Question

#6. A car insurance company has high-risk, medium-risk, and low-risk clients, who have, respectively, probabilities.04,.02, and.01 of filing claims within a given year. The proportions of the numbers of clients in the three categories are .15, .25, and.60, respectively (a) What is the probability that a random client doesn't file a claim? (b) What proportion of the claims filed each year come from high-risk clients? (c) What is the probability that a random client who didn't file a claim is low- risk?

Answer 1

Answer :

P(high chance documents a claim)=0.04;

P(high hazard does not record a case) =1-0.04

=0.96

P(med hazard records a claim)=0.02;

P(med chance does not document a case) =1-0.02

=0.98

P(low hazard records a claim)=0.01;

P(low chance does not document a case) =1-0.01

=0.99

P(high chance claim)=0.15;

P(med chance claim)=0.25;

P(low hazard claim)=0.6

(a) now we need to find out the probability that a random client doesn't file claim :

P(one does not record a claim)= P(high chance does not document a claim)P(high chance claim)+ P(med chance does not document a claim)P(med hazard claim)+P(low chance does not document a claim)P(low chance case)

= 0.96*0.15+0.98*0.25+0.99*0.6

= 0.144+0.245+0.594

= 0.983

(b) now find out the claims field each year come from high risk clients :

P(high chance given one records claim)= [P(high chance documents a claim)P(high hazard claim)]/[P(high chance document a claim)P(high chance claim)+ P(med chance document a claim)P(med chance claim)+P(low chance record a claim)P(low chance claim)]

=0.04*0.15/(0.04*0.15+0.02*0.25+0.01*0.6)

= 0.006/(0.006+0.005+0.006)

= 0.006/0.017

= 0.3529

(c). now we need to find out the probability that a random client who didn't file a claim is low risk :

P(low hazard given one doesn't record claim)=[P(low chance does not document a claim)P(low chance claim)]/[P(high chance does not document a claim)P(high chance claim)+ P(med chance does not record a claim)P(med chance claim)+P(low chance does not document a claim)P(low chance claim)]

=0.99*0.6/0.983

= 0.594/0.983

=0.60427