At first, there is only a limited supply of 1000 doses of See-Ya-Corona. The government has to develop a strategy for allocating vaccine doses. You have been hired to determine how the government should allocate vaccine doses. All available doses of See-Ya-Corona will be allocated to health care workers and professors. Assume the government wants to minimize the expected number of new COVID-19 infections (or equivalently, maximize the expected number of COVID-19 cases prevented). Let x1 denote the number of health care workers vaccinated and x2 denote the number of professors vaccinated. Each health care worker has a 20% chance of becoming infected and each professor has a 10% chance of becoming infected. An infected health care worker infects 10 new people, while an infected professor infects 5 new people. A single dose of See-Ya-Corona provides 100% protection against COVID-19.

1. (a)What is the function that represents the government's preferences?
2. (b)Because of repeated contact with COVID-19 patients, health care workers now require 2 booster shots in addition to 1 dose of See-Ya-Corona. Everything else remains the same as in part (a). Let x3 denote booster shots. Show function that represents the government's preferences.
3. (c)Assume everything is the same as in part (a), except for the probability that a professor becomes infected with COVID-19. What does the probability that a professor becomes infected have to be for the government to vaccinate both health care workers and professors in an optimal allocation?