Prelims Revision Set: Problems involving Overbooking Q2

In a certain airline route, there are a maximum of \(250\) seats and each ticket is sold at \(\$350\). If costs \(\$1050\) for the airline for every overbooked passenger if they show up for the flight for compensations. Assume that a booked passenger will not show for the flight with probability \(8\%\) and that the number of passengers who show up for each flight follows a binomial distribution. The airline has sold \(254\) tickets for a particular flight.

1. What is the probability that there will not be enough seats for the flight?
2. Find the expected amount of revenue for selling \(254\) tickets.

Assume that the airline sells \(n\) tickets with \(n>250\) as they decide to overbook.

3. Find the expected amount of revenue for selling \(n\) tickets in the form of \(\mathrm{f}\left( n \right)\mathrm{P}\left( X\le 250 \right)+\sum\limits_{r=251}^{n}{\left( \left[ \mathrm{f}\left( n \right)-\mathrm{g}\left( r \right) \right]P\left( X=r \right) \right)}\), where \(\mathrm{f}\left( n \right)\) and \(\mathrm{g}\left( r \right)\) are functions in terms of \(n\) and \(r\) respectively to be determine.

4. Hence, find the number of tickets to be overbooked in order to maximise the revenue.