Please answer $($a$)$ through $($g$).$

Neighborhood Insurance sells fire insurance policies to local homeowners. The premium is $$120,$ the probability of a fire is $0\mathrm{.}1\%,$ and in the event of a fire, the insured damages $($the payout on the policy$)$ will be $$110,000.$

Required:

a$.$ Make a table of the two possible payouts on each policy with the probability of each.

Note: Negative answers should be indicated with a minus sign.

b$.$ Suppose you own the entire firm, and the company issues only one policy. What are the expected value, variance, and standard

deviation of your profit?

Note: Do not round intermediate calculations. Round your standard deviation to the nearest whole number.

c$.$ Now suppose your company issues two policies. The risk of fire is independent across the two policies. Make a table of the three

possible payouts along with their associated probabilities.

Note: Negative answers should be indicated with a minus sign. Round your "Probability" answers to $4$ decimal places.

d$.$ What are the expected value, variance, and standard deviation of your profit?

Note: Do not round intermediate calculations. Round your standard deviation to the nearest whole number.

e$.$ Compare your answers to $($b$)$ and $($d$).$ Did risk pooling increase or decrease the variance of your profit?

f$.$ Continue to assume the company has issued two policies, but now assume you take on a partner, so that you each own one$-$half of

the firm. Make a table of your share of the possible payouts the company may have to make on the two policies, along with their

associated probabilities.

Note: Negative answers should be indicated with a minus sign. Round your "Probability" answers to $4$ decimal places.

g$.$ What are the expected value and variance of your profit?