Question

An auto insurance company concludes that 30% of policyholders with only collision coverage will have a claim next year, 40% of policyholders with only comprehensive coverage will have a claim next year and 50% of policyholders with both collision and comprehensive coverage will have a claim next year. Records show 60% of policyholders have collision coverage 70% have comprehensive coverage and all policyholders have at least one of these coverages.

Calculate the percentage of policyholders expected to have an accident next year.

1. 10%

2. 20%

3. 31%

4. 36%

5. 40%

Answer 1

We are given here that:

P( collision coverage ) = 0.6
P( comprehensive coverage ) = 0.7

Therefore using law of addition of probability, we get here:

P( collision coverage and comprehensive coverage ) = P( collision coverage ) + P( comprehensive coverage ) - P( collision coverage or comprehensive coverage )

P( collision coverage and comprehensive coverage ) = 0.6 + 0.7 - 1 = 0.3

Therefore, we have here:

P( collision coverage only) = P( collision coverage ) - P( collision coverage and comprehensive coverage )

P( collision coverage only) = 0.6 - 0.3 = 0.3

P( comprehensive coverage only) = P( comprehensive coverage ) - P( collision coverage and comprehensive coverage )

P( comprehensive coverage only) = 0.7 - 0.3 = 0.4

Now using law of total probability, we get here:

P( accident ) = P( accident | collision coverage only)P( collision coverage only) + P( accident | comprehensive coverage only)P( comprehensive coverage only) + P( accident | collision coverage and comprehensive coverage only)P( collision coverage and comprehensive coverage only)

P( accident ) = 0.3*0.3 + 0.4*0.4 + 0.5*0.3 = 0.4

Therefore 40% is the required percentage here.