If I have two independent exponential distributions, one with $\lambda_1$ and the other with $\lambda_2$. That is expected time to failure/success for the first is $\frac{1}{\lambda_1}$ and the other is $\frac{1}{\lambda_2}$.
How can I find the expected time to that either one fails/succeed?
Possible approach 1:
The probability that neither fails/succeeds by time $t$ is $e^{-\lambda_1 t} e^{-\lambda_2 t}$ so you can find the probability either fails/succeeds, and then differentiate and use the density to find the expected time.
Possible approach 2:
Treat these distributions as first times for independent Poisson processes with rates $\lambda_1$ and $\lambda_2$. Combined they make a Poisson process with rate $\lambda_1 + \lambda_2$, and the expected first time of the combined process is obvious