I'm aware there are similar questions, but I haven't been able to find what I'm asking.
Say we have an $n$-sided die, labeled with 1 thru $n$ and roll it $N$ times. We take the average, call it $m$.
The die is fair, so the expectancy for the die roll is $E=\frac{n+1}{2}$.
How large must $N$ be for the average to be within $\epsilon$ from $E$ with a probability, say, $p$?
For example, 20-sided die: $E=10.5$, choose $\epsilon = 0.01$, and $p=0.99$.
So how many times do I have to roll the 20-sided die for the average to lie in the interval $[10.49, 10.51]$ with 99% probability?