A home mortgage loan officer for a large mortgage bank has kept records on the c

Question

A home mortgage loan officer for a large mortgage bank has kept records on the customers who visited her office in the past 5 years. Her records indicate that half of her visitors have been women and 20% of the women who visited her ultimately took a mortgage loan from the bank. On the other hand, only 15% of the men who visited her office ultimately took a mortgage loan from the bank.

If a customer visited the loan officer and decided to take a home mortgage loan from her bank, what is the probability that the customer was a man? Show the necessary work.

If a customer visited the loan officer and decided not to take a home mortgage loan from the bank, what is the probability that the customer was a woman? Show the necessary work.

Please calculate the probability that a customer who visited the home mortgage officer will take a home mortgage loan from the bank clearly showing how you arrived at your answer.

Explanation / Answer

P(Woman visits) = 0.50

P(Man visits) = 1 - 0.50 = 0.50

P(Mortgage | Woman) = 0.20

P(Mortgage | Man) = 0.15

By theorem of total probability,

P(Mortgage) = P(Mortgage | Woman)*P(Woman visits) + P(Mortgage | Man)*P(Man visits)

= 0.20 * 0.50 + 0.15 * 0.50 = 0.175 (Answer to Part 3)

By Bayes' theorem,

P( Man | Mortgage) = P(Mortgage | Man)*P(Man visits) / P(Mortgage) = 0.15*0.50 / 0.175

= 0.4286 = 3/7

P(Woman | Mortgage) = 1 - P( Man | Mortgage) = 4/7 = 0.5714

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