Let $[q] = \{0,1,\ldots ,q-1\}$, let $\Delta [q]$ denote the simplex of probability measures on $[q]$, and let $\gamma$ denote the Lebesgue measure normalized on $\Delta [q]$. We prove that for any symmetric monotone function ${\kern1pt}f \colon{\kern-1pt} [q]^n \to [q]$ and any $a \in [q]$, we have

\begin{equation*} \gamma (\{\mu \in \Delta [q]\;\vert \;\mathbb{P}_{x\sim \mu ^{\otimes n}}[\,f(x)=a] \in (\varepsilon ,1-\varepsilon )\}) = O(1/\log n)\text{.} \end{equation*}

We also show that this bound is tight. This improves Kalai and Mossel's previous bound of $O(\!\log \log n/\log n)$ and answers their question completely.