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arXiv · 2006.12866

Possible probability and irreducibility of balanced non-transitive dice

Injo Hur·Yeansu Kim
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Abstract

We construct irreducible balanced non-transitive sets of $n$-sided dice for any positive integer $n$, which was raised in \cite[Question 5.2]{SS17}. One main tool of the construction is to study so-called fair sets of dice. Furthermore, we also study the distribution of the probabilities of balanced non-transitive sets of dice. For a lower bound, we show that the probability could be arbitrarily close to $\frac{1}{2}$ and for a upper bound, we construct a balanced non-transitive set of dice whose probability is $\frac{1}{2} + \frac{13-\sqrt{153}}{24} \approx \frac{1}{2} + \frac{1}{9.12}.$

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Injo Hur, Yeansu Kim. 2020-06-23. Possible probability and irreducibility of balanced non-transitive dice. https://arxiv.org/abs/2006.12866

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