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Bumped from your Flight
Posted:
Jul 3, 1996 11:02 PM


One possible miniproject that can be used to illustrate the expected value of a discrete distribution would be to have the class look at why airlines overbook. Here's a rough sketch of a problem that would require everyone to contribute a chunk in order for the class as a whole to find a solution.
Setup: An airline runs a jet with suchandso many seats (say, 125) between these two cities. Suppose that each of the tickets cost a certain fixed amount (say, $159) (an unrealistic assumption that all seats cost the same; you're welcome to refine this) and the ticket is fully refundable. Past experience informs the airline that there is a certain likelihood (say 5%) that any given ticketholder won't show up. (Another unrealistic assumption; here, independence  no families, etc., travel together.) If the airline overbooks and has to deny boarding to confirmed ticketholders, then they give the bumped passenger a compensation package (cash, meal, hotel, guaranteed seat on next plane,...) worth $x (say, $400). So, if more people show up than the jet has seats, the airline loses $x per bumpee, and if fewer people show up, the airline loses ($159 or whatever) on each empty seat.
Give individuals or small groups a certain number of confirmed ticketholders, different numbers to each person/group, and ask questions such as these:  How many should show up for the airline to make the most off the flight?  If a certain number t tickets are sold, what is the probability distribution for the number of folks who actually show up (easy)  If X represents the possible values of the airline's 'loss' on the flight, what values can X take on? What is the probability dist of X? (And what does 'loss' mean here?) (easy, but definitely less so) Note: The individual/small group is still working with only the single number of tickets sold that you assigned.  If that many tickets are sold, what is the airline's expected 'loss' on the flight? What does this mean?
Have all the individuals/groups pool their results and decide how many tickets the airline should sell for this flight in order to maximize their revenue.
Thus, overbooking. (In case you ever wondered.)
 Peter Blaskiewicz McLennan Community College Waco, Tx pjb@mcc.cc.tx.us



