How and when to use the "law of total expectation", a.k.a. "tower property"? I know what it is and how to prove it. However, I am not able to use it for problem solving. I would really appreciate if you can point out some kind of problems where it is usually applied. Thanks.
 A: Example
Consider a sequence of iid random variables $X_1,X_2,\ldots$ and a discrete random variable $N$, independent of $\{X_i\}$. Now consider the compound sum
$$
S_N = \sum_{i = 1}^N X_i
$$ 
and find the expected value of it. That is,
$$
E[S_N] = E\left[ \sum_{i = 1}^N X_i\right].
$$
From here, we will apply the tower property,
$$
E\left[ \sum_{i = 1}^N X_i\right] = E\left[E\left[ \sum_{i = 1}^N X_i \;\middle \vert \; N\right]\right] = E\left[  E[N \cdot X_1\mid N]\right],
$$
where we used that the $X_i$'s are iid ($X_i \stackrel d= X_1$ for all $i$) in the last equality. If you are not convinced of the second equality, try to do, informally, the steps with $N = n$ fixed instead of a random variable. 
From here on, use that a random variable is measurable to itself, i.e. $E[N\mid N] = N$ and the independence between $X_1$ and $N$, giving $E[X_1\mid N] = E[X_1]$ to finally get
$$
E[S_N] = E\left[  E[N \cdot X_1\mid N]\right] = E[N]E[X_1].
$$
A: Are you referring to the case when the entire sample space can be partitioned into a discrete set of events? If yes, here is an example:
Suppose that adult male lions weight an average of 420lb, while adult female lions weight an average of 280lb. Suppose also that adult male lions constitute 20% of each pride, while adult female lions constitute the remaining 80%. You are interested in finding the expected weight of a randomly-selected adult lion:
$$E(W) = E(W|M) \cdot P(M)+E(W|F) \cdot P(F) = (420)\cdot(0.2)+(280)\cdot(0.8) = 308$$
In this case, the entire sample space of adult lions can be divided into two discrete events: picking a male and picking a female.
Hope that helps!
