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Question
An insurance company insured 3000 cyclists, 6000 scooter drivers, and 9000 car drivers. The probability of an accident involving a cyclist, a scooter driver, and a car driver are 0⋅3, 0⋅05 and 0⋅02 respectively. One of the insured persons meets with an accident. What is the probability that he is a cyclist?
Solution
Let E1, E2, and E3 be the respective events that the driver is a cyclist, a scooter driver, and a car driver.
Let A be the event that the person meets with an accident.
There are 3000 cyclists, 6000 scooter drivers, and 9000 car drivers.
Total number of drivers = 3000 + 6000 + 9000 = 18000
P(E1) = P(driver is a cyclist) `=3000/1800=1/6`
P(E2) = P(driver is a scooter driver) `=6000/18000=1/3`
P(E3) = P(driver is a car driver)`=9000/18000=1/2`
P (A|E1) = P (cyclist met with an accident) = 0.3
P (A|E2) = P (scooter driver met with an accident) = 0.05
P (A|E3) = P (car driver met with an accident) = 0.02
The probability that the driver is a cyclist, given that he met with an accident, is given by P (E1|A).
P (E1|A) `=("P"("E"_1)."P"("A"|"E"_1))/("P"("E"_1)."P"("A"|"E"_1)+"P"("E"_2)."P"("A"|"E"_2)+"P"("E"_3)."P"("A"|"E"_3)`
`=(1/6xx0.3)/(1/6xx0.3+1/3xx0.05+1/2xx0.02)`
`=(1/6xx30/100)/(1/6xx30/100+2/6xx5/100+3/6xx2/100)`
`=30/46=15/23`
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