Suppose that MegaInsurance classifies people into one of three categories: good

Question

Suppose that MegaInsurance classifies people into one of three categories: good risks, average risks,
and bad risks. Historical data indicates that the probability that a good risk person is accident-free during a 1-year
period is 0.95; for an average-risk person, it is 0.85, and for a bad-risk person it is 0.70. Suppose that 22% of people
are good risks, 50% are average risks, and 28% are bad risks. What fraction of the population is involved in an
accident in any one year? If Joe (a policyholder at MegaInsurance) had no accidents in 2011, what is the probability
Joe is a good-risk person? an average-risk person?

Explanation / Answer

ood risk = 5 : 95 (20%) average risk = 15 : 85 (50%) bad risk = 30 : 70 (30%) q1 P(involve in accident) = 0.05*0.2 + 0.15*0.5 + 0.3*0.3 = 0.010 + 0.075 + 0.09 = 0.175 / 1 = 7 / 40 q2 P(good risk | no accidents) = 0.95*0.2 / (0.95*0.2 + 0.85*0.5 + 0.7*0.3) = 0.190 / (0.190 + 0.425 + 0.21) = 0.190 / 0.825 = 0.2303 q3 P(average risk | no accidents) = 0.85*0.5 / (0.95*0.2 + 0.85*0.5 + 0.7*0.3) = 0.425 / (0.190 + 0.425 + 0.21) = 0.425 / 0.825 = 0.5152

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