Probability that a system will last more than 500 hours. 
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?

Comment: Verification, by simulation (2 or3-place accuracy).
m = 10^6
a = rexp(m, .002)
b = rexp(m, .002)
c = rexp(m, .002)
mean((a > 500) & (b > 500 | c > 500))     ## '&' means 'and';  '|' means 'or'
## 0.221315

