Many healthcare clinics now employ overbooking" where they schedule more patients than they have appointment slots, since patients do not always show up. How many patients should be scheduled? Consider the following basic model to help answer this problem. There is only one doctor, and each patient requires exactly 20 minutes for their appointment with the doctor. Appointments are scheduled from 8:30am - 12:30pm (12 appointments) and from 1pm - 5pm (another 12 appointments). We ignore the time dynamics in the sense that if 24 or fewer patients show up, then we assume that the doctor can see them and finish by 5pm, but if more than 24 patients show up, then the first 24 are completed by 5pm, while the remaining are seen, one at a time, from 5pm onwards. For ex, if 28 patients show up then the last patient would finish their appointment at 6:20pm. This is slightly unrealistic, but greatly simplifies the model. (More complicated and realistic models are possible and used in practice.)
The number of patients scheduled for tomorrow is 32. Assume that the probability that any individual patient shows up is 0.75, and that patients show up (or not) independently from one another.

(a) describe why X, the random number of patients who show up tomorrow is a binomial random variable (think number of trials, probability of success, independence), give its parameters and its mean and variance.
(b) What is the probability that the doctor will finish serving patients by 5:20pm or earlier?
(c) Assume that the doctor goes home after the last patient is served, or at 5pm if all patients are finished before 5pm. What is the expected time that the doctor will go home?
(d) Suppose that the clinic is trying to decide n, where n is the number of patients to book in one day. What is the maximum value of n that still ensures that with probability 0.8 or more, all patients are completed by time 6pm? (Notice that here n takes the place of the 32 patients we previously assumed were scheduled.)
(e) Suppose that 5 doctors decide to work together in a single clinic with a single pool of patients. Suppose that each patient is willing to see any of the 5 doctors. Let m be the number of patients to book in one day. Now what is the maximum value of m that still ensures that with probability 0.8 or more, all patients are completed by time 6pm? (FYI: The answer is more than 5 times the answer to the previous problem due to risk pooling - more on that later in the course. Risk pooling can reduce costs.)