Let $(p_\pi)_{\pi\in S_3}$ be given nonnegative reals such that $\sum_{\pi \in S_3} p_\pi = 1$. What are necessary and sufficient conditions for there to exist independent random variables $X_1,X_2,X_3$ such that, for each $\pi$, $p_\pi$ is the probability of $X_{\pi(1)} < X_{\pi(2)} < X_{\pi(3)}$?
