Law of total expectation with conditional expectations I am confused between these two equations which use the law of total expectation:
$$E(X|Y)=E(X|Y,Z)P(Z)+E(X|Y,Z')P(Z')$$
$$E(X|Y)=E(X|Y,Z)P(Z|Y)+E(X|Y,Z')P(Z'|Y)$$
where Z' is the complement of Z.
The first one makes sense to me because if one defines the random variable $A=X|Y$, then it is simply using the law of total expectation. The other also makes sense as it is as if we are applying the law of total probability on $X$ but then reducing the universe to the "given Y" subspace. Which one is right?
In general, is thinking of X given Y and Z the same as X given Y, given Z?
$$P(X|Y,Z)=P((X|Y)|Z)$$ 
 A: Your second equation is correct, and the first one is wrong.  Here is an example.
In my example, $X,Y,Z$ are events, i.e. subsets of the sample space $\Omega$.  If you wish to have them be random variables, then consider their indicator variables.  As is well known, $P(A) = E[1_A]$


*

*Pick a number $\in \Omega = \{1,2,3\}$, uniformly randomly, i.e each with equal prob ${1 \over 3}$.

*$X = \{1\}, Y = \{1,2\}, Z = \{2,3\}$

*$Y\cap Z = \{2\}$ and $Y \cap Z' = \{1\}$

*$P(X\mid Y) = 1/2$

*$P(X\mid Y,Z) = 0$ and $P(X \mid Y,Z') = 1$

*1st RHS $= P(X \mid Y,Z) P(Z) + P(X\mid Y,Z') P(Z') = 0\times \frac23 + 1 \times \frac13 = \frac13 \neq P(X \mid Y)$

*2nd RHS $= P(X \mid Y,Z) P(Z \mid Y) + P(X\mid Y,Z') P(Z' \mid Y) = 0 \times \frac12 + 1 \times \frac12 = \frac12 = P(X \mid Y)$

More generally, lets expand:
$$
\begin{array}{}
P(X \mid Y) &= {P(X, Y) \over P(Y)}\\
&={P(X,Y,Z) + P(X,Y,Z') \over P(Y)}\\
&={P(X \mid Y,Z) P(Y,Z) + P(X \mid Y,Z') P(Y, Z') \over P(Y)}\\
&=P(X\mid Y,Z) {P(Y,Z) \over P(Y)}+ P(X \mid Y,Z') {P(Y,Z') \over P(Y)}\\
&=P(X \mid Y,Z) P(Z \mid Y) + P(X \mid Y,Z') P(Z' \mid Y)
\end{array}
$$ 
The way I remember is that conditioning on $Y$ creates its own probability law, so the correct equations look like Law of Total Probability/Expectation but within the event $Y$, i.e. every term is conditioned on $Y$.
