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See below chart. The yellow blocks are two islands. It's connected by a grid of cables. There are 16 vertical cables and 9 horizontal cables and 12 nodes (blue highlighted) in between.

Hurricane strike and each cable has 1/2 probability of breaking. Two islands lost contact when there is no path to go from one island to the other on the grid. What's the probability that these two islands lost contact?

I am thinking of something like letting $f(1,1)$ be the probability that the top left node cannot be reached from top island. Then $f(1,1) = 0.5 \cdot (0.5 \cdot (1-f(1,2)) + f(1,2))$. Reason is you need the direct cable from top island to first node broke, and you either need cable from $(1,1)$ to $(1,2)$ broke, or $f(1,2)$ broke. But the equation starts to get a bit messy.

It looks like this might be able to expressed in matrix form or Markov Chain or some sort though..

enter image description here

R.Yeh
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  • This is a classical problem which uses the dual lattice, as illustrated in the answer below. Just for reference, this self-duality of the square lattice leads to a very elegant approach to the percolation theory for the bond percolation on the square lattice, which deals with various generalizations of this kind of question. (Yes, I am advertising some area of probability theory.) – Sangchul Lee Dec 04 '20 at 05:08

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This is an old chestnut (though I couldn't readily locate an MSE duplicate) and best tackled by a symmetry argument.

To illustrate, consider the following problem: Say all the cables are low, so a boat wanting to go from left to right will get stopped by them. Now, if enough cables are broken, there will be a path for such a boat to cross. In fact the possible paths of the boat (marked in your diagram in red as below) is a rotated copy of the cable network itself. enter image description here

Note the symmetry of the boat crossing problem with your island connectivity problem, and also consider that these events are disjoint and one of them is bound to hold true. It follows that the answer to the probability of either occurring is $\frac12$.

Macavity
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