Awesome riddle including independence and exponential distribution The life cycles of 3 devices $A, B$ and $C$ are independent and exponentially distributed with parameters $\alpha,\beta,\gamma$. These three devices form a system that fails if not only device A fails but also device B or C fails too. Maybe  $a \land (b \lor c) $ is easier to understand.
Calculate the probability that the system fails before time $t$.
This riddle is driving me insane, I have spent like 5 hours thinking about it and I just can't seem to find the answer. I am pretty sure thought that there is a simple solution, anyone creative here?

We first go after the complementary event, the event the system is still alive at time $t$. This event can happen in two disjoint ways: (i) $A$ is alive or (ii) $A$ is dead but $B$ and $C$ are alive.
The probability of (i) is $e^{-\alpha t}$.
The probability of (ii) is $(1-e^{-\alpha t})e^{-\beta t}e^{-\gamma t}$. 
Thus the probability the system is dead is 
$$1-\left[e^{-\alpha t}+(1-e^{-\alpha t})e^{-\beta t}e^{-\gamma t}\right].$$
