Expectation of a random variable over the variable's conditional value.

Question

Let $X$ is a binary variable. $T$ is a continuous variable which is a function of $X$.

What will be the following two expectation:

$(1)$ $\mathbb E_X[P(T\le t\mid X=0)]$, and

$(2)$ $\mathbb E_X[T\mid X=0]$ ?

What I am trying to ask is that: If we take expectation on random variable $X$, Will the expectation work on conditional $X=0$? Or, conditional $X$ is always a constant though the expectation is over $X$?

Will $(1)$ $\mathbb E_X[P(T\le t \mid X=0)]=P(T\le t\mid X=0)$? or $\mathbb E_X[P(T\le t \mid X=0)]$ become unconditional on $X$?

Answer 1

Given $X=0$, $(T\mid X=0)$ is no more function of $X$. So $\mathbb E_X[P(T\leq t \mid X=0)] = P(T\leq t\mid X=0)$ and $\mathbb E_X[T\mid X=0]= (T\mid X=0)$, (which is putting $X=0$ in $T$).