/items/th-total-probability

---

Let a collection of sets \(\mathcal{A}= \lbrace A_i:A_i \subseteq \Omega\rbrace\) be such that,

1. each \(x\in \Omega\) occurs in exactly one \(A_i \in \mathcal{A}\) (partition);
2. and \(P(A_i) > 0\) for each \(A_i\in\mathcal{A}\),

then for any event \(B\subseteq \Omega\):

\[P(B) = \sum_{i}^{} P(A_i \cap B) = \sum_{i}^{}{P(B\given A_i) P(A_i)}\]