Patients arrive to a hospital emergency room at a rate of 2 per hour. A doctor works a 12-hour shift from 6 a.m. until 6 p.m. Answer the following questions by approximating the arrival-counting...

Patients arrive to a hospital emergency room at a rate of 2 per hour. A doctor works a 12-hour shift from 6 a.m. until 6 p.m. Answer the following questions by approximating the arrival-counting process as a Poisson process:

(a) If the doctor has seen six patients by 8 a.m., what is the probability that the doctor will see a total of nine patients by 10 a.m.?

(b) What is the expected time between the arrival of successive patients? What is the probability that the time between the arrival of successive patients will be more than 1 hour?

(c) What is the expected time after coming on duty until the doctor sees her first patient? What is the probability that she will see her first patient in 15 minutes or less after coming on duty?

(d) What is the probability that the doctor will see her thirteenth patient before 1 p.m.?

(e) Of patients admitted to the emergency room, 14% are classified as “urgent.” What is the probability that the doctor will see more than six urgent patients during her shift?

(f) The hospital also has a walk-in clinic to handle minor problems. Patients arrive at this clinic at a rate of 4 per hour. What is the probability that the total number of patients arriving at both the emergency room and clinic from 6 a.m. until 12 noon will be greater than 30?