One possible mini-project 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.
Set-up: An airline runs a jet with such-and-so 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 firstname.lastname@example.org