4. An insurance company divides its policyholders into the tollowing three categ

Question

4. An insurance company divides its policyholders into the tollowing three categories, high risk (5%), moderate risk (30%), low risk (65%), of being involved in at least one car accident in any given year. In addition, assume that the probability a policyholder is involved in a car accident is 0.6, 0.3 and 0.01 for, respectively, policyholders that are at high, moderate and low risk of being involved in one or more car accidents over a year. Given a policyholder was involved in at least one car accident over one year, what is the probability he/she is classified in the moderate or low risk category?

Explanation / Answer

We are given here that:

P( high ) = 0.05
P( moderate ) = 0.3
P( low ) = 0.65

Also, we are given that:

P( accident | high ) = 0.6
P( accident | moderate ) = 0.3
P( accident | low ) = 0.01

Therefore using law of total probability, we get here:

P( accident ) = P( accident | high ) P( high ) + P( accident | moderate ) P( moderate ) + P( accident | low )P(low )
P( accident ) = 0.05*0.6 + 0.3*0.3 + 0.65*0.01 = 0.1265

Therefore now using bayes theorem, given that there is at least one accident probability that it is from a low or moderate risk category is computed as:

= (0.3*0.3 + 0.65*0.01 ) / 0.1265

= 0.7628

Therefore 0.7628 is the required probability here.

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