Question

An insurance company offers a comprehensive policy with car, house, and accidental injury components. Over a five-year period, the probability that a policy holder files a claim under at least one of the three categories is 0.44. A policy holder is twice as likely to file a car claim as a house claim and three times as likely to file a car claim as an accidental injury claim. Whether a policy holder files a house claim is independent of whether the policy holder files either a car claim or an accidental injury claim. The probability that a policy holder files both a house claim and an accidental injury claim is 0.015. The probability that a policy holder who files a car claim also files an accidental injury claim is 0.2. What is the probability that a policy holder files claims under all three categories over a five-year period?

Answer 1

let probability of filing a car ; house and injury claim are P(A), P(B) and P(C)

here P(A u B u C)=0.44

P(A)=2*P(B) ; P(A)=3*P(C) ; therefore P(B)=1.5P(C)

also as B is independent of A and C

P(B n A)=P(A)*P(B)

P(B n C)=P(B)*P(C)=0.015

or 1.5*P(C)*P(C)=0.015

P(C)=0.1 ;

hence P(A)=0.3 and P(B)=0.15

also as olicy holder who files a car claim also files an accidental injury claim is 0.2

therefore P(C|A)=0.2

P(C n A)=0.2*P(A)=0.2*0.3=0.06

hence probability that a policy holder files claims under all three categories over a five-year period

P(A n B n C)=P(A u B u C)-(P(A)+P(B)+P(C)-P(A n B)-P(B n C)-P(A n C))

=0.44-(0.3+0.15+0.1-0.3*0.15-0.15*0.1-0.06)= 0.01