2017 PJC P2 Q11

Based on past statistical data, there is a \(7\%\) chance that a passenger with reservation for a flight will not show up. In order to maximise revenue, airline companies accept more reservations than the passenger capacity of its planes. State \(2\) assumptions needed such that the number of passengers who do not show up for a flight may be well modelled by a Binomial distribution. [2]

An airline company operates a flight from Singapore to Maluku on Boeing \(737-200\) planes, which has capacity of \(232\) passengers each.

1. Find the probability that when \(245\) reservations are accepted, the flight is overbooked, i.e. there is not enough seats available for the passengers who show up.[2]
2. Find the maximum number of reservations that should be accepted in order to ensure that the probability of overbooking is less than \(1\%\).[3]

This flight operates once daily throughout the year and \(245\) reservations are accepted for each flight

3. Find the probability that no flight is overbooked in a week.[2]
4. Taking a year as \(52\) weeks, estimate the probability that the mean number of flights that is overbooked in each week for the year is not more than \(1\).[3]