Please help me solve all parts of this problem with an explanation Thank you

There are three types of residents in Bruintown. The health gurus (H ), regular people (R) and people prone to sickness (P). Assume each group represents one third of the population. The utility of each resident of Bruintown is given by u(w} = 210 0.05192; where 10 represents the wealth of a particular resident. Initially, each resident has a wealth of 12. Residents face a health shock, that is, there is some positive probability that a resident will get sick. A visit to the doctor costs 8 dollars and fully heals the resident. Each group has a different propensity to being sick: health gurus have a probability of 0.1; regular people have a probability of 0.35 and people prone to sickness have a probability of 0.6. 1. What is the expected utility of each type of resident? 2. Ashe Center is promoting a health insurance plan. Let's assume that that every resident of Bruintown will take up this plan, because it will be fully paid for by the government. What is the actuarial fair premium (that is, the premium that yields zero prots to the insurance company)? 3. Now assume that individuals choose whether to have insurance or not, and pay for the insur- ance premium themselves. If Ashe Center charges the premium you computed in the previous question, which types of residents will sign up? 4. If the individuals sign up for insurance in the way you described in the previous item, what is the expected prot for Ashe Center? Interpret your result