1. Airline Overbooking. On any flight, it is likely that a small number of ticket-holders won't show up. In order to fly a full plane, airlines overbook - that is, they sell more tickets for the flight than there are seats on the plane. The problem is, the airline has no idea in advance how many people will be no-shows. By how many seats should the airline overbook? If it sells too many tickets above the plane's capacity, it may have to offload passengers at boarding, causing distress and annoyance. If it sells too few, the plane may have to fly with empty seats. In order to figure out what the airline should do, we model passengers' propensity to show up for flights using the binomial distribution. We assume that each ticket holder has a fixed probability p of showing up for a flight, independently of all other ticket holders, and that this probability is the same for all ticket-holders. Granted these are somewhat unrealistic assumptions, but they will do as a first pass at modeling the situation. Let's say that the airline sells N tickets for a certain flight. In this binomial model, each ticket-holder is treated like a coin that has probability p of coming up heads, independently of all the other coins. Each of the N coins is flipped once, and the airline wants to predict how many coins come up heads; in other words, how many ticket-holders show up for the flight. On past flights, about 5% of ticket-holders have missed their flights, so we assume that each ticket-holder has a 5% chance of being a no-show, hence a 95% chance of showing up. We'll also assume there are 100 seats on the flight. Let SN be the random variable representing the number of ticket-holders that actually show up for the flight when the airline sells N tickets. We want to find the probability that SN exceeds 100, as the airline would like to keep that probability low. (a) Write down expressions for the following. Your expressions will involve the quantity N; you do not need to simplify them. . P(SN = 100) . P(SN 100) (b) The airline is trying to decide how many tickets, N, to sell for the flight, to keep the risk of having to offload passengers reasonably low. Of course if N s 100, this probability will be 0, but then there may well be empty seats on the flight because of no-shows. They think up to a 4% chance of having more than 100 passengers show up for the flight is a reasonable risk, as that would mean offloading passengers on at most 1 out of every 25 flights. For N = 101, 102, .... 110, find the probability that the number of people who actually show up for the flight exceeds 100 when the airline sells N tickets. Use Excel: The box at the end of this assignment describes Excel's BINOM.DIST function. Based on the probabilities that you find, how many tickets should the airline sell for the flight above the plane's capacity of 100 seats? Please explain