Let $n=p\cdot q$ with $p,q$ two distinct odd primes, suppose that they are large enough.
Let $x,y\in(\frac{n}{8},\frac{7n}{8})$ two elements of $\mathbb{Z}_n^\star$.
Can you provide some constant bound on the probability that $x\cdot y\ (\text{mod}\ n)\in(\frac{n}{8},\frac{7n}{8})$?
It seems to me that it is possible to provide a bound which is independent of $n$ but I failed doing so.