You have 100 business clients who own businesses that you insure against floods. The probability density function of the number of claims k per year for these 100 clients when rainfall is at or below average (which happens 50% of the time) is described by a binomial distribution in which floods are independent across the clients and the probability of a flood for each is 10% (p=.1) per year. Your insurance covers the first $100,000 in damage. For all your clients, if there is damage, it will very likely exceed $100,000. Thus, for simplicity assume that when there is a claim, you have to pay out $100,000 on each claim. You currently charge $20,000 per year for this coverage.
1. When rainfall is at or below average, what is the expected number of claims?
2. When rainfall is at or below average, what is the standard deviation of the number of claims?
3. When rainfall is at or below average, what is the probability that you will have 5 or fewer claims? Show work.
4. When rainfall is at or below average, what is the probability that you will lose money on your pool of policies? Show work.
5. When rainfall is above average (which happens 50% of the time), the pdf for the number of claims is given by the binomial with p=.20. What is the probability that you will lose money if rainfall is above average?
6. Given that rainfall is above average half the time and below average half the time, what is the unconditional probability that you will lose money if you charge $20,000 per $100,000 in coverage?