HW XXXXXXXXXXStrategic Abstention XXXXXXXXXXYou and your two friends, Anthony and Renee, are trying to determine whether to watch an action move or a romantic comedy. Each of you can express a...

Please do both questions


HW 8 (+18) Strategic Abstention 1. (+8) You and your two friends, Anthony and Renee, are trying to determine whether to watch an action move or a romantic comedy. Each of you can express a preference or not. If more of you express a preference or one movie or the other, you will all watch that movie. If two of you express opposite preferences and the other expresses no preference, or if none of you express any preference, you will flip a coin and watch either movie with equal probability. From past experience, you expect that Anthony prefers action movies and is 70% likely to express his preference. Renee prefers romantic comedies but is only 40% likely to express her preference. Suppose that their expressions of preference are statistically independent. If you express a preference for the action movie, how much will it increase the probability that the three of you watch an action movie? If you express a preference for the romantic comedy, how much will it increase the probability of watching a romantic comedy? How does this illustrate the underdog effect? 2. (+10) Avery, Betty, Christina, Danielle, and Evelyn form a study group, and are trying to correctly answer a true/false question that is equally likely to be true or false. Before meeting, each tries independently to answer the question. Avery, Betty, and Christina are familiar with the material, and each has probability ???? = .8 of getting the answer right. Danielle and Evelyn are less familiar with the material, and each has probability ???? = .6 of getting the answer right. Which is more likely to be right, the majority opinion of Avery, Betty, and Christina, or the majority opinion of all five? (Hint #1: it suffices to restrict attention to situations in which the committee of three and the committee of five come to opposite conclusions. Hint #2: depending on how you solve the problem, it might be useful to recall that there are ??23 = 3! 2!1! = 3 ways for a three-person vote to split 2-against-1.) Does your answer change if ???? = .7? HW 8 (+18) Strategic Abstention