I am wondering about a following problem, which seems to be intuitively trivial, but I am not sure how to approach it (due to my lack of knowledge, I believe).
Imagine I have two machines working. Each of them have may break in an exponentially distributed moments in time with mean $1/\lambda_1$ and $1/\lambda_2$.
My question is, is, can these machines break together in the same instant of time if we consider that they are independent from each other?
My intuition tells me that the probability of them breaking in the same moment is 0, but I don't know how to prove it mathematically. Could anyone shed some light on this issue?