An automobile dealer has kept records on the customers who visited his showroom.

Question

An automobile dealer has kept records on the customers who visited his showroom. Forty percent of the people who visited his dealership were women. Furthermore, his records show that 37% of the women who visited his dealership purchased an automobile, while 21% of the men who visited his dealership purchased an automobile

(a.) What is the probability that a customer entering the showroom will buy an automobile?

(b.) Suppose a customer visited the showroom and purchased a car. What is the probability that the customer was a woman?

(c.) Suppose a customer visited the showroom but did not purchase a car. What is the probability that the customer was a man?

Explanation / Answer

Let

W, M = women and men, resp
B = will buy

Thus,

a)

By Bayes' Rule,

P(B) = P(W) P(B|W) + P(M) P(B|M) = 0.40*0.37 + (1-0.40)*0.21 = 0.274 [answer]

****************

b)

P(W|B) = P(W) P(B|W) / P(B) = 0.40*0.37/0.274 = 0.540145985 [answer]

****************

c)

P(M|B') = P(M) P(B'|M) / P(B')

As

P(B') = 1 - P(B) = 1 - 0.274 = 0.726

Then

P(M|B') = P(M) P(B'|M) / P(B') = 0.60*(1-0.21)/0.726 = 0.652892562 [ANSWER]

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