We want move from the [[Conditional Expectation]] to the unconditional [[Expectation]]. Therefore we write the conditional expectation in its functional form $g(Y)=\mathbb E[X\vert Y]$. $ \begin{align} \mathbb{E} \big[ \mathbb E[X \vert Y] \big] &= \mathbb E[g(Y)] \tag{1}\\[8pt] &= \sum_Y g(y)*p_Y(y) \tag{2}\\ &= \sum_Y \mathbb{E}[X \vert Y=y]* p_Y(y) \tag{3}\\ &= \mathbb{E}[X] \tag{4} \end{align} $ where: - (1) Substituting the function $g(Y)$ with the conditional expectation. - (2) Writing out the [[Expectation#^32925b|expectation]] as a sum. - (3) Substituting back the conditional expectation for the function $g(Y)$. - (4) Recognizing this as the unconditional expectation by [[Total Expectation Theorem#^38d58d|total expectation theorem]]. Thus: $\mathbb{E} \big [ \mathbb{E}[X \vert Y] \big ] = \mathbb{E}[X] $