An automobile insurance company divides customers into three categories, good risks, medium risks, and poor risks Assume that 70% Of tfl6 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.

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?

Solution 20E

Step1 of 3:

We have an automobile insurance company it divides customers into three categories they are

1).Good risk(G)

2).Medium risk(M)

3).Poor risk(P).

Also we have,

P(Good risk) = 70%

= 0.70

P(Medium risk) = 20%

= 0.20

P(Poor risk) = 10%

= 0.1

P(Claim/Good risk) = 0.005

P(Claim/Medium risk) = 0.01

P(Claim/Poor risk) = 0.025

We need to find,

a).We need to find the probability that the customer is a good risk and has filed a claim?

b).We need to find the probability that the customer has filed a claim?

c).We need to find the probability that the customer is a good risk? When customer has filed a claim,

Step2 of 3:

a).

The probability that the customer is a good risk and has filed a claim is given by

P(The probability that the customer is a good risk and has filed a claim) = P(GC)

=

=

= 0.0035

Therefore, The probability that the customer is a good risk and has filed a claim is 0.0035.

b).

The probability that the customer has filed a claim, is given by

P(The probability that the customer has filed a claim) = P(C)

= P(GC) + P(MC) + P(PC) .....(1)

Where

P(GC) =

=

= 0.0035

Hence, P(GC) = 0.0035

Now

P(MC) =

=

= 0.002

Hence, P(MC) = 0.002 and

P(PC) =

=

= 0.0025

Hence, P(PC) = 0.0025.

Substitute these values in equation (1) we get

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

= 0.0035 + 0.002 + 0.0025

= 0.008

Therefore, The probability that the customer has filed a claim is 0.008.

Step3 of 3:

c).

The probability that the customer is a good risk is given by

P(The probability that the customer is a good risk) = P(G/C)

=

=

[= 0.0035 from part(a) and P(C) = 0.008 from part(b)]

= 0.4375

Therefore, The probability that the customer is a good risk is 0.4375.

Conclusion:

a).The probability that the customer is a good risk and has filed a claim is 0.0035.

b).The probability that the customer has filed a claim is 0.008.

c).The probability that the customer is a good risk is 0.4375.