$\begin{align*} \mathbb{P}(S_3 = 3) = \mathbb{P}(X_1 + X_2 + X_3 = 3) = \mathbb{P}(X_1 = 1, X_2 = 1, X_3 = 1) = \mathbb{P}(X_1 = 1) \mathbb{P}(X_2 = 1) \mathbb{P}(X_3 = 1) = \frac{1}{8} \end{align*}$