American Airlines Flight 179 from New York to San Francisco uses a Boeing 767-300 with 213 seats. Because some people with reservations don't show up, American Airlines can overbook by accepting more than 213 reservations. If the flight is not overbooked, the airline will lose revenue due to empty seats, but if too many seats are sold and some passengers are denied seats, the airline loses money from the compensation that must be given to the bumped passengers. Assume that there is a 0.0995 probability that a passenger with a reservation will not show up for the flight (based on the data from the IBM research paper "Passenger-Based Predictive Modeling of Airline No-Show Rates," by Lawrence, Hong, and Cherrier). Also assume that the airline accepts 236 reservations for the 213 seats that are available.
Procedures:
1. Find the probability that when 236 reservations are accepted for Flight 179, there are more passengers showing up than there are seats available. (That is, find the probability of more than 213 people showing up with reservations, assuming that 236 reservations were accepted. Calculations with the binomial probability formula would be extremely tedious, so the best approach is to use the TI-83/84 Plus calculator. See pg. 230 in section 5-3 of the textbook for instructions on Using a TI-83/84 Plus Calculator for the Binomial Probability Distribution.)
2. Is the probability of overbooking small enough so that it does not happen very often, or does it seem too high so that changes must be made to make it lower?
3. Now use trial and error to find the maximum number of reservations that could be accepted so that the probability of having more passengers than seats is 0.05 or less.