Question: Great Air Commuter Service The Great Air Commuter Service Company started in 1984 to provide efficient and inexpensive commuter travel between Boston and New York City. People in the airline industry know Peter Wilson, the principal owner and operating manager of the company, as "a real promoter." Before founding Great Air, Peter operated a small regional airline in the Rocky Mountains with varying success. When Cascade Airlines offered to buy his company, Peter decided to sell and return to the East. Peter arrived at his office near Fenway Park in Boston a little later than usual this morning. He had stopped to have a business breakfast with Aaron Little, his long-time friend and sometime partner in various business deals. Peter needed some advice and through the years has learned to rely on Aaron as a ready source, no matter what the subject. Peter explained to Aaron that his commuter service needed a promotional gimmick to improve its visibility in the business communities in Boston and New York. Peter was thinking of running a contest on each flight and awarding the winner a prize.
The idea would be that travelers who commute between Boston and New York might just as well have fun on the way and have a chance to win a nice prize. As Aaron listened to Peter outlining his contest plans, his mind raced through contest ideas. Aaron thought that a large variety of contests would be needed, because many of the passengers would likely be repeat customers and might tire of the same old thing. In addition, some of the contests should be chance-type contests, whereas others should be skill based. "Well, what do you think?" asked Peter. Aaron finished his scrambled eggs before responding. When he did, it was completely in character. "I think it will fly," Aaron said, and proceeded to offer a variety of suggestions. Peter felt good about the enthusiastic response Aaron had given to the idea and thought that the ideas discussed at breakfast presented a good basis for the promotional effort. Now back at the office, Peter does have some concerns with one part of the plan. Aaron thought that in addition to the regular in-flight contests for prizes (such as free flights, dictation equipment, and business periodical subscriptions), each month on a randomly selected day a major prize should be offered on all Great Air flights.
This would encourage regular business fliers to fly Great Air all the time. Aaron proposed that the prize could be a trip to the Virgin Islands or somewhere similar, or the cash equivalent. Great Air has three flights daily to New York and three flights returning to Boston, for a total of six flights. Peter is concerned that the cost of funding six prizes of this size each month plus six daily smaller prizes might be excessive. He also believes that it might be better to increase the size of the large prize to something such as a new car but use a contest that will not guarantee a winner. But what kind of a contest can be used? Just as he is about to dial Aaron's number, Margaret Runyon, Great Air's marketing manager, enters Peter's office. He has been waiting for her to return from a meeting so he can run the contest idea past her and get her input. Margaret's response is not as upbeat as Aaron's, but she does think the idea is worth exploring. She offers an idea for the largeprize contest that she thinks might be workable. She outlines the contest as follows. On the first of each month, she and Peter will randomly select a day for that month on which the major contest will be run. That date will not be disclosed to the public.
Then, on each flight that day, the flight attendant will have passengers write down their birthdays (month and day). If any two people on the plane have the same birthday, they will place their names in a hat and one name will be selected to receive the grand prize. Margaret explains that because the capacity of each flight is 40 passengers plus the crew, there is a very low chance of a birthday match and, therefore, the chance of giving away a grand prize on any one flight is small. Peter likes the idea, but when he asks Margaret what the probability is that a match will occur, her response does not sound quite right. She believes the probability for a match will be 40/365 for a full plane and less than that when there are fewer than 40 passengers aboard. After Margaret leaves, Peter decides that it would be useful to know the probability of one or more birthday matches on flights with 20, 30, and 40 passengers. He realizes that he will need some help from someone with knowledge of statistics.
Required Tasks: 1. Assume that there are 365 days in a year (in other words, there is no leap year). Also assume there is an equal probability of a passenger's birthday falling on any one of the 365 days. Calculate the probability that there will be at least one birthday match for a flight containing exactly 20 passengers.
2. Repeat (1) above for a flight containing 30 passengers and a flight containing 40 passengers. Again, it will be easier to compute the probabilities of one or more matches if you first compute the probability of no birthday matches.
3. Assuming that each of the six daily flights carries 20 passengers, calculate the probability that the airline will have to award two or more major prizes that month.