Please help solve the following question:
Suppose that you consider insuring some property against damage. There is only one insurance company. After examining the risks and the premium, P, you find that you have no clear preference between buying insurance and not. Then, you hear about a new policy called probabilistic insurance supplied by the same company. In this policy, you pay half of the regular premium, i.e., you pay P/2. In case of damage, with probability 1/2 you pay the rest of the premium and the company covers all the damage, and with probability 1/2 the insurance company returns your premium and you suffer all the loss form the damage, would you buy the probabilistic insurance if:
a. If a risk averse person is indifferent between buying the insurance or not, can it be actuarially fair (the insurance pays full cost of damage when it happens)?
b. Many risk averse people answered that they would not buy the probabilistic insurance in the situation described above. Then the researchers concluded that such an answer by a risk averse individual violates the expected utility theory. Prove that any risk averse person in the situation described above should buy the probabilistic insurance.
If a risk averse person is indifferent between buying the insurance or not, can it be actuarially fair (the insurance pays full cost of damage when it happens)?