Law of Iterated Expectations

It is also know as LIE. X is a random variable. Y is another variable. We have:

$$\operatorname {E} (X)=\operatorname {E} (\operatorname {E} (X\mid Y))$$

In $\operatorname {E} (X\mid Y)$, $Y$ is a specific value. In $\operatorname {E} (\operatorname {E} (X\mid Y))$, we need to enumerate all the possible value of $Y$ and multiply $\operatorname {E} (X\mid Y)$ with the corresponding probability of that $Y$ and sum together. Therefore, they are equal.