The law of total expectation, also known as the law of iterated expectations (or LIE) and the “tower rule”, states that for random variables $X$ and $Y$,
\[\Ex(X) = \Ex\{ \Ex(X \mid Y) \},\]provided that the expectations exist. A common special case involves conditioning on a partition of the sample space:
\[\Ex(X) = \sum_i \Ex(X \mid A_i) \Pr(A_i).\]In this case, the law can also be written as
\[\Ex(X) = \sum_i \Ex\{ X 1(A_i) \},\]where $1(A_i)$ is an indicator function. This is simply the definition of expected value with the integral broken up into the partition defined by $\{A_i\}$.