1. (28 marks) This question concerns a male driver aged less than 22 years of age who is seeking comprehensive insurance for his car. Such insurance fully covers the cost of any damages from an accident the driver has, both to their own car and any other car(s) involved.

Assume that such a customer has a 1% chance of having a serious accident in any given year, a 3% chance of having a medium accident and a 7% chance of having a minor accident. Thus their total probability of having any sort of accident is 11%. Furthermore assume the average cost for repairs from a serious accident is $30000, a medium accident is $8000 and a minor accident is $1000.

Note: when someone has an accident, there is the choice to either use insurance or just pay the costs yourself. Insurance companies have all sorts of ways to encourage you to just pay small claims yourself (eg you get a discount on your payments if you didnt make a claim last year etc). We are going to ignore all these sorts of considerations and we will ignore the possibility of a driver having more than one accident in a year.

1. (a) Let C be the cost to the insurance company in a year for a male driver aged less than 22 years of age. What is the pmf and mean of C? [4 marks]

2. (b) Typically insurance companies charge a fee, called the premium, payable whether or not you have an accident. Then there is an additional payment, called an excess, payable only if you actually have an accident. Let Y be the customers annual cost, derive the pmf of Y and calculate the mean and variance of Y in each of the following scenarios.

   (i) A policy has a premium of $650 and an excess of $900. [6 marks] (ii) A policy has a premium of $300 and an excess of $2000. [6 marks]

3. (c) Explain whether or not there is a best policy to choose out of the two. If not explain what sort of people might prefer each type of policy. [3 marks]

4. (d) Now imagine a customer compares the two policies above but they do not know their total probability of having any sort of accident. What probability of them having any accident would make the two policies equal in terms of expected cost? [5 marks]

5. (e) If the insurance company charges such a customer an annual premium of $650 and an excess of $900 if they have an accident, calculate the insurance companys average profit for each male customer aged less than 22 years. [4 marks]