Given $Y \sim \mathrm{Unif}[1,3]$ and $X \sim \mathrm{Unif}[1,3]$ where $X$ and $Y$ are IID random variables.
a) Compute the value of $P(|X-Y|>1/3)$
b) Find $E[Y \mid X]$
c) Find $E[ E[Y \mid X] ]$
The last one was a no-brainer for me and I used the formula to find E[Y]=2. First one I got 2/3(not sure if it is right). I am completely unaware of computing the expectation
It is not enough to know both the distribution of $X$ and the distribution of $Y$ to answer (a) and (b). You need to know the joint distribution. If it is the case that $X$ and $Y$ are independent, then the joint density function would be $$f_{XY}(x,y)=f_X(x)f_Y(y)=\begin{cases}\frac14&1\le x,y\le3\\0&\text{otherwise}\\\end{cases}.$$
Anyway, if $X$ and $Y$ are independent, then $E(Y|X)=E(Y)$. But for (a) you have to compute $$P(|X-Y|>\frac13)=\iint_{|x-y|>\frac13}f_{XY}(x,y)dA.$$
But if you cannot assume that $X$ and $Y$ are independent, then there is some missing information in the problem.