# 3, 25 pts An insurance company believes that people can be divided into two cl

Question

# 3, 25 pts An insurance company believes that people can be divided into two classes those that are accident prone and those that are not. Their statistics show that an accident-prone persorn will have an accident at some time within a fixed 1-year period with probability .4, whereas this probability decreases to .2 for a non-accident-prone person. If we assume that 30 percent of the population is accident prone, 1. what is the probability that a new policy holder will have an accid purchasing a policy? 2

Explanation / Answer

a) Here, we are given that:

P( accident within first year | accident prone ) = 0.4,
P( accident within first year | not accident prone ) = 0.2,

Also, we are given here that: P( accident prone ) = 0.3

Therefore using law of total probability, we get:

P( accident within first year ) = P( accident within first year | accident prone )P( accident prone ) + P( accident within first year | not accident prone )P( not accident prone )

P( accident within first year ) = 0.4*0.3 + 0.2*0.7 = 0.26

Therefore 0.26 is the required probability here.

2) Now here given that there was an accident within first year, probability that he was accident prone is computed as: ( Using bayes theorem )

P( accident prone | accident within first year ) = P( accident within first year | accident prone )P( accident prone ) / P( accident within first year )

P( accident prone | accident within first year ) = 0.4*0.3 / 0.26 = 0.4615

Therefore 0.4615 is the required probability here.

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