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Problem 9

Who goes first - more on dice sums

Due: January 19
Points: 2

The previous problem asks about the probability that two pairs of dice add up to the same number between 2 and 12.

Note that this is a common method to decide who gets to go first in a game such as Monopoly: each of \(n\) players rolls 2 dice, and whoever gets the highest sum goes first. If there is a tie for the highest sum, those players have to roll again.

If \(n=2\), we have a two player game, and the probability from the previous question applies. For this problem, I want you to come up with a formula that, for any \(n\), gives the probability that there is a tie for the highest dice sum. That is, if \(n\) players each roll 2 dice and take the sum, what is the probability that more than one player ties for the highest sum?

(Hint: If you're stuck, try starting with \(n=3\), then \(n=4\), and work your way up until you see the pattern.)