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A partition of the sample space $\Omega$ is a collection of disjoint events $S_{1}, \ldots, S_{n}$ such that $\Omega=\cup_{i=1}^{n} S_{i}$
(a) Show that for any event $A$, we have
$$\mathbf{P}(A)=\sum_{i=1}^{n} \mathbf{P}\left(A \cap S_{i}\right)$$
(b) Use part (a) to show that for any events $A, B$, and $C$, we have
$$\mathbf{P}(A)=\mathbf{P}(A \cap B)+\mathbf{P}(A \cap C)+\mathbf{P}\left(A \cap B^{c} \cap C^{c}\right)-\mathbf{P}(A \cap B \cap C)$$