2024 SAJC H1 Q9

At a clinic, records show that \(60\)% of patients turn up for their appointments every day. Over a period of \(50\) days, there are \(48\) patients who make appointments to see the doctor each day. The number of patients who turn up for their appointments each day is independent of other days.

1. Find the probability that the mean number of patients who turn up for their appointments per day is at least \(29\).[3]

To better utilise appointment slots, the clinic decides to allow for overbooking of appointment slots by increasing the number of appointments that can be made on each day. It is known that on any day, the doctor has time to see at most \(40\) patients.

2. Find the greatest number of appointments that can be made so that there is a probability of at least \(0.85\) that the doctor has time to see all patients who turn up.[3]