Question

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?

Answer 1

Answer:

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%)