real valued random variable inclusion-exclusion

Question

I have the following question and I need help with it. Let $X_1, X_2, X_3$ be realvalued Random Variables with densities and for all i,j in {1,2,3} such that $i \neq j$ we have $a_{ij} = P(X_i > X_j)$.

I must prove that min{$a_{12},a_{23},a_{31}$} $\leq 2/3$.

There is a hint to use the inclusion-exclusion principle but I don't know how.

Answer 1

Suppose that $\min\{a_{12},a_{23},a_{31}\}>2/3$. Then \begin{align} 2<a_{12}+a_{23}+a_{31}&=\mathsf{P}(\{X_1>X_2\}\cup \{X_2>X_3\}\cup \{X_3>X_1\}) \\ &\quad+\mathsf{P}(X_1>X_2>X_3) \\ &\quad+\mathsf{P}(X_3>X_1>X_2) \\ &\quad+\mathsf{P}(X_2>X_3>X_1)\le 2. \end{align}