Randomly color the squares of an $m\times n$ chessboard red or black (each square has a fifty-fifty chance of being red or black). A monochromatic region is a set of squares that are connected along their sides and all of the same color, that is, for any $2$ squares in the region there is a path connecting the $2$ that stays on the same color.
What is the expected size of a monochromatic region, given $m,n$?