Hi Michelle!
Here is a helpful formula:
See if you can prove it!
Here is the proof that E [ E [ X | Y ] ] = E [ X ] {\displaystyle \mathrm {E} [\mathrm {E} [X|Y]]=\mathrm {E} [X]\,} for any X {\displaystyle X\,} :
Let X {\displaystyle \mathbb {X} } be the range of X {\displaystyle X\,} and Y {\displaystyle \mathbb {Y} } the range of Y {\displaystyle Y\,} . Then