The following is a well-known theorem (see e.g. *The Chebyshev Polynomial* by Rivlin):

If $p(x) = x^n + a_{n_1} x^{n-1} + \ldots + a_0$, then $\max_{-1\leq x \leq 1} |p(x)| \geq 2^{1-n}$ for $n \geq 1$ with equality being attained only if $p$ is the $n$th Chebyshev polynomial normalized so that the leading coefficient is $1$.

In my work I came across a question that can be seen as a refined version of the question answered by the above theorem:

Let $p$ be a polynomial as above and let $0 \leq \epsilon \leq 2^{1-n}$. Suppose that $x$ is a uniformly distributed random variable on $[-1, 1]$. What is $\sup_p \Pr[|p(x)|\leq \epsilon]$ as a function of $\epsilon$ and $n$ (sup is over all possible choices of the deg-$n$ monic polynomial $p$)? A good upper bound on this quantity will also be useful. Note that for $\epsilon=2^{1-n}$ this quantity is equal to $1$ by the theorem above.