Assume you have an $n\times n$ grid, and a set W of white and a set B of black tiles that are placed randomly on this grid. 

I think that at least one of the sets W, B must include a connected path of tiles from one side of the grid to the opposite side of the grid. 

What I mean by connected path: the tiles have the same color, and they are pairwise neighbors to each other (each pair of them shares an edge or a vertex). 

I think such a connected path, connecting one side of the grid with the opposite side, must be included in B or W, regardless of the distribution of tiles. 

I suspect it is sufficient to prove this for the case when W and B have equal size $n^2/2$. I also suspect that one could start from a chess board style pattern, and go from there to cover all other distributions of black and white tiles. Finally, I suspect the Pigeon Hole Principle might prove it in one go; but I have not found the right entry point to this route. 
Does anybody know a short simple proof?