Intuitive explanation of the tower property of conditional expectation I understand how to define conditional expectation and how to prove that it exists.
Further, I think I understand what conditional expectation means intuitively. I can also prove the tower property, that is if $X$ and $Y$ are random variables (or $Y$ a $\sigma$-field) then we have that
$$\mathbb E[X] = \mathbb{E}[\mathbb E [X | Y]].$$
My question is: What is the intuitive meaning of this? It seems quite puzzling to me.
(I could find similar questions but not this one.)

For simple discrete situations from which one obtains most basic intuitions, the meaning is clear.
I have a large bag of biased coins.  Suppose that half of them favour heads, probability of head $0.7$.  Two-fifths of them favour heads, probability of head $0.8$.  And the rest favour heads, probability of head $0.9$.
Pick a coin at random, toss it, say once.  To find the expected number of heads, calculate the expectations, given the various biasing possibilities.  Then average the answers, taking into consideration the proportions of the various types of coin.  
It is intuitively clear that this formal procedure "should" give about the same answer as the highly informal process of say repeating the experiment $1000$ times, and dividing by $1000$.  For if we do that, in about $500$ cases we will get the first type of coin, and out of these $500$ we will get about $350$ heads, and so on.  The informal arithmetic mirrors exactly the more formal process described in the preceding paragraph. 
If it is more persuasive, we can imagine tossing the chosen coin $12$ times.
