Two players alternate throwing a six-sided die. The first player who fails to roll a higher number than the preceding roll loses. What is the probability that the first player wins?
For an n-sided die, the probability the first player wins is 1 - (1-1/n)^n. When n is 6 this is 31031/46656 = .665102023319616...
(See http://www.physics.harvard.edu/probweek/sol54.pdf).
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