According to an insurance company, 1/5 of people are at high risk of a certain t

Question

According to an insurance company, 1/5 of people are at high risk of a certain type of accident, and the other 4/5 are at low risk. The people at high risk have a probability of 40% of getting into this type of accident in any given month, while the people at low risk have only a 10% chance. When an insured person gets into an accident of this type, the insurance company pays them $200 (maximum of one accident per month). Mark is a new customer who has not been determined to be at high or low risk yet. (a) If the company would like their expected gain from Mark to be S100 in the first month, how much should they set his premium to be? (The premium is how much the customer pays the company in any month.) (b) Mark ended up getting into an accident on the first month. If the company still wants the expected gain from him to be $100 in the second month, what do they need to increase the premium to?

Explanation / Answer

Let the event that customer has high risk of accident be denoted by H and low risk of accident by L.

And the event that customer gets into an accident be denoted by A.

Now, P(H) = 1/5, P(L) = 4/5

P(A|H) = 40% = 0.4

P(A|L) = 10% = 0.1

Thus, P(A) = P(A|H)*P(H) + P(A|L)*P(L) = 0.4*0.2 + 0.1*0.8 = 0.16

Probability of not getting into an accident, P(A') = 0.84

Let x be the premium.

And amount paid by the company in case of an accident, y = $200

(a) Expected gain = P(A)*(-y) + P(A')*x

-> 100 = 0.16*(-200) + 0.84*x

-> x = 157.14

Thus, the premium should be set to $157.14

(b) Now, when mark ended up getting into an accident on the first month, the insurance company had a loss of $(200 - x). As they need to pay $200 and earned a premium of $x.

So to get an expected gain of $100 in second month, the total expected gain should be $100 + (loss in first month)

-> 100 + (200 - x) = 0.16*(-200) + 0.84*x

-> x = 180.43

Thus, the premium need to be increased to $180.43

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