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?

Let $T_i$ be the lifetime of component $i$. Then the lifetime of the system is $$T = T_1\wedge(T_2\vee T_3)$$
The $T_i$ are independent with distribution function $F(t)=1-e^{-\lambda t}$, so $T_2\vee T_3$ has distribution function $F^2(t)=\left(1-e^{-\lambda t}\right)^2$. It follows then that $T$ has survivor function
\begin{align}
\overline G(t) &=(1-F(t))(1-F^2(t))\\ 
&= e^{-\lambda t}\left(1-\left(1-e^{-\lambda t}\right)\right)\\
&= 2e^{-2\lambda t} - e^{-3\lambda t}
\end{align}
A straightforward computation yields
$$\mathbb P(T>500) = \overline G(500) = 2e^{-2}-e^{-3}\approx 0.220883.$$
