Problem:
Raashan, Sylvia, and Ted play the following game. Each starts with . A bell rings every seconds, at which time each of the players who currently has money simultaneously chooses one of the other two players independently and at random and gives to that player. What is the probability that after the bell has rung times, each player will have ? (For example, Raashan and Ted may each decide to give to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have , Sylvia will have , and Ted will have , and that is the end of the first round of play. In the second round Raashan has no money to give, but Sylvia and Ted might choose each other to give their to, and the holdings will be the same at the end of the second round.)
Answer Choices:
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Solution:
No player can ever end up with at the end of a round, because that player had to give away one of the dollars in play. Therefore the only two possible distributions of the money are -- and --. Suppose that a round of the game starts at --. Without loss of generality, assume that Raashan gives his dollar to Sylvia. Then the only way for the round to end at -- is for Ted to give his dollar to Raashan (otherwise Sylvia would end up with ) and for Sylvia to give her dollar to Ted; the probability of this is . Next suppose that a round starts at --; without loss of generality, assume that Raashan has and Sylvia has . Then the only way for the round to end at -- is for Sylvia to give her dollar to Ted (otherwise Raashan would end up with ) and for Raashan to give his dollar to Sylvia; the probability of this is . Thus no matter how the round starts, the probability that the round will end at -- is . In particular, the probability is that at the end of the th round each player will have .
Note: It may seem counterintuitive that an uneven distribution is more likely than an even distribution. But in a situation with larger initial bankrolls evenly distributed to a larger number of players, inequality reigns after many rounds. See this website:
decisionsciencenews.com/?s=happens+next
The problems on this page are the property of the MAA's American Mathematics Competitions