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Raashan, Sylvia, and Ted and Chip Firing

Source: 2019 AMC 12B #19

February 14, 2019
2019 AMC 12Bprobability2019 AMCAMCAMC 12

Problem Statement

Raashan, Sylvia, and Ted play the following game. Each starts with $1\$1. A bell rings every 1515 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 $1\$1 to that player. What is the probability that after the bell has rung 20192019 times, each player will have $1\$1? (For example, Raashan and Ted may each decide to give $1\$1 to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $0\$0, Sylvia would have $2\$2, and Ted would have $1\$1, and 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 $1\$1 to, and and the holdings will be the same as the end of the second [sic] round.
<spanclass=′latex−bold′>(A)</span>17<spanclass=′latex−bold′>(B)</span>14<spanclass=′latex−bold′>(C)</span>13<spanclass=′latex−bold′>(D)</span>12<spanclass=′latex−bold′>(E)</span>23<span class='latex-bold'>(A) </span> \frac{1}{7} \qquad<span class='latex-bold'>(B) </span> \frac{1}{4} \qquad<span class='latex-bold'>(C) </span> \frac{1}{3} \qquad<span class='latex-bold'>(D) </span> \frac{1}{2} \qquad<span class='latex-bold'>(E) </span> \frac{2}{3}
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