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Bond percolation the square lattice

On an infinite square lattice one places bonds on the edges of the lattice with probability 1/2. What is the probability that two nearest neighbor vertices are connected?

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documentclass[a4paper]{article}
usepackage{graphicx,amsmath,amssymb}
newcommand{haak}[1]{left(#1right)}
newcommand{rhaak}[1]{left [#1right]}
newcommand{lhaak}[1]{left | #1right |}
newcommand{ahaak}[1]{left{#1right}}
newcommand{gem}[1]{leftlangle #1rightrangle}
newcommand{gemc}[2]{leftlangleleftlangleleft. #1right | #2
rightranglerightrangle}
newcommand{geml}[1]{leftlangle #1right.}
newcommand{gemr}[1]{left. #1rightrangle}
newcommand{haakl}[1]{left(#1right.}
newcommand{haakr}[1]{left.#1right)}
newcommand{rhaakl}[1]{left[#1right.}
newcommand{rhaakr}[1]{left.#1right]}
newcommand{lhaakl}[1]{left ...

Solution Summary

It is explained how the problem can be solved using duality.

$2.19