Total Expectation Theorem

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

Let $x$ be a discrete random variable on $\mathcal E$.

Let $\left\{{B_1 \mid B_2 \mid \cdots}\right\}$ be a partition of $\omega$ such that $\Pr \left({B_i}\right) > 0$ for each $i$.

Then:
 * $\displaystyle E \left({X}\right) = \sum_i E \left({X \mid B_i}\right) \Pr \left({B_i}\right)$

whenever this sum converges absolutely.

In the above:
 * $E \left({X}\right)$ denotes the expectation of $X$
 * $E \left({X \mid B_i}\right)$ denotes the conditional expectation of $X$ given $B_i$.

Also known as
Some sources refer to this as the partition theorem, which causes ambiguous as that name is used for other things as well.