Question

The probability that a credit score is equal to or higher than 700 is 0.45. The probability that a mortgage will be approved given a credit score of 700 or higher is 0.9. The probability that a mortgage application will be denied given a credit score of less than700 is 0.7.

1. What is the probability a random mortgage application will be approved?

2. What is the probability a credit score was 700 or higher, given that the mortgage application was approved?

Answer 1

We are given here that:

P( higher score ) = 0.45
P( approved | higher score ) = 0.9, P( denied | higher score ) = 0.1
P( denied | lesser score ) = 0.7, therefore P( approved | lesser score ) = 0.3

1) Using law of total probability, we get here:

P( approved ) = P( approved | higher score )P( higher score ) + P( approved | lesser score )P( lesser score )

P( approved ) = 0.9*0.45 + 0.3*(1 - 0.45 ) = 0.570

Therefore 0.570 is the required probability here.

2) Using Bayes theorem, we get:

P( higher score | approved ) = P( approved | higher score ) P( higher score ) / P( approved )

P( higher score | approved ) = 0.9*0.45 / 0.57 = 0.7105

Therefore 0.7105 is the required probability here.