Suppose we have a rectangular floor, $a$ units long by $b$ units wide, which we need to tile with black and white unit square tiles. We flip a coin to decide whether the first tile will be black or white, and lay it down in the top left corner. The second and all subsequent tiles will also have a 50-50 chance of being black and white.
A 'region' is defined as a contiguous area of same-colour tiles, touching each other by their sides (just a corner is not enough). Thus, the maximum number of regions possible is $ab$ (checkerboard tiling), while the minimum number is 1 (the whole floor is either black or white).
What is the expected number of regions on the board, as a function of $a$ and $b$?