We consider a sequence of random variables $(X_n : \Omega \longrightarrow \{-1,1\})_{n \in \mathbf{N}}$ defined on the same probability space $(\Omega, \mathscr{A}, P)$, taking values in $\{-1,1\}$, mutually independent and centered. For every $n \in \mathbf{N}^*$, we denote $S_n = \sum_{k=1}^n X_k$. Let $\ell \in \mathbf{Z}$ be an integer and $n \geqslant 1$ be another integer. By distinguishing the case where the integer $\ell - n$ is even or odd, calculate $\mathbf{P}(S_n = \ell)$.