An automobile dealer has kept records on the customers who v

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?

Solution

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]

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b)

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

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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]