A bank uses Bayes Theorem to help determine whether to give a customer a new credit card. A customer is assessed as either a good credit risk (G) or a bad credit risk (Gc). From historical data the bank has found that 35% of customers considered as a bad credit risk default on their credit card payments. Only 6% of customers considered as a good credit risk default on their payments. Let the symbols G;Gc;D;Dc represent the events of a good credit risk, a bad credit risk, default of payment and not defaulting, respectively.
A new customer applies for a credit card and the bank believes from the initial paper work and checks that there is a 70% chance she is a good credit risk, i.e. Pr(G) = 0:7. After one year the bank has new information as the new customer did not default on any of her payments. Using this new information, nd the probability Pr(GjDc) which is the probability of the customer being a good credit risk given that she didn't default. How does the bank view this customer now?
I'm not sure how to set out the answer OR what they're really looking for...
This is a straightforward application of Bayes' Theorem.
The problem tells you to find P(G|D
c) which by Bayes' Theorem is P(G|D
c) = P(D
c|G) P(G) / P(D
c).
We know the probabilities of P(D
c|G) and P(G), but we don't know the straight-up chance of the customer not defaulting, so we need to figure this out. We use the law of total probability:
P(D
c) = P(D
c|G) P(G) + P(D
c|G
c) P(G
c) = .94 * .7 + .65 * .3 = 0.853
Substituting this back in:
P(G|D
c) = P(D
c|G) P(G) / P(D
c) = .94 * .7 / 0.853 ≈ 0.7713
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