At a mortgage company, 60% of calls are answered by an attendant. The remaining 40% of callers leave their phone numbers. Of these 40%, 75% receive a return phone call the same day. The remaining 25% receive a return call the next day.

Of those who initially spoke to an attendant, 80% will apply for a mortgage. Of those who received a return call the same day, 60% will apply. Of those who received a return call the next day, 40% will apply.

Calculate the probability that a person initially spoke to an attendant, given that he or she applied for a mortgage.

• 0.06
• 0.26
• 0.48
• 0.60
• 0.69

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: E

Define the events as follows:

A = applies for a mortgage

S = initially spoke to an attendant

R = call returned the same day

N = call returned the next day

Then, using Bayes’ Theorem,

[[math]] \begin{align*} \operatorname{P}(S | A) &= \frac{\operatorname{P}( A | S ) \operatorname{P}( S )}{\operatorname{P}( A | S ) \operatorname{P}( S ) + \operatorname{P}( A | R) \operatorname{P}( R) + \operatorname{P}( A | N ) \operatorname{P}( N )} \\ &= \frac{0.8(0.6)}{0.8(0.6) + 0.6(0.4)(0.75) + 0.4(0.4)(0.25)} \\ &= 0.69. \end{align*} [[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.