At the emergency department in one of the public hospitals in South East Queensland, patients arrive according to a Poisson distribution with an average of 15 minutes between arrivals. i. What is the probability that the hospital experiences a busy period and more than three patients arrive in half an hour? ii. Last Saturday, between the hours of 10am and 12pm, there were 10 emergency arrivals. A nurse on shift-work was required to triage all emergency arrivals between 9am and 1pm that day. How many patients were expected to arrive over that time? iii. An on-call doctor is attempting to nap in between emergency arrivals. The doctor is going to set their alarm such that the probability of a patient arriving in that time is less than 40%. How long can the doctor set their alarm for?