# What is the probability of tie in a game of rock paper scissors for multiple people?

In the game of RPS, a tie happens when every player chooses the same symbol (i.e all rock) or there is at least 1 rock, 1 paper, and 1 scissors chosen. A tie results in the reply of the round. What is the probability of a tie happening when 2, 3, 4...n people play the game? Is there any general formula to be used? Assume that the probability of each symbol is 1/3.

What I have thought so far is with 4 people. By assuming that P1 plays rock, there are 3 players left yet to choose their symbol. My line of thinking is to find the probabilities of having at least 1 paper and at least 1 scissors using the binomial theorem and then multiplying together to find their 'and' probability. However, I suppose that my method uses 2 different sets of data, 1 for the paper and 1 for scissors.

This has boggled my brain.

Thanks very much

Hints, for general $$n$$: