Mario, Yoshi, and Toadette play a game of "nonconformity": They each choose rock, paper, or scissors. If two of the three people choose the same symbol, and the third person chooses a different symbol, then the one who chose the different symbol wins. Otherwise, no one wins.
If they play 4 rounds of this game, all choosing their symbols at random, what's the probability that nobody wins any of the 4 games? Express your answer as a common fraction.
To find the probability that nobody wins any of the 4 games, we need to consider the possible outcomes for each game.
In each game, each player can choose one of three symbols: rock, paper, or scissors. Since they are choosing at random, there are 3 choices for each player in each game.
For the first game, there are two scenarios where nobody wins:
1. All three players choose the same symbol. There are 3 ways this can happen: (rock, rock, rock), (paper, paper, paper), or (scissors, scissors, scissors).
2. Each player chooses a different symbol. There are also 3 ways this can happen: (rock, paper, scissors), (paper, scissors, rock), or (scissors, rock, paper).
Therefore, the probability of nobody winning the first game is (3 + 3) / (3^3) = 6 / 27 = 2 / 9.
Since each game is independent, the probability of nobody winning any of the 4 games is (2 / 9)^4 = 16 / 6561.
So the probability that nobody wins any of the 4 games is 16 / 6561, which is the final answer.