Each of m + 2 players pays 1 unit to a kitty inorder to

Chapter 7, Problem 7.60

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Each of m + 2 players pays 1 unit to a kitty inorder to play the following game: A fair coin is tobe flipped successively n times, where n is an oddnumber, and the successive outcomes are noted.Before the n flips, each player writes down a predictionof the outcomes. For instance, if n = 3,then a player might write down (H,H,T), whichmeans that he or she predicts that the first flipwill land on heads, the second on heads, and thethird on tails. After the coins are flipped, the playerscount their total number of correct predictions.Thus, if the actual outcomes are all heads, then theplayer who wrote (H, H, T) would have 2 correctpredictions. The total kitty of m + 2 is then evenlysplit up among those players having the largestnumber of correct predictions.Since each of the coin flips is equally likely toland on either heads or tails, m of the players havedecided to make their predictions in a totally randomfashion. Specifically, they will each flip oneof their own fair coins n times and then use theresult as their prediction. However, the final 2 ofthe players have formed a syndicate and will usethe following strategy: One of them will make predictionsin the same random fashion as the otherm players, but the other one will then predictexactly the opposite of the first. That is, when therandomizing member of the syndicate predicts anH, the other member predicts a T. For instance,if the randomizing member of the syndicatepredicts (H, H, T), then the other one predicts (T,)(a) Argue that exactly one of the syndicate members (c) With X as defined in part (b), argue thatE[payoff to the syndicate] = (m + 2)*E1X + 1(d) Use part (c) of to conclude thatE[payoff to the syndicate] = 2(m + 2)m + 1*1 12m+1and explicitly compute this number whenm =1, 2, and 3. Because it can be shown that2(m + 2)m + 11 12m+1> 2it follows that the syndicates strategy alwaysgives it a positive expected profit.will have more than n/2 correct predictions.(Remember, n is odd.)(b) Let X denote the number of the m nonsyndicateplayers that have more than n/2 correctpredictions. What is the distribution of X?T, H).