Let (A_{1}, A_{2}, ldots, A_{n}) be random events. Prove the formula [ begin{aligned} Pleft{sum_{k=1}^{n} A_{k} ight}=sum_{i=1}^{n} Pleft(A_{j}

Let \(A_{1}, A_{2}, \ldots, A_{n}\) be random events. Prove the formula

\[ \begin{aligned} P\left\{\sum_{k=1}^{n} A_{k}\right\}=\sum_{i=1}^{n} P\left(A_{j}\right)- & \sum_{1 \leqslant i