United flight 15 from New York’s JFK to San Francisco uses a Boeing 757-200 with 182 seats. Because some people with reservations don’t show up, United can overbook by accepting more than 182 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 bumped passengers. Assume that there is a 0.0975 probability that a passenger with a reservation will not show up for the flight. Also assume that airline accepts 200 reservations for the 182 seats that are available.
1. Find the probability that when 200 reservations are accepted for United flight 15, there are more passengers than seats available. Show your calculation.
2. Is the probability of overbooking small enough that it does not happen very often (ie, unusual), or is it high enough to consider making changes to lower the probability of overbooking? Justify your answer using a complete sentence and proper grammar.
3. Use trial and error (and your calculator) 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. Make a table like the one below:
Number of reservations
P(more passengers than seats available)