I have two independent, identically distributed normal random variables $X \sim N(0,\sigma^2)$, $Y \sim N(0,\sigma^2)$. I want to know
$$Pr[X-Y>4\sigma \text{ and } X<3\sigma \text { and } Y<3\sigma]$$
Of course, the condition $Y<3\sigma$ seems to be redundant, since we must have $Y<-\sigma$ in order to have $3\sigma>X>4\sigma+Y$. I also know the fact that $X-Y$ is itself a normal random variable with $X-Y \sim N(0,2\sigma^2)$. However, $X-Y$ and $X$ are not independent, so I don't know how to separate out the "and"s in the probability condition...