\begin{fact}
	 Let $f$ be an $S$-valued random variable on $S^k$ with $1 < \# S \leq k$ and \\$\phi_a(j) = P(f(a) = j) = \frac{ \phi(j)  \lb 1-\bs 1_a(j) r(j) \rb}{1-\sum_{x\in a} \phi(x)r(x)}$ for arbitrary, zero-free PDF $\phi$ on $S$ and non-constant function $r:S \to (0,1)$. Then $\phi_a(j) > \phi(j)$ for some $j \in S$ whenever $\textnormal{set}(a) = S$.
	\end{fact}