Suppose we have a system like the one in the picture. Each component works and fails independently, and each component's duration is given by a random variable with exponential distribution with $\lambda = 0.002$.
What's the probability that the system will last more than $500$ hours?
I tried the following, but I'm not sure if its correct
Let $T_i$ be the duration of each component. We know that the system fails if component $1$ fails, or if both, components $2$ and $3$ fail (the two in parallel).
Therefore $P(T<t)= P(T_1<t) + P(T_2,T_3<t)- P(T_1,T_2,T_3<t)$, this yields (through independence), $P(T<t)=(1-\exp(-\lambda t))+\left(1-\exp(-\lambda t)\right)^2-(1-\exp(-\lambda t))^3$.
And now, I just do $P(T>500)=1-P(T<500)\approx 0.22$.
Is this correct?