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I know it's different depending on who goes first, so for my example, I went first. This is what I came up with.

Total[(5/6)^x×1/6 where x = {1, 3, 5}]

This gives me the sum of the probability of winning on the first, second, and third round. The problem is, I want the overall probability of winning. The number of rounds may vary by a lot. You could roll a 6 on the first roll, or you may have 20 rounds before either of you roll it.

I just don't understand how I would come up with an answer to this.

Any help is appreciated.