question archive An insurance company writes policies for a large number of newly-licensed drivers each year
An insurance company writes policies for a large number of newly-licensed drivers each year
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An insurance company writes policies for a large number of newly-licensed drivers each year. Suppose 40% of these are low-risk drivers, 40% are moderate risk, and 20% are high risk. The company has no way to know which group any individual driver falls in when it writes the policies. None of the low-risk drivers will have an at-fault accident in the next year, but 10% of the moderate-risk and 20% of the high-risk drivers will have such an accident. If a driver has an at-fault accident in the next year, what is the probability that he or she is high-risk?
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Given,
P(Low risk) = 0.40
P( Moderate risk ) = 0.40
P( High risk) = 0.20
P(At-fault accident | Low risk) = 0
P(At-fault accident | Moderate risk) = 0.10
P(At-fault accident | High risk) = 0.20
Here, we want to calculate P( High risk | at-fault accident) = ?
Using Bayes' conditional probability theorem,
P(High risk | at-fault accident) = ( P( High risk) * P(At-fault accident | High risk) ) / {P( Low risk) * P(At-fault accident | Low risk) +P( Moderate risk) * P(At-fault accident | Moderate risk) + P( High risk) * P(At-fault accident | High risk)}
= (0.20 * 0.20) / (0.40 * 0 + 0.40 * 0.10 + 0.20 * 0.20)
= 0.04/0.08
= 0.5
Therefore, P( High risk | at-fault accident) = 0.50
