  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.

Combinatorics;Probability