Question

A bank classifies borrowers as "high risk" or "low risk," and 17% of its loans are made to those in the "high risk" category. Of all the bank's loans, 5% are in default. It is also known that 56% 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?

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

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

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

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

Answer 1

(a)

From the given data, the following Table is calculated:

	High risk (H)	Low risk ()	Total
Default (D)	0.05X0.56=0.028	0.05-0.028=0.022	0.05
Not default ()	0.17-0.028=0.142	0.83-0.022=0.808	0.95
Total	0.17	1-0.17=0.83	1.00

The probability that a randomly selected loan is in default and issued to a high-risk borrower = 0.028

So,

Answer is:

0.028

(b)

The probability that a loan will default, given that it is issued to a high-risk borrower =

P(Default/ High-risk borrower) = P(Default AND High-risk borrower)/ P( High-risk borrower)

                                              = 0.028/0.17

                                           = 0.1647

So,

Answer is:

0.1647

(c)

The probability that a randomly selected loan is either in default or issued to a high-risk borrower =

   P(Default OR High-risk borrower) = P(Default) + P(High risk borrower) - P(Default AND High-risk borrower)

                                              = 0.05 + 0.17 - 0.028

                                             = 0.192

So,

Answer is:

0.192

(d)

P(Default/ Not high risk) = P(Default AND Not high risk)/ P(Not high risk)

                                    = 0.022/0.83

                                    = 0.0265

So,

Answer is:

0.0265

(e)

P(D/H) = 0.1647

P(D) = 0.05

So,

Correct option:

No, because P(D | H) ≠ P(D)