Assume the following: you have a pool of $N$ d6s. You roll the dice, tally up the number of 6s rolled, and tally up the number of 1s rolled. Then, subtract the number of 1s from the number of 6s. Call this result $X.$ How does the probability that $X$ is greater than or equal to a given threshold ($T$) scale with the size of the pool, $N$? I'm also interested in how $T$ changes the distribution. Ignore the cases where $N < T.$
I'm looking for a simple summary of how it scales, for example, "For $T=5,$ the probability that $X\geq T$ scales linearly with $N.$"
$X$ is the sum of $N$ independent variables $D_i$, each having pmf $$f_D(x)=\begin{cases}\frac16&x=+1\\\frac16&x=-1\\\frac23&x=0\end{cases}$$ (A $+1$ result corresponds to rolling a six and a $-1$ result to rolling a one.) Each $D_i$ has mean $0$ and variance $\frac13$. Therefore, by the central limit theorem, $X$ tends to a normal distribution with mean $0$ and variance $\frac N3$ as $N$ increases, and $$\Pr(X\ge T)\to\frac12(1-\operatorname{erf}(T\sqrt{3/2N}))\to\frac12$$ This is true for all $T\in\mathbb R$. $T$ is something you choose and is therefore independent of $X$.