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 $S_n = \sum_{k=1}^n X_k$. We fix the integer $n \geqslant 1$. A path is any $2n$-tuple $\gamma = (\varepsilon_1, \cdots, \varepsilon_{2n})$ whose components $\varepsilon_k$ equal $-1$ or $1$. An equality index of a path is any integer $k \in \llbracket 1, 2n \rrbracket$ such that $\sum_{i=1}^k \varepsilon_i = 0$. For every integer $i$ between $1$ and $n$, the event $A_i$ is defined by: $$A_i = \left\{\omega,\; 2i \text{ is an equality index of } (X_1(\omega), \cdots, X_{2n}(\omega))\right\}.$$ Calculate the probability $\mathbf{P}(A_i)$, for every integer $i$ between $1$ and $n$.
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 $S_n = \sum_{k=1}^n X_k$. We fix the integer $n \geqslant 1$. A path is any $2n$-tuple $\gamma = (\varepsilon_1, \cdots, \varepsilon_{2n})$ whose components $\varepsilon_k$ equal $-1$ or $1$. An equality index of a path is any integer $k \in \llbracket 1, 2n \rrbracket$ such that $\sum_{i=1}^k \varepsilon_i = 0$. For every integer $i$ between $1$ and $n$, the event $A_i$ is defined by:
$$A_i = \left\{\omega,\; 2i \text{ is an equality index of } (X_1(\omega), \cdots, X_{2n}(\omega))\right\}.$$
Calculate the probability $\mathbf{P}(A_i)$, for every integer $i$ between $1$ and $n$.