I have struggled to understand the parts, why cant we let p=clumsy group, why grace group?

* Q25. There are two groups of equal size: the clumsy and the graceful. Each group has a utility function given by U = VM where M = 100 is the initial wealth level for every individual. Each member of the clumsy group faces accidentally damaging their own property and incurring a loss of 36 with probability 0.5. Each member of the graceful group faces the same loss with probability 0.1. a) What is the most a member of each group would be willing to pay to insure against this loss? Let x denote the reservation price for the clumsy group; x must satisfy: 100 - Xc = 0.5v100 + 0.5v64 = 9, which yields xc = 19. If we let x, denote the reservation price for the graceful group, we get 100 -X, = 0.9V100 + 0.1V64 = 9.8, which yields x, = 3.96. b) In part (a), if it is impossible for outsiders to discover which individuals belong to which group, will it be practical for members of the graceful group to insure against this loss in a competitive insurance market? (For simplicity, you may assume that insurance companies charge only enough in premiums to cover their expected benefit payments.) Explain. If members of the two groups are indistinguishable, an insurance company will have to charge the same premium to each. If its policyholders consisted of equal numbers of people from each group, this premium would have to cover the expected loss, which is [(0.5) (36) + (0.1)(36)]/2 = 10.8. Since this exceeds the reservation price of members of the graceful group, nobody from that group would buy insurance. And with only clumsy members remaining in the insured pool, the premium would have to rise to 18 in order to cover the expected loss for members of that group. c) Now suppose that the insurance companies in part (b) have a test for identifying which persons belong to which group (they force them to make a video of themselves attempting ballet). The test is good but not perfect - if the test says that a person belongs to a particular group, the probability that he really does belong to that group is p