For this question, suppose that you and 5 of your friends have agreed to play 9 consecutive games of Among Us, with a fixed number of 2 imposters per game. Let X denote the number of games in which you play as the imposter, such that X ∼ Bin(9, 1 3 ). (a) Calculate P(X ≤ 1). (2 marks) (b) Calculate E(X) and the standard deviation of X. Hint: You can use the table at the end of Readings 5.2. (2 marks) (c) Suppose that you have played 5 out of 9 games, and have been an imposter 3 times so far. One friend, who has done an introductory statistics class, claims that it is almost certain that you won’t be an imposter more than once more. Your other friends agree it is unlikely. However you are enjoying the imposter role, and think you have decent odds of playing in this role a couple more times. The group decides to offer a wager - if you end up being imposter a total of 2 or more times out of the remaining 4 games, you will win the wager, and they will chip in together to buy you a voucher for a store of your choice. If however you are only an imposter for at most 1 more game, then you will lose the wager, and have to buy your friends a round of drinks. Assume that you will only agree to the wager if you have a more than 50% chance of winning. Do you agree to the wager? Explain your reasoning clearly. (3 marks) (d) Given the success of your Among Us gaming session, you agree to reconvene the following weekend. When the weekend arrives, more friends have joined your Discord group, so you now have 12 players in total. You decide to play 12 games, with a fixed number of 3 imposters per game. You are hoping to play as the imposter for several games, but would also like to play a few games as a crew member. What is the probability that you play between 4 and 7 games (inclusive) as the imposter?

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