Beat My Roll

Question

You and I each roll one fair die. What is the probability that your roll is strictly higher than mine?

Accepted Answer

By symmetry P(textyou higher) = P(textme higher), and the three outcomes partition all possibilities: 2,P(textyou higher) + P(texttie) = 1. A tie requires your die to match mine: P(texttie) = frac636 = frac16. Hence P(textyou higher) = frac1 - frac162 = frac512 approx 0.4167. The symmetry-plus-ties trick generalizes instantly: for any i.i.d. pair, P(X > Y) = frac1 - P(X=Y)2, far faster than enumerating 36 cells, and it works for 100-sided dice just as well.

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