2025 TGM P2 Q11

Airlines overbook their flights primarily to compensate for the predictable behavior of passengers missing or canceling their reservations. By overbooking, they aim to ensure maximum seat occupancy and revenue, given that a certain percentage of ticketed passengers typically do not show up. This practice helps in offsetting operational costs and maximizing profit, even though it can occasionally lead to challenges if too many passengers arrive for a flight.

In a certain airline route, there are a maximum of \(150\) seats. Each ticket is sold at \(\$350\). Assume that a booked passenger will not show for the flight with probability \(3\%\) and that the number of passengers who show up for each flight follows a binomial distribution.

1. The airline has sold \(155\) tickets for a particular flight.
   1. Show that the expectation of the number of passengers that will show up is \(150.35\) and find the variance.[2]
   2. What is the probability that there will not be enough seats for the flight?[3]

Another flight with the same route sold \(152\) tickets. It costs \(\$1050\) for the airline for every overbooked passenger if they show up for the flight for compensations. Assuming that the compensations are the only cost incurred for each flight,

2. find the expected amount of profit from selling \(152\) tickets.[4]

Assume that the airline sells \(n\) tickets with \(n > 150\) as they decide to overbook and \(Y\) is the number of passengers who will show up for the flight out of \(n\) passengers.

3. Find the expected amount of profit for selling \(n\) tickets in the form of

   \(\mathrm{f}(n)\mathrm{P}(Y \le 150) + \sum_{r=151}^{n} \left\{ \left[ \mathrm{f}(n) - 1050 \mathrm{g}(r) \right] \mathrm{P}(Y = r) \right\},\) where \(\mathrm{f}(n)\) and \(\mathrm{g}(r)\) are functions in terms of \(n\) and \(r\) respectively to be determined.

   [3]
4. Hence, find the number of tickets to be overbooked to maximise the expected profit.[1]