You can calculate the three probabilities.
That's probably not a clever way, but it must work.
Then get
and
Then compare to see if P(AB)=P(A)P(B).
Let X and Y be random variables that both have an exponential distribution with parameters lambda and mu respectively. X and Y are independent.
Define Z = min{X, Y}
(1) Prove that the event {X < Y} is independent of the event {Z > z}
(2) Find E[ e^(-z) | X ].