A property and casualty insurance company categorizes its policyholders into three groups: low risk, medium risk,...

A property and casualty insurance company categorizes its policyholders into three groups: low risk, medium risk, or high risk. 25% of the policyholders are low risk and 30% are high risk. The high risk policyholders are three times as likely as medium risk policyholders to file an insurance claim and the medium risk policyholders are twice as likely as the low risk policyholders to file an insurance claim. Determine the proportion of the insurance claims filed by medium risk policyholders.

Solutions

Expert Solution

Let events

LR= a randomly chosen policyholder is low risk
MR= a randomly chosen policyholder is medium risk
HR= a randomly chosen policyholder is high risk
C= a randomly chosen policyholder has filed an insurance claim

We know the following probabilities

The probability that a randomly chosen policyholder is low risk is P(LR) = 0.25
The probability that a randomly chosen policyholder is high risk is P(HR) = 0.30
The probability that a randomly chosen policyholder is medium risk is
- P(MR) = 1- (P(LR)+P(HR) = 1-(0.25+0.30)=0.45

The conditional probability that a policy holder files an insurance claim given that the policyholder is high risk is

The conditional probability that a policy holder files an insurance claim given that the policyholder is medium risk is

The conditional probability that a policy holder files an insurance claim given that the policyholder is low risk is

The high risk policyholders are three times as likely as medium risk policyholders to file an insurance claim. This means

the medium risk policyholders are twice as likely as the low risk policyholders to file an insurance claim. This means

The joint probability that a randomly selected policy holder files for an insurance claim and is high risk is

The joint probability that a randomly selected policy holder files for an insurance claim and is medium risk is

The joint probability that a randomly selected policy holder files for an insurance claim and is low risk is

The marginal probability that a policy holder files a claim is

the proportion of the insurance claims filed by medium risk policyholders is

ans: the proportion of the insurance claims filed by medium risk policyholders is 30.51%

orchestra answered 1 year ago

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