Hi, first of all thanks for reading this. I'm 47 and trying to teach myself Stats from an excellent textbook as part of my course in maths and economics which I'm studying independently.

Glad to hear you keep on learning!

I've been picking my way through probability and have hit a problem which I have a answer but I don't trust my method.

i'm given 3 facts

1 only 15% loans made to high risk customers.

2 5% of loans made are in default.

3 of those in default 40% were made to high risk customers.

Let's define each of these events in terms of A and B.

Using what you already started with, given that the loan is in default (A), 40% are high risk (B).

So as you noted below, P(B|A) =.4

The "|" means "given".

I'm taking 3 to be a conditional prob of those being in default being high risk ie (B|A)

See just above.

I'm asked to find probability of high risk borrower defaulting?

the method I used was to say that P(B|A)= P(AnB)/P(A)

therefore 0.4 x 0.15 = P(AnB) = 6%

If we stick with the events defined as above, A is default and B is high risk. So, they want to know: given that the borrower is high risk (B), what is the probability he defaults (A); P(A|B).

We know P(A) = .05, P(B)= .15, P(B|A) = .4.

[P(B|A)*P(A)]/P(B) = P(A|B) --> [.4*.05]/.15 = 0.02/.15 = .13

Probability of default GIVEN that the borrower is high risk, P(A|B), is .13 (13%).

Another way to approach problems like this might be with a tree diagram or with natural frequencies (I believe that's the term).

We know that out of every 100 loans:

A) 15 are made to high risk people

B) 5 are in default

A|B) 2 are from high risk people of the 5 total in default, (40% are due to high risk people)

So, 15 are from high risk people, 2 are defaults from high risk people, so 2/15 is probability of default, given that they are high risk loans.

Had to remove a frequency table option due to poor formatting :shakehead

It's pretty early near me, so if another person could confirm, that would be great. Although, I believe this is correct.