2019 AMC 12B Problem 19
Problem 19 of 25HarderCounting & Probability
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 have 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
From each of the three players gives to one of two others, so there are equally likely outcomes; only the cyclic gift patterns return to a probability of
From a state the broke player gives nothing, and checking the equally likely choices of the other two shows exactly one yields again probability
So after any ring the probability of is including after rings.
Thus, B is the correct answer.