Expected lifetime of system

[wtrow]

Consider the system shown in Figure 1. Each second, every component fails with probability
p independently of other components. Once a component fails, it will not come back up. The
system as a whole works if there is a path from the input to the output containing properly
functioning components. Find the expected lifetime of the system in seconds.

Figure one looks like:
____ 1 ________ 2 ___
|___ 3 ________ 4 ___|

(1 and 2 in series, 3 and 4 in series, 12 and 34 in parallel)

So the probability of success on 1 and 2 is (1-p)^2, and the same for 3 and 4.

For the total, the probability of success is I think 1-(1-(1-p)^2)^2

Each second I assume I would multiply p^2 to 1,2 and 3,4.

After that I dont know what to do.

 

[tkhunny]

To keep things simpler, let's distinguish the two parallel paths.

p = p(1 failing) = p(2 failing)
q = p(3 failing) = p(4 failing)

In a given second we have:

p(1 and 2 surviving) = (1-p)^2 = s

p(3 and 4 surviving) = (1-q)^2 = t

Further,

p(both independent parallel circuits failing) = (1-s)*(1-t)
p(both independent parallel circuits surviving) = (s)*(t)
p(either but not both) = the rest

Where does that leave us?