Let $n\in\mathbb{N}$ and let $q\in[n,2n]$ be a prime number.
In addition, let $s,s':\mathbb{F}_q\to\mathbb{F}_q$ be polynomials of degree $\sqrt{n}$ such that $s\neq s'$.
From the Schwartz–Zippel lemma (or simply because $s-s'$ is also a polynomials of degree $\sqrt{n}$, thus is has at most $\sqrt{n}$ roots) we know that $$\Pr_{r\in \mathbb{F}_q}[s(r)=s'(r)]\leq\frac{\sqrt{n}}{q}\leq\frac{1}{\sqrt{n}}$$
What is the best upper bound I can give for the probability over $r$ which is uniformly picked out of $\{1,\ldots,\sqrt{n}\}$ ? That is $$\Pr_{r\in \{1,\ldots,\sqrt{n}\}}[s(r)=s'(r)]\leq\quad?$$