Abstract In the study of nontransitive dice, seemingly paradoxically, the probability that one die rolls higher than another is not transitive. We prove a more general result which implies that a cyclic sum of ordering probabilities for n random variables can be much greater than 1, and that each probability can be much greater than 1 n . In particular, we establish an upper bound for this sum and give necessary and sufficient conditions under which it is attained. We also prove that, given an absolutely continuous univariate probability distribution and e > 0 , there are random variables X 1 , X 2 , … , X n , each of which has this distribution, and for which each of the probabilities P r X 1 > X 2 , P r X 2 > X 3 , … , P r X n − 1 > X n , and P r X n > X 1 exceeds 1 − e .

中文翻译：

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循环概率和的上限

摘要 在对非传递骰子的研究中，看似矛盾的是，一个骰子比另一个骰子掷得更高的概率不是传递性的。我们证明了一个更一般的结果，这意味着 n 个随机变量的排序概率的循环总和可以远大于 1，并且每个概率可以远大于 1 n 。特别是，我们为这个总和建立了一个上限，并给出了达到它的充分必要条件。我们还证明，给定绝对连续的单变量概率分布且 e > 0 ，存在随机变量 X 1 , X 2 , … , X n ，每个变量都具有这种分布，并且对于其中每个概率 P r X 1 > X 2 , P r X 2 > X 3 , … , P r X n - 1 > X n ，并且 P r X n > X 1 超过1 - e 。