Question

A bank classifies borrowers as "high risk" or "low risk," and 23% of its loans are made to those in the "high risk" category. Of all the bank's loans, 8% are in default. It is also known that 46% of the loans in default are to high-risk borrowers. Let H represent the event that a randomly selected loan is issued to a "high risk" borrower. Let D be the event that a randomly selected loan is in default. Round your answers to 4 decimal places. Your work for this entire problem will be hand-graded, see instructions below.

a. What is the probability that a randomly selected loan is in default and issued to a high-risk borrower?

Tries 0/5

b. What is the probability that a loan will default, given that it is issued to a high-risk borrower?

Tries 0/5

c. What is the probability that a randomly selected loan is either in default or issued to a high-risk borrower?

Tries 0/5

d. A loan is being issued to a borrower who is not high-risk. What is the probability that this loan will default?

Tries 0/5

e. Are events D and H independent? Justify the answer.
Yes, because P(D) ≠ P(H)
Yes, because P(H | D) > P(H) + P(D)
No, because P(D ∩ H) ≠ P(D) × P(H)
No, because P(D | H) ≠ P(H)
Not enough information to determine

Answer 1

We have, P(H) = 0.23

P(D) = 0.08, P(H | D) = 0.46

Let L be the event of a loan being in low risk category.

Thus, P(L) = 1 - 0.23 = 0.77

a. The probability that a randomly selected loan is in default and issued to a high-risk borrower = P(D H) = P(H | D) P(D) = 0.46 * 0.08 = 0.0368 (Ans).

b. The probability that a loan will default, given that it is issued to a high-risk borrower = P(D | H) = [P(H | D) * P(D)] / P(H) = (0.46 * 0.08)/0.23 = 0.16 (Ans).

c. The probability that a randomly selected loan is either in default or issued to a high-risk borrower = P(D H) = P(D) + P(H) - P(D H) = 0.08 + 0.23 - 0.0368 = 0.2732 (Ans).

d. If a loan is being issued to a borrower who is not high-risk, the probability that this loan will default = P(D | L) = P(D | H') [As, H' = L] = P(D H') P(H') = [P(D) - P(D H)] P(H') [As, P(D) = P(D H) + P(D H')]

= (0.08 - 0.0368) * 0.77 = 0.5877 (Ans).

e. We have, P(D) * P(H) = 0.08 * 0.23 = 0.0184 P(D H)

Hence, D and H are not independent. (Ans).