Law of total expectation? Apparently $E[X] = E[E[X\mid Y]]$ but I don't understand what this really means. I looked at https://en.wikipedia.org/wiki/Law_of_total_expectation but need another explanation.
Isn't this the same as $E[X] = E[X\mid Y]$? Why the extra $E[ \cdot ]$? And ultimately why does the $Y$ not seem to even matter if we say that $X$ depends on $Y$?

A concrete example:
Suppose $X$ can take the values $0$ or $1$. Suppose $Y$ can take the values $a$ or $b$ each with probability $1/2$. Suppose that if $Y=a$ then the probability $X=0$ is $1$ while if $Y=b$ then the probability $X=0$ is $0$. Then $E[X\mid Y=a]=0$ while $E[X\mid Y=b]=1$. Then $$E[X]=E[E[X\mid Y]]=\frac{1}{2}E[X\mid Y=a]+\frac{1}{2}E[X\mid Y=b]=\frac{1}{2}0+\frac{1}{2}1=\frac{1}{2}.$$
You can see that $Y$ "disappears" because when you take the expectation of $E[X\mid Y]$ you are just weighting the conditional expectations of $X$ given each $Y$ by the probability of that value of $Y$.
