A large operator of timeshare complexes requires anyone interested in making a p

Question

A large operator of timeshare complexes requires anyone interested in making a purchase to first visit the site of interest. Historical data indicates that 40% of all potential purchasers select a day visit, 60% choose a one-night visit. In addition, 10% of day visitors ultimately make a purchase while 30% for one-night visitors. It might be helpful to draw a probability tree. A. What is the probability that a randomly selected visitor made a purchase? B. If you learn that a visitor made a purchase, how likely is it that he or she has a day visit? C. Consider the event A = {a visitor had a day visit and the event B = {a visit made a purchase}. Are these two events independent? Justify your answer for full credits.

Explanation / Answer

P(day visit) = 0.4

P(night visit) = 0.6

P(make a purchase | day visit) = 0.1

P(make a purchase | night visit) = 0.3

a) P(make a purchase) = P(make a purchase | day visit) * P(day visit) + P(make a purchase | night visit) * P(night visit)

                                      = 0.1 * 0.4 + 0.3 * 0.6

                                      = 0.22

b) P(day visit | make a purchase) = P(make a purchase | day visit) * P(day visit) / P(make a purchase)

                                                     = 0.1 * 0.4 / 0.22

                                                     = 0.1818

c) P(A) = P(day visit) = 0.4

P(B) = P(make a purchase) = 0.22

P(A | B) = P(day visit | make a purchase) = 0.1818

Since P(A | B) is not equal to P(A), A and B are not independent events

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