# 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 $$P(T, this yields (through independence), $$P(T.

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 $$\mathbb P(T>500) = \overline G(500) = 2e^{-2}-e^{-3}\approx 0.220883.$$
m = 10^6