You are trying to classify drivers in terms of accident risk. You define the top

Question

You are trying to classify drivers in terms of accident risk. You define the top 25% of drivers to be Good, the bottom 25% to be Bad, and the rest as Medium. From historical data you believe that Good drivers have a 95% chance of no accident in any given year, Medium drivers have a 90% chance, and Bad drivers have a 75% chance. You also believe that the chance of having multiple accidents in a given year is 3% for Good drivers, 5% for Medium drivers, and 15% for Bad drivers. Let G, B, M respectively denote good, bad, and medium drivers. Then P(G)= 0.25 P(B) =0.25 P(M) = 0.5 In 2012, the average auto liability claim for property damage was $3,073; the average auto liability claim for bodily injury was $14,653. (These are claims paid to individuals considered “Not at Fault.”) · In 2012, the average collision claim was $2,950; the average comprehensive claim was $1,585. (These are claims paid to the “At Fault” individuals covered by the insurance policy.)

***Create a plan to deal with the uncertainty in the decision of calculating a reasonable auto premium. This plan should include the data figures for Good, Medium, and Bad drivers calculated in Part 1, as well as the costs associated with the average costs of auto accidents.***

Explanation / Answer

For any driver,

P( 1 accident | driver is of class X) =P( less than or equal to 1 accident | class X driver)-P(0 accident | class X driver)=(1-P( more than 1 accident | class X driver)) - P(0 accident | class X driver)

= (1 - P( multiple accidents | class X driver)) - P(no accidents | class X driver)

so, P( 1 accident and driver is good) = P(1 accident | Good driver)*P(Good driver) = (1 - 0.03 - 0.95)*0.25 =0.02*0.25= 0.005

P( 1 accident and driver is medium) = P( 1 accident | medium driver )*P(medium driver)

= (1 - 0.05-0.9)*0.5=0.05*0.5=0.025

P( 1 accident and driver is bad ) =P(1 accident | driver is bad)*P(driver is bad)=(1-0.15-.75)*.25=0.10*.25=0.025

So,

P( 1 accident | Good)=0.02 P( 1 accident and good) = 0.005

P( 1 accident | medium) = 0.05 P( 1 accident and medium ) = 0.025

P( 1 accident | bad ) = 0.10 P( 1 accident and bad ) = 0.025

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