The empirical mean and empirical variance of i.i.d. normal samples are independent and follow known distributions, which are respectively normal and chi-squared. This indicates that $ \mathrm P\left(|\bar X-\mu|\gt S\right)=\mathrm P\left((n-1)Z_1^2\gt n(Z_2^2+\cdots+Z_n^2)\right), $ where $(Z_k)_{1\leqslant k\leqslant n}$ is i.i.d. and standard normal. More simply, this is $\mathrm P(|T_{n-1}|\gt 1)$, where the distribution of $T_{n-1}$ is the Student's $t$-distribution with $n-1$ degrees of freedom. Hence, $ \mathrm P\left(|\bar X-\mu|\gt S\right)=\mathrm P\left(|T_{n-1}|\gt1\right)=I_{\frac{n-1}n}\left(\frac{n-1}2,\frac12\right), $ where $I$ denotes the regularized incomplete beta function. The cases $n=2$, $3$, $4$ and $\infty$ are somewhat explicit.
Edit: Recall that the empirical mean $\bar X$ and the empirical variance $S^2$ of the sample $(X_k)_{1\leqslant k\leqslant n}$ are defined as $ \bar X=\frac1n\sum\limits_{k=1}^nX_k,\qquad\qquad S^2=\frac1{n-1}\sum\limits_{k=1}^n(X_k-\bar X)^2. $