Two random variables i.i.d and same distribution function, compute $P(X\leq Y|Y)$

Question

I am struggling with the concept of regular conditional probability and conditional expectation, maybe someone can help me in order to have things more clear. I have two random variables X,Y i.i.d and same distribution function $F(t)$, and I have to compute $P(X\leq Y|Y)$... any idea? Thanks you so much in advance!

Answer 1

$\mathbb P\left[X \le Y |Y\right]$ is the (random) probability of $X$ is smaller than $Y$ knowing the (random) value of $Y$. For instance if $X$ and $Y$ are independant, $$\mathbb P\left[X \le Y |Y\right] = F(Y)$$