Fairness of rolling a die 3 times

Question

I have a pair of sneakers to give away and 3 friends would like to have it. I decided to have them each roll a dice and the person with the highest number wins the sneakers. Does the first person to roll the dice have an advantage over the other two?

Answer 1

No, but if you or your friends doubt that you can try and get three dice and have all of them roll at the same time.

BTW can you explain a bit why you would expect the first player to have an advantage? Then it would be a bit easier for us to explain why it isn't the case.

For instance, if it is because you believe that the average outcome of dice throws get lower over time, I could advice you to test this believe by throwing dice 1000s of times before your friends come in. If throws get lower and lower you would expect to see after, say, an hour to only get 1's and 2's. This is easy to test (and hence debunk) experimentally. But perhaps you already didn't believe this and had a different reason to worry. Can you tell us what it is?

Answer 2

I agree with @Vincent (+1), that the first person to roll has no advantage. (That is, unless you had an an unfair rule that the first person to roll the highest number wins, in case of a tie.)

My concern is with @Henry's question: how to pick a winner if the top two get the same number, which has probability almost $1/4.$ (That's not obvious: simulation in R below is accurate to about 2 decimal places. Maybe you can get an exact combinatorial solution.)

A partial solution might be to say that the lower number wins if the the top two are tied. It's easy to see that the probability all three are tied is only $1/36.$ In that case you'd have everyone roll again.

Simulation: In $100\,000$ 3-roll experiments keep track of the top two in a $2\times 100\,000$ matrix top2. Then find the proportion of columns having duplicate numbers.

set.seed(2020)
top2 = replicate(10^5, sort(sample(1:6, 3, rep=T))[2:3])
mean(top2[1,]==top2[2,])
[1] 0.23763