Strategy to solve conditional probability question

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.

The problem is asking me to find the percentage of policyholders expected to have an accident next year.

Maybe I'm overthinking it but having difficulty approaching this problem. I have .30 of collision only that file a claim but not the total collision-only holders (same with comprehensive). Tried doing a chart, diagram but finding I don't have enough info. Any help/advice on how to approach this would be appreciated!

Answer 1

Records show 60% of policyholders have collision coverage 70% have comprehensive coverage and all policyholders have at least one of these coverages.

Given this information, we know that 30% of policyholders hold both policies, 30% hold collision coverage alone and 40% hold comprehensive coverage alone. We know this because 60% + 70% = 130% which means that exactly 30% of the policyholders are double-counted.

This information also suggests a way to attack the problem. Since every policy holder either a) holds a comprehensive policy alone, b) holds a collision policy alone, or c) holds both policies, we can consider each of those cases independently. We want to know $P(a) P(f|a) + P(b) P(f|b) + P(c) P(f|c) $ where $f$ denotes the act of filing a claim in the next year and $P$ denotes the probability.

If we want to know the overall proportion of policyholders that file a claim, we can compute it as a sum of products as shown below

coverage_type   proportion    probability_filing   product
comprehensive   .4            .4                   .16
collision       .3            .3                   .09
both            .3            .5                   .15

So, the overall sum is .4 or 40% .