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?

Answer 1

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`