Consider the following resource capacity allocation problem that arises in intensive care units of congested hospitals. An intensive care unit has a limited number of beds. To simplify, let us that two types of patients arrive randomly each day: critical (c) and highly critical (hc). Assume that on a given day 12 beds are available at the unit. Critical patients arrive during the morning and highly critical patients arrive in the afternoon after all critical patients. The probability distribution of number of critical arrivals on that specific day is discrete uniform between 3 and 8 (P(D_c=i)=1/6, i=3,4,8) and the probability distribution of the number of highly critical arrivals on the same day is given by: P(D_hc=5)=0.1, P(D_hc=6)=0.2, P(D_hc=7)=0.3, P(D_hc=8)=0.4.

1. Assume that 5 beds are reserved for highly critical patients on that day. What is the expected number of beds that will be taken by critical patients?
2. Assume that 5 beds are reserved for highly critical patients on that day. What is the probability that at least one highly critical patient cannot be admitted? Note that this depends on the number of critical patient arrivals.
3. Medical experts place a value of 2 on highly critical patients that are admitted and a value of 1 on critical patients that are admitted. How many beds should be reserved for highly critical patients to maximize the expected total value of patients admitted on that given day?