Bayes Theorem - Validating Questionable Results

Question

Supposed we have a large data set regarding a users Credit History (1-Good, 0-Bad) and whether or not their Loan (1-Yes, 0-No) has been approved.

The probabilities are calculated and they look like this:

|---------------------|------------------|------------------|
|                     |        Yes       |       No         |
|---------------------|------------------|------------------|
|    Loan Approved?   |         68%      |       32%        |
|---------------------|------------------|------------------|
|    Good Credit?     |         84%      |       16%        |
|---------------------|------------------|------------------|

And we apply Bayes' Theorem:

P(Approved|Good Credit) = (0.68)(0.84) / [(0.68)(0.84) + (0.32)(0.16)] = 91.77%

Great! That seems to make sense.

But now...

P(Denied|Good Credit) = (0.32)(0.84) / [(0.32)(0.84) + (0.68)(0.16)] = 71.18%

This cannot be correct but I'm not sure how to validate this outcome (they are mutually exclusive). How does one validate results when they don't seem intuitive (or reasonable)?

Answer 1

Your table is bad or at least extremely questionable. The top row is whether the loan was approved. The second row was the probability of good credit. What you need is a table that looks like the following:

You cannot calculate a problem using Bayes theorem as above unless they are independent of each other which would not be logical unless creditworthiness was only discovered after approval.