Theorem of Total Probability
The main tool in this section is the following statement, usually referred to as the theorem or law of total probability:

Theorem: Let \( \left\{ B_n \,:\, n = 1,2,3,\ldots \right\} \) be a finite or countably infinite partition of a sample space into pairwise disjoint events whose union is the entire sample space, and each event Bn has a probability. Then for any event A from the same sample space

\[ \Pr [A] = \sum_n \Pr \left[ A \, | \, B_n \right] \Pr [B_n ] \]
or, alternatively, \[ \Pr [A] = \sum_n \Pr \left[ A \cap B_n \right] , \]
where, for any n for which Pr[Bn] = 0, these terms are simply omitted from the summation, because \( \Pr \left[ A \, | \, B_n \right] \) is finite.    ▣