Given $p$ a prime and a random $\alpha\in\Bbb Z_p$ with $\alpha\approx\sqrt{p}$ suppose we pick a random $\beta\approx\log p$ then what is the probability that remainder $\alpha\beta^{-1}\bmod p$ is close to $p$ in size asymptotically? Is the odds close to $1$?
More formally:
Do there exist $f(p)=o(\sqrt p)$, $g(p)=o(\log p)$ and $h(p)=o(p)$ such that if $|\alpha-\sqrt p|\leq f(p)$ and $|\beta-\log p|\leq g(p)$ then the probability that $|p-\alpha\beta^{-1}\bmod p|\leq h(p)$ is $1-o(1)$ as $p\to\infty$?