You live in an area that has a possibility of

You live in an area that has a possibility of incurring a massive earthquake, so you are considering buying earthquake insurance on your home at an annual cost of $180. The probability of an earthquake damaging your home during one year is 0.001. If this happens, you estimate that the cost of the damage (fully covered by earthquake insurance) will be $160,000. Your total assets (including your home) are worth $250,000.
(a) Apply Bayes’ decision rule to determine which alternative (take the insurance or not) maximizes your expected assets after one year.
(b) You now have constructed a utility function that measures how much you value having total assets worth x dollars (x ≥ 0). This utility function is U(x) = √x. Compare the utility of reducing your total assets next year by the cost of the earthquake insurance with the expected utility next year of not taking the earthquake insurance. Should you take the insurance?