Probability of difference between sets that are both dependent on a stochastic variable. I'm reviewing some statistics for my exams and have a hard time figuring out what exactly the meaning of the question and solution is: The variables are $X\sim exponential(2)$ and $X\sim exponential(2)$, and they are both independent The probability that should be found is: $P(\\{X\le 1\\}\backslash\\{Y\le 1\\})$ And i know the solution is $(1-e^{-2})e^{-2}$ How do you arrive at this solution, and more detailed - how do you rewrite the probability into something tractable.

We have:

$$\mathbb{P}(\\{X\leq 1\\}\setminus \\{Y\leq 1\\})$$ $$=\mathbb{P}(\\{X\leq 1\\}\cap \\{Y >1\\})= \mathbb{P}(X\leq 1)\mathbb{P}(Y>1)$$ and now it is just a matter of calculating the integrals. The last equality follows because $X,Y$ are independent.