Conditional and Total Variance Why does $ \text{Var}(Y) = E(\text{Var}(Y|X))+ \text{Var}(E(Y|X))$? What is the intuitive explanation for this? In laymen's terms it seems to say that the variance of $Y$ equals the expected value of the conditional variance plus the variance of the conditional expectation. 

Geometrically it's just the Pythagorean theorem. We may measure the "length" of random variables by standard deviation.
We start with a random variable Y. E(Y|X) is the projection of this Y to the set of random variables wich may be expressed as a deterministic function of X. 
We have a hypotenuse Y with squared length Var(Y).
The first leg is E(Y|X) with squared length Var(E(Y|X)).
The second leg is Y-E(Y|X) with squared length Var(Y-E(Y|X))=...=E(Var(Y|X)).
