Affect of noise on Random variable separation

Question

We have two random variables $X$ and $Y$. Suppose $P_1$ is the probability that $Pr[X > Y]$. $Z_1$ and $Z_2$ are two i.i.d. (identical and independent) random variables, and let $P_2$ be the probability that $Pr[X+Z_1 > Y+Z_2]$.

Can $P_2$ be greater than $P_1$?

If we think $Z_1$ and $Z_2$ as noise, then in words, can adding i.i.d noise make two random variables more separable?

Answer 1

It seems that you need more assumptions on $X$ and $Y$. Otherwise, take both to be constant and $X = x < Y = y$, so that $P_1 = 0$. Then it is very easy to find examples such that $P_2 > 0$.

However, this does not mean that the two random variables are made more separable...