# Why stochastic dominance is "stochastic"?

I think the CDF is pretty much fixed, so the FOSD (first order stochastic dominance) is pretty much non-stochastic. Why does it have a "stochastic" in its name?

In the below figure, CDF $$F(\cdot)$$ is first-order stochastically dominated by $$G(\cdot)$$. But $$X_1$$ and $$X_2$$ fall within the support of both distributions. So it would be possible to draw $$X_1$$ from $$F$$ and $$X_2$$ from $$G$$, or to draw $$X_2$$ from $$F$$ and $$X_1$$ from $$G$$.
More generally, if $$X_G$$ is a draw from $$G$$ and $$X_F$$ is a draw from $$F$$ then $$X_F-X_G$$ will sometimes be positive and sometimes negative. In this sense, the dominance is only stochastic: $$G$$ produces larger draws than $$F$$ on average, but not all of the time.