Q:

probability of above-average outcome (N uniform dice)

Sorry if my question is trivial, it's been decades since I did any "real" math, so I'm quite rusty. 
If I roll $N$ uniform $Y$-sided dice ($1\dots Y$), for the sum I get approximately a gaussian distribution that peaks at $A = \frac{N(Y+1)}2$. 
How do I calculate the probability to get a sum of, say, $x A$ (where $1 < x < N\, Y/A$) ?

A:

Let $S_N$ denote the sum of $N$ results. Call $m_N=\max\{\mathbb P(S_N=k)\mid k\in\mathbb Z\}$, then $m_N$ behaves like $1/\sqrt{N}$ and, in particular, $\lim\limits_{N\to\infty}m_N=0$.
A quite different question is to estimate the behaviour or $q_N(x)=\mathbb P(S_N\geqslant xN)$ when $N\to\infty$, for some fixed real number $x$. Let $x^*=\frac12(Y+1)$. Then the following asymptotics hold. If $x\lt x^*$, $q_N(x)\to1$. If $x\gt x^*$, $q_N(x)\to0$. And $q_N(x^*)\to\frac12$.

