Possible probability and irreducibility of balanced non-transitive dice

2020/06/23 by Injo Hur, Hur, Injo, Yeansu Kim +1 · 1 citation
Engineering · Mathematics · #05A99 #91A60 #Combinatorics (math.CO) #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Probability (math.PR) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2006.12866

openalex publication_date 2020/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 (1)/(2) and for a upper bound, we construct a balanced non-transitive set of dice whose probability is (1)/(2) + (13-√(153))/(24) ≈ (1)/(2) + (1)/(9.12).

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