Problem

Each of m + 2 players pays 1 unit to a kitty in order to play the following game: A fair c...

Each of m + 2 players pays 1 unit to a kitty in order to play the following game: A fair coin is to be flipped successively n times, where n is an odd number, and the successive outcomes arc noted. Before the n flips, each player writes down a prediction of the outcomes. For instance, if n = 3, then a player might write down (H, H, T), which means that he or she predicts that the first flip will land on heads, the second on heads, and the third on tails. After the coins are flipped, the players count then total number of correct predictions. Thus, if the actual outcomes are all heads, then the player who wrote (H, H, T) would have 2 correct predictions. The total kitty of m + 2 is then evenly split up among those players having the largest number of correct predictions.

Since each of the coin flips is equally likely to land on either heads or tails, m of the players have decided to make their predictions in a totally random fashion. Specifically, they will each flip one of their own fair coins n times and then use the result as their prediction. How-ever, the final 2 of the players have formed a syndicate and will use the following strategy: One of them will make predictions in the same random fashion as the other m players, but the other one will then predict exactly the opposite of the first. That is, when the randomizing member of the syndicate predicts an H, the other member predicts a T. For instance, if the randomizing member of the syndicate predicts (H, H, T), then the other one predicts (T, T, H).

(a) Argue that exactly one of the syndicate members will have more than n/2 correct predictions. (Remember, n is odd.)
(b) Let X denote the number of the m nonsyndicate players who have more than n/2 correct predictions. What is the distribution of X?
(c) With X as defined in part (b), argue that
E[payoff to the syndicate] =
(d) Use part (c) of Problem 1 to conclude that
E[payoff to the syndicate] =
and explicitly compute this number when m = 1, 2, and 3. Because it can be shown that
it follows that the syndicate’s strategy always gives it a positive expected profit.

Problem 1

There are n + 1 participants in a game. Each person independently is a winner with probability p. The winners share a total prize of 1 unit. (For instance, if 4 people win, then each of them receives , whereas if there are no winners, then none of the participants receives anything.) Let A denote a specified one of the players, and let X denote the amount that is received by A.

(a) Compute the expected total prize shared by the players.
(b) Argue that .
(c) Compute E[X] by conditioning on whether Ais a winner, and conclude that
when B is a binomial random variable with parameters n and p.