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?