An automobile insurance company divides customers into three categories, good
risks, medium risks, and poor risks. Assume that 70% of the customers are good
risks, 20% are medium risks, and 10% are poor risks. Assume that during the
course of a year, a good risk customer has probability 0.005 of filing an accident claim, a medium risk customer has probability 0.01, and a poor risk customer has probability 0.025. A customer is chosen at random.
Given that the customer has filed a claim, what is the probability that the
customer is a good risk?
I think the total probability that the customer has filed a claim is (0.7*0.005+0.2*0.01+0.1*0.025)
So the probability that the customer is a good risk is (0.7*0.005)/(0.7*0.005+0.2*0.01+0.1*0.025).
Am I right?
Best Answer
You are correct, this is a direct application of Bayes' theorem. You just need to finish the math:
$$\begin{split} P(Good|Claim) &= \frac{P(Good \cap Claim)}{P(Claim)} \\ &= \frac{0.7 \cdot 0.005}{0.7 \cdot 0.005 + 0.2 \cdot 0.01 + 0.1 \cdot 0.025} \\ &= \frac{0.0035}{0.008} \\ &= 0.4375 \end{split}$$