Classify drivers in terms of accident risk: You define the top 25% of drivers to

Question

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.

What is the probability that a driver has EXACTLY ONE ACCIDENT in a year for each:

i.Good Driver

ii.Medium Driver

iii.Bad Driver

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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