Total Probability Theorem

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\set {B_1, B_2, \ldots}$ be a partition of $\Omega$ such that $\forall i: \map \Pr {B_i} > 0$.

Then:
 * $\displaystyle \forall A \in \Sigma: \map \Pr A = \sum_i \map \Pr {A \mid B_i} \, \map \Pr {B_i}$

Also known as
This theorem is also called the Partition Theorem, but as there are already quite a few theorems with such a name, prefers to use this somewhat more distinctive name.

Other names include:
 * Law of Alternatives
 * Law of Total Probability
 * Theorem of Total Probability

and appear to be dismissive of them:


 * This theorem has several other fancy names, such as 'the theorem of total probability'; ... from

Also see

 * Bayes' Theorem