关于P（AB）

问题如下：

A credit risk analyst at a mortgage lending firm is updating a borrower risk profile table
used as a reference in originating and evaluating residential mortgage loa ns. The
analyst assesses how a borrower's access to government-sponsored mortgage
enterprises (US agencies such as Fannie Mae, Freddie Mac, and Ginnie Mae) affects
their likelihood of defaulting on a residential mortgage. The analyst gathers the following
information:
• The probability of a borrower having an agency secure their residential mortgage loa n is 62%.
• 32% of borrowers are classified as millennials.
• 17% of borrowers with residential mortgage loans secured by an agency are classified as millennials.
What is the probability that a borrower has an agency-secured mortgage loan given that the borrower is a millennial?

选项：

A.

17%

B.

20%

C.

33%

D.

52%

解释：

C is correct. First, we identify and define the relevant probabilities:
A is the presence of a secured home mortgage loa n via agency
B is the prevalence of millennial borrowers
P(A) is the probability of borrower having an agency secure their home mortgage loan. In this case, it is 62% or 0.62.
P(B) is the probability of being a millennial borrower. Here it is 32% or 0.32.
P(A|B) is the probability of having an agency-secured home mortgage loan given a millennial status, which is what we want to calculate.
P(B|A) is the probability of being a millennial give n the existence/presence of an agency-secured home mortgage loa n. Here it is 17% or 0.17.
Recall that Bayes' Rule gives us

Thus, P(A|B) = [0.17 * 0.62] / 0.32 = 0.329 ≈ 0.33 or 33%.
A is in correct. This is just P(B|A).
B is in correct. This is the result of P(A)*P(B).
D is in correct. This is the result when the formula is miscalculated as P(B)/P(A)= 0.32/0.62 = 0.516.
 
Explain and apply Bayes' rule.
Global Association of Risk Professionals. Quantitative Analysis. New York, NY: Pearson,
2022. Chapter 1. Fundamentals of Probability.