The random variable $X$ [stochastically dominates][1] the random variable $Y$, written $X\succeq Y$, if $\Pr(X\ge y)\ge\Pr(Y\ge y)$ for all $y$. This relation *is* transitive.

Let's write $X\gtrsim Y$ if $\Pr(X\ge Y)\ge 1/2$. This relation is *not* transitive as the example of intransitive dice shows. However it is "total" in the sense that $X\lesssim Y$ or $Y\lesssim X$ holds.


----------


Theorem. $X\succeq Y$ implies $X\gtrsim Y$.

Proof: In the notation of continuous random variables with pdfs $f_X$, $f_Y$,
\begin{align}
\Pr(X\ge Y)&=\int_{-\infty}^\infty \int_y^\infty f_X(x)f_Y(y)\,dx\, dy
=\int_{-\infty}^\infty \Pr(X\ge y) \,f_Y(y)\, dy\\
&\ge\int_{-\infty}^\infty \Pr(Y\ge y) \,f_Y(y)\, dy=\int_{-\infty}^\infty \int_y^\infty f_Y(x)f_Y(y)\,dx\, dy= \frac12.
\end{align}


----------


By the Theorem, the intransitive dice phenomenon cannot occur when stochastic domination totally orders the elements of $\mathcal X$.

  [1]: https://en.wikipedia.org/wiki/Stochastic_dominance#First-order_stochastic_dominance