On the restrictiveness of the hazard rate order

Abstract: Every element $\theta=(\theta_1,\ldots,\theta_n)$ of the probability $n$-simplex induces a probability distribution $P_\theta$ of a random variable $X$ that can assume only a finite number of real values $x_1 < \cdots < x_n$ by defining $P_\theta(X=x_i) = \theta_i, 1\leq i \leq n$. We show that if $\Theta$ and $\Theta'$ are two random vectors uniformly distributed on $\Delta^n$, then $P(P_\Theta\leq_{\rm hr} P_{\Theta'})=\frac{1}{2^{n-1}}$ where $\leq_{\rm hr}$ denotes the hazard rate order.