Expert: Paul Klarreich - 9/23/2011

Question
A bank has a test designed to establish the credit rating of a loan applicant. of the persons, who default (D), 90% fail the test (f). of the persons, who will repay the bank (ND), 5% fail the test. Futhermore, it is given that 4% of the population is not worthy of credit i.e. P(D)=0.04. Given that someone failed the test,what is the probability that he actually will default(when given a loan)?

Answer
Questioner:devesh
Country:India
Category:Advanced Math
Private:No
Subject:probability
Question:

A bank has a test designed to establish the credit rating of a loan applicant. of the persons, who default (D), 90% fail the test (f). of the persons, who will repay the bank (ND), 5% fail the test. Futhermore, it is given that 4% of the population is not worthy of credit i.e. P(D)=0.04. Given that someone failed the test, what is the probability that he actually will default(when given a loan)?

Let events be designated:

F = a person fails the test.
D = a person defaults. (pays is -D)

Now:

"of the persons, who default (D), 90% fail the test (f)"  means:

p(F,given D) = 0.90

"of the persons, who will repay the bank (ND), 5% fail the test"

p(F,given -D) = 0.05

By definition:  P(A, given B) = p(A and B)/p(B)

And:  p(A and B) + p(A and -B) = p(A)

If p(F,given D) = 0.90,
then  p(F and D)/p(D) = 0.90

and  p(F and D) = 0.90 p(D)

If p(F, given -D) = 0.05
then  p(F and -D)/p(-D) = 0.05

p(F and -D) = 0.05 p(-D)
p(F and -D) = 0.05 (1 - p(D))

p(F and -D) = 0.05  - 0.05 p(D)
p(F and D) =  0.90 p(D)
-------------------------------------
p(F and -D) + p(F and D) = 0.05 + 0.85 p(D)

p(F) = 0.05 + 0.85 p(D)

But you said that  p(D) = 0.04, so

p(F) = 0.05 + 0.85 (0.04)  <<< You compute this.

Finally, "Given that someone failed the test, what is the probability that he actually will default" means:

find p(D, given F)

p(D, given F) = p(F and D) / p(F)

I think you can finish this up.