An automobile insurance company divides customers into three categories, good risks, medium risks, and poor risks. Assume taht 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 a probability of 0.01, and a poor risk customer has probability 0.025. a customer is chosen at random.

a) What is the probability that the customer is a good risk and has filed a claim?

b) What is the probability that the customer has filed a claim?

c) Given that the customer has filed a claim, what is the probability that the customer is a good risk?

a) the probability is P (G∩C) = 0.0035 (0.35%)

b) the probability is P (C) = 0.008 (0.8%)

c) the probability is P (G/C) = 0.4375 (43.75%)

Step-by-step explanation:

defining the event G = the customer is a good risk, C = the customer fills a claim then using the theorem of Bayes for conditional probability

a) P (G∩C) = P (G) * P (C/G)

where

P (G∩C) = probability that the customer is a good risk and has filed a claim

P (C/G) = probability to fill a claim given that the customer is a good risk

replacing values

P (G∩C) = P (G) * P (C/G) = 0.70 * 0.005 = 0.0035 (0.35%)

b) for P (C)

P (C) = probability that the customer is a good risk * probability to fill a claim given that the customer is a good risk + probability that the customer is a medium risk * probability to fill a claim given that the customer is a medium risk + probability that the customer is a low risk * probability to fill a claim given that the customer is a low risk = 0.70 * 0.005 + 0.2 * 0.01 + 0.1 * 0.025

= 0.008 (0.8%)

therefore

P (C) = 0.008 (0.8%)

c) using the theorem of Bayes:

P (G/C) = P (G∩C) / P (C)

P (C/G) = probability that the customer is a good risk given that the customer has filled a claim

replacing values

P (G/C) = P (G∩C) / P (C) = 0.0035 / 0.008 = 0.4375 (43.75%)