car, what is the probability that he or she bought a Toyota? (2 pts) It is observed that 65% of all customers buy an automatic shift car. However, 80% of white Toyota buyers purchase an automatic shift. d. What is the probability that the customer buys a white Toyota with an automatic shift? 2 pts) What is the probability that the customer bought a white Toyota given that he or she bought an automatic shift car? (2 pts) e.

Answer 1

a.
Let F , H, T denote the events that the customer buys Ford, Honda ot Toyota car respectively.
Let W be the event that the customer buys white car.
P(F) = 0.35, P(T) = 0.45
P(W | T) = 0.4
P(W | H) = 0.3
P(W | F) = 0

b.
P(H) = 1 - (0.35 + 0.45) = 0.2
By law of total probability,
P(W) = P(F) * P(W | F) + P(H) * P(W | H) + P(T) * P(W | T)
= 0.35 * 0 + 0.2 * 0.3 + 0.45 * 0.4
= 0.24

c.
By Bayes theorem,
P(T | W) = P(W | T) * P(T) / P(W)
= 0.4 * 0.45 / 0.24
= 0.75

d.
Let A be the evnet that the customer buys automatic shift car.
P(A) = 0.65
P(A | T and W) = 0.80
P(T and W) = P(T | W) * P(W) = 0.75 * 0.24 = 0.18
Probability that the customer buys white Toyota with an automatic shift = P(T and W and A)
= P(A | T and W) * P(T and W)
= 0.80 * 0.18
= 0.144

e.
By Bayes theorem,
Probability that the customer buys white Toyota given that it is an automatic shift = P(T and W | A)
= P(A | T and W) * P(T and W) / P(A)
= 0.80 * 0.18 / 0.65
= 0.2215