A bit of economic history. A salient feature of passenger air travel is that a small percentage of the ticket holders for any given flight does not actually show up (check-in) in time for the flight. Since airlines run nearly identical flight patterns each day, by now these percentages are fairly well known by the individual airlines (e. g., United knows that on average 91% of ticketed passengers show for United's weekday 6:32am Chicago-San Francisco flight). As a result, the airlines know that with high probability a slightly oversold flight will not actually be overbooked at departure. Suppose that on a particular flight there are 225 seats, and that each ticket-holder (independently) shows up with a 91% chance. Each passenger pays a $300 airfare, but the airline refunds all but $75 of the fare to any passenger that fails to show up for the flight. (That is, the airline collects $300 from every passenger that shows, and keeps $75 from every passenger that does not.) (a) Using software of your choice, find the expected total revenue if the airline sells 225 tickets. (b) Now suppose the airline sells more than 225 tickets. In this case, the airline would have to bump passengers if 226 or more ticket-holders show up. Assume the airline keeps a bumped passengers $300 fare and puts him or her on a later flight for no additional charge, but incurs a gross cost (in terms of lost goodwill, vouchers, and internal re-ticketing costs) of $900 per bumped passenger. Thus, if 226 or more ticket-holders show up, the airlines net revenues are 225 × $300 from the seated passengers less $600 per bumped passenger plus $75 per no-show, if any. Using software of your choice, find the expected profit if the airline sells two extra (i. e., 227) tickets for this flight. (N. B.: This requires explicitly evaluating the probability and payoff for each possible outcome of the random variable "number who show". This might not be obvious at first, but make sure you understand why.) (c) Find the optimal number of seats to overbook – that is, the number of tickets to sell in order to maximize expected profit on this flight. Plot the expected profit versus the number of tickets sold, for 225 through 260 tickets sold. (d) What is the expected profit under the optimal overbooking strategy? What is this as a percentage gain over the naive strategy of selling only 225 tickets?