Total Probability Theorem

Theorem
Let $$\left({\Omega, \Sigma, \Pr}\right)$$ be a probability space.

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

Then:
 * $$\forall A \in \Sigma: \Pr \left({A}\right) = \sum_i \Pr \left({A | B_i}\right) \Pr \left({B_i}\right)$$

Proof
$$ $$ $$ $$

Also see

 * Bayes' Theorem, which is related.

Alternative Names
This theorem is also called the Partition Theorem, but as there are already quite a few theorems with such a name (with some guy's name appended to it), it can be argued that it is a good idea to use this somewhat more distinctive name. Grimmett and Welsh appear to be dismissive of them.

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