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?



Affect of noise on Random variable separation
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...


