Let $\displaystyle I_A$ be the indicator function of the event A
$\displaystyle E[I_A]=E[E[I_A|A]]=E[E[1]]=1$
$\displaystyle E[I_A]=E[E[I_A|A^C]]=E[E[0]]=0$
:S
What's going on here?
Hello,
I guess the mistake is that in the iterated expectations, you have $\displaystyle \mathbb{E}[X]=\mathbb{E}[\mathbb{E}[X|Y]]$, where Y is a random variable, and not an event. And X has to be a random variable too.
Here, $\displaystyle I_A$ is indeed a random variable. But A is defined as an event.
A few facts :
X|A is a random variable. E(X|A) is a number.
X|Y is a random variable. E(X|Y) is a random variable. E(E(X|Y)) is a number.