Abstract

We give an explicit geometric proof that for each n there exists a cycle of n independent random variables such that for each random variable from this cycle the probability that it takes a value smaller than the next random variable from the cycle is at least $1-1/\left(4{\hspace{0.17em}\text{cos}\hspace{0.17em}}^2\frac{\pi }{n+2}\right)$. We also show that this bound cannot be replaced by any larger number. This generalizes the famous paradox of nontransitive dice (or Efron’s dice) presented by Martin Gardner in 1970.

中文翻译：

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非传递随机变量和非传递骰子

摘要

我们给出了一个明确的几何证明，对于每个n，存在一个由n个独立随机变量组成的循环，这样，对于该循环中的每个随机变量，其取值小于该循环中的下一个随机变量的概率至少为$\mathrm{1个}-\mathrm{1个}/（4{\hspace{0.17em}\text{cos}\hspace{0.17em}}^{\mathrm{2个}}\frac{\pi }{ñ+\mathrm{2个}}）$。我们还表明，不能用任何更大的数字来代替此界限。这概括了马丁·加德纳（Martin Gardner）在1970年提出的著名的非传递性骰子（或埃夫隆骰子）悖论。