2019 AMC 12B Problems/Problem 19

Problem

Raashan, Sylvia, and Ted play the following game. Each starts with 1 dollar. A bell rings every 15 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 1 dollar to that player. What is the probability that after the bell has rung 2019 times, each player will have 1 dollar?

There is a long example given following the problem statement: For example, Raashan and Ted may each decide to give 1 dollar to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have no money, Sylvia will have 2 dollars, and Ted will have 1 dollar, 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 dollar to, and the holdings will be the same at the end of the second round.