An insurer offers auto insurance, fire insurance, and flood insurance. It offers

Question

An insurer offers auto insurance, fire insurance, and flood insurance. It offers only these policies. The company offers flood insurance only to fire insurance policyholders. That is, policyholders with fire insurance can add flood insurance. You are given: • 80% of all policyholders have auto insurance. • 40% of all policyholders have fire insurance. • 25% of policyholders with auto insurance also have fire insurance. • 50% of policyholders with fire insurance also have flood insurance. • 50% of policyholders with flood insurance also have auto insurance. Of the policyholders that have fire insurance, determine the fraction that have neither auto insurance nor flood insurance.

Explanation / Answer

here let probability of fire insuarance is P(F) ; flood insuarance F(FL) and auto insuarance =P(A)

hence P(A) =0.8 ; P(F) =0.4 ; P(F|A) =0.25 ; P(FL|F) =0.5 ; P(A|FL) =0.5

here P(F n FL ) =P(F)*P(FL|F) =0.4*0.5 =0.20

also P(F nA) =P(A)*P(F|A)= 0.8*0.25=0.20

P(F n A n FL) =P(F)*P(FL|F)*P(A|FL) =0.4*0.5*0.5=0.10

therefore P(F n(A U FL) =P(F n A)+P(F n FL)-P(A n F n FL) =0.2+0.2-0.10=0.30

therefore  fraction that have neither auto insurance nor flood insurance given fire insuarance=1-P(fraction that have at least one of them given fire insurance)

=1-P(F n(A U FL)/P(F) =1-0.3/0.4 =1-0.75 =0.25

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