Question

You are the Claims Manager for a large auto insurance firm. You know the following facts: • 25% of your policy-holders are younger (age 16-34). Within this group, 55% are men and 45% are women. Only some of these policy holders make an accident claim. The men have a 10% probability of making a claim, while the women have a 5% probability of making a claim. • 50% of your policy-holders are middle-aged (ages 35-59). Of these, 50% are men and 50% are women. Only some of these policy holders make an accident claim. The men and women within this age group have 4% and 3% probabilities of making a claim, respectively. • 25% of your policy-holders are older (ages 60 and above). Of these, 55% are women and 45% are men. Only some of these policy holders make an accident claim. The men and women each have a 2% probability of making a claim. (a) What is the probability that a policy-holder who makes a claim is in the “16-34” age group? (b) What is the probability that a policy-holder who does not make a claim is in the “60 and above” age category?

Answer 1

a)

P(   age 16-34   ) =    0.25
P(   age 35-59   ) =    0.5
P(    age 60 and above   ) =    0.25

P(   claim   | age 16-34) =0.55*0.1+0.45*0.05 =   0.0775
P(   claim   | age 35-59)=0.5*0.04+0.5*0.03 =   0.035
P(   claim   | age 60 and above) =0.55*0.02+0.45*0.02 =   0.02

P(claim) = P(age 16-34) * P(claim| age 16-34) + P(age 35-59) *P(claim| age 35-59) + P( age 60 and above)*P(claim| age 60 and above) =                                    0.25*0.0775+0.5*0.035+0.25*0.02=               0.041875  

P(age 16-34| claim) = P(age 16-34)*P(claim| age 16-34)/P(claim)=               0.25*0.0775/0.041875=       0.4627

b)

P(no claim) = 1-0.041875 = 0.958125

P(≥ 60 | no claim) = (1-0.02)*0.25/0.958125 = 0.255707763

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