Suppose there is a circuit from A to B.

The current can flow from 1-2, or from 3-4. (it makes sense to draw a picture)

Each node (1, 2, 3, 4) has a .9 percent chance of success.

This means that the system has a .9*.9 + .9*.9 - .9^4 chance of success (.9639)

The question is this: what is the probability of nodes 1 and 3 working, given that the current is flowing.

I've been working on this for at least an hour now. I've had a few methods that I believe should work, but none of them are giving me the correct answer of .916.

On the surface, the problem sounds simple.

I tried finding the probability of node 1 working, given the system working and multiplying it by the probability of node 2 working, given the system is working

(.9 / .9639) * (.9 / .9639), but for some reason I can't discern, this isn't right.

I've tried using Venn Diagram, and I get a number which I'm convinced must be right (.81 / .9639) * (.81 / .9639) = .706, and that doesn't work

I tried phrasing the question in formal conditional probability:

P((1 and 3) given ((1 and 2) or (1 and 4) or (2 and 3) or (2 and 4)))

and the whole thing cancels out on me, leaving .9 * .9.

Could anyone shed some insight here? Three different ideas I have, each of which make sense to me, give different answers, and are all wrong.