I know nothing about percolation theory. But I would like to learn how one solves the following basic problem.
Consider a grid of size $(n+1)\times n$. Each edge is assigned 0 or 1 with equal probability, independently. A crossing is a path that joins the left and right sides of the rectangle, and consists entirely of edges valued 1. Show that such grid has crossing with probability 1/2.
I feel that we need to use symmetry of this problem in some way but not sure how.