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Magazine Problem 123
A history teacher has given her class a list of seven essay questions to study before the next test. The teacher announced that she will choose four of the seven questions to give on the test, and each student will have to answer three of those four questions. a. In how many ways can the teacher choose four questions from the set of seven? b. Suppose that a student has enough time to study only five questions. In how many ways can the teacher choose four questions from the set of seven so that the four selected questions include both questions that the student did not study? c. What is the probability that the student in part b will have to answer a question that he or she did not study? That is, what is the probability that the four questions on the test will include both questions that the student did not study?
Problem 124
Consider the following three games. Which one would you be most likely to play? Which one would you be least likely to play? Explain your answer mathematically. Game I: You toss a fair coin once. If a head appears you receive \(\$ 3\), but if a tail appears you have to pay \(\$ 1\). Game II: You buy a single ticket for a raffle that has a total of 500 tickets. Two tickets are chosen without replacement from the \(500 .\) The holder of the first ticket selected receives \(\$ 300\), and the holder of the second ticket selected receives \(\$ 150 .\) Game III: You toss a fair coin once. If a head appears you receive \(\$ 1,000,002\), but if a tail appears you have to pay \(\$ 1,000,000\).
Problem 128
Customers arrive at the checkout counter of a supermarket at an average rate of 10 per hour, and these arrivals follow a Poisson distribution. Using each of the following two methods, find the probability that exactly 4 customers will arrive at this checkout counter during a 2 -hour period. a. Use the arrivals in each of the two nonoverlapping 1 -hour periods and then add these. (Note that the numbers of arrivals in two nonoverlapping periods are independent of each other.) b. Use the arrivals in a single 2 -hour period.
