We're throwing a die 3600 times. Let $X_i$ be the number rolled, and $S_n=X_1+...+X_n$. By the law of large numbers, we know $\mu_Χ=3.5$. We want to approximate the probability that $\frac{S_n}{n}$ differs from 3.5 by more than 0.05. My idea is to use the Central Limit Theorem for this. However, I run into very unreasonable numbers, so I'm wondering where I am wrong.
We want to approximate $1-P(3.45\leq\frac{S_n}{n}\leq3.55)$. By independence, $Var(\frac{S_n}{n})=\frac{1}{n^2}(\sum_i^nVar(X_i))=\frac{n}{n^2}Var(X_i)\approx0.0008$. Hence, $\sigma_{S_n}\approx0.028$.
By the CLT, $\lim_{n\to\infty}P(a\leq\frac{S_n-n\mu}{\sqrt{n\sigma^2}}\leq b)=\Phi(b)-\Phi(a)$. Since $n\mu=3.5\times 3600=12600$, we get $$-107.14\leq\frac{S_n-n\mu}{\sqrt{n\sigma^2}}\leq 107.14$$.
I can show steps if needed. This approximation, $\Phi(107.14)-\Phi(-107.14)$, cannot be right. This makes me reconsider whether the CLT is the way to go. Thanks a lot for any input.