With $5$ hours left before college apps are due, Billy needs to figure out a strategy to maximize his expected admissions rate. He has $20$ essays to work on for one college conveniently numbered $1 \ldots 20$. He has completed $(5n) \%$ of essay $n$ and he can complete essay $n$ at a rate of $\frac{5}{n}$ essays per hour. Given that Billy’s chance of acceptance is the product of the completion of all essays, find Billy’s maximum chance of acceptance.