I am trying to understand better the concept of conditional expectation, and I am facing the following exercise:

I have two independent dice $X,Y$ with four faces s.t. $$P(X=i)=1/4=P(Y=i) , i\in\{1,2,3,4\}$$
What is the conditional expectation of the maximum of the two, given the value of one of them?

First of all, what is it asking me to compute? $\mathbb{E}(max(X,Y)|Y)$ or $\mathbb{E}(max(X,Y)|Y=y)$ 

I am kind of lost here: $$\mathbb{E}(max(X,Y)|Y=y)= \sum\_{i=1}^4 i\cdot P(\{max = i\}|\{Y=y\})$$ 

because the value of $P(\{max = i\}|\{Y=y\})$ changes according to $i$ and to $y$ so I can not find a generic form of the value. 
For instance, is it true that $P(\{max=1\}|\{Y=y\}) = \begin{cases} 1/4 &\text{if} &y=1 \\ 0 &else \end{cases}\\
P(\{max=2\}|\{Y=y\}) = \begin{cases} 1/4 &\text{if} &y=1 \\ 1/2 &\text{if} &y=2 \\ 0 &else \end{cases}$

or am I doing something wrong?