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| Main Author: | |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.16156 |
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Table of Contents:
- An $n$-sided die is an $n$-tuple of positive integers. We say that a die $(a_1,\dots,a_n)$ beats a die $(b_1,\dots,b_n)$ if the number of pairs $(i,j)$ such that $a_i>b_j$ is greater than the number of pairs $(i,j)$ such that $a_i<b_j$. We show that for a natural model of random $n$-sided dice, if $A, B$ and $C$ are three random dice then the probability that $A$ beats $C$ given that $A$ beats $B$ and $B$ beats $C$ is approximately 1/2. In other words, the information that $A$ beats $B$ and $B$ beats $C$ has almost no effect on the probability that $A$ beats $C$. This proves a statement that was conjectured by Conrey, Gabbard, Grant, Liu and Morrison for a different model.