Law of Total Probability

Definition

For some positive integer $k$ , let the sets ${B_{1}, B_{2}, \dots, B_{k}}$ be such that:
- $S = B_{1} \cup B_{2} \cup \dots \cup B_{k}$
- $B_{i} \cap B_{j} = \emptyset$ for $i \neq j$
This is a partition of the sample space

Theorem 20
- If ${B_{1}, B_{2}, \dots, B_{k}}$ is a partition of $S$ .
- Where $P (B_{i}) > 0$ for $i = 1, \dots, k$
- Then for any event $A$ , $P (A) = \sum_{i = 1}^{k} P (A | B_{i}) P (B_{i})$