Question

An automobile insurance company divides customers into three categories, good risks, medium risks, and poor risks. Assume that 74% of the customers are good risks, 20% are medium risks, and 6% 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.

What is the probability that the customer is a good risk and has filed a claim? Round the answer to four decimal places.

What is the probability that the customer has filed a claim? Round the answer to four decimal places.

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

Answer 1

Given

P(Good risk) = 74%

= 0.74

P(Medium risk) = 20%

= 0.20

P(Poor risk) = 6%

= 0.06

P(Claim/Good risk) = 0.005

P(Claim/Medium risk) = 0.01

P(Claim/Poor risk) = 0.025

G : Good risk

C : Claim

i)

P( good risk and has filed a claim) = P(GC)

= P(G) * P(C/G)

= 0.74 * 0.005

= 0.0037

  

Therefore, The probability that the customer is a good risk and has filed a claim is 0.0037.

ii)

P( customer filed a claim) = P(C)

   = P(GC) + P(MC) + P(PC) .....(1)

Where

P(GC) = P(G) * P(C/G)

= 0.74 * 0.005

=0.0037

P(MC) = P(M) * P(C/M)

= 0.20 * 0.01

= 0.002

P(PC) = P(P) * P(C/P)      

= 0.06 * 0.025

= 0.0015

From (1)

P(C) = P(GC) + P(MC) + P(PC)

= 0.0037 + 0.002 + 0.0015

= 0.0072

Therefore, the probability that the customer has filed a claim =0.0072

iii)

= 0.5139

Therefore,

given that the customer has filed a claim, the probability that the customer is a good risk = 0.5139