2020 ASRJC P2 Q10

A ferry company operates a 9am ferry, with a passenger capacity of \(108\), from Island \(X\) to Island \(Y\) daily. To maximise profits, \(113\) tickets are sold online each day because it was found that on average, \(p\%\) of customers who have purchased a ticket do not turn up.

1. It is known that there is a probability of \(0.012\) that at most \(1\) customer will not turn up for the ferry. Write down an equation in terms of \(p\), and hence find \(p\) correct to \(3\) decimal places.[3]

It is now given that \(p=6\).

2. Find the probability that there are no empty seats if every customer who turns up get a seat on the 9am ferry.[3]
3. Find the probability that, in a randomly chosen week, every customer who turns up gets a seat on the 9am ferry on at least \(5\) days of the week.[2]
4. Find the maximum number of tickets that could be sold so that the probability of a customer not getting a seat on the ferry is no more than \(0.01\).[3]
5. A period of forty days are randomly selected. Find the probability that the mean daily number of customers who purchased a ticket and managed to turn up during this period is at most \(106\).[2]