The question has been asked before here.
There are three independent exponential variables $X$, $Y$ and $Z$ with parameter $\lambda$, $\mu$ and $\eta$. Find $P(X<Y<Z)$.
Part of the solution is that the event $X<Y<Z$ can be split into two independent events $X<min\{Y,Z\}$ and $Y<Z$.
My initial intuition was that the first of the two resulting events should be $X<Y$ instead. And I'm still having trouble to understand why this is incorrect.
Can someone point out why using $X<Y$ instead of $X<min\{Y,Z\}$ is wrong?
