Let $X_1, X_2, ... $ be a sequence of i.i.d random variables with mean equal to $0$, and finite variance. Find the variance $\operatorname{Var}(X_1)$ if $$\lim \limits_{n \to +\infty} P\left(\frac{S_n}{\sqrt n} > 1\right)=\frac{1}{3}$$
I've tried researching any theorems that could help me solve this, but can't find any. Never solved anything like this, with limits and sequences of random variables, so I have $0$ idea how to even approach it.