Uniform samples needed to hit a small window

Independent $\text{Unif}(0,1)$ values are generated, $n$ of them. How large must $n$ be, at minimum, so that with probability $0.99$ or more some value lands inside the open window from $0.48$ to $0.52$?

$113$. Each draw misses the window with probability $0.96$, so we need $0.96^n\le0.01$, i.e. $n\ge\tfrac{\ln 0.01}{\ln 0.96}\approx112.8$.