It is a nice exercise with rational generating functions (or equivalently, linear recurrence relations) to show that for a random domino tiling of a $2\times n$ rectangle, with $n$ large, we can expect about $\frac{1}{\sqrt{5}}\approx 44.7\%$ of the tiles to be vertical. In the spirit of https://mathoverflow.net/questions/86118/non-enumerative-proof-that-there-are-many-derangements, I wonder if there is an easy way to see, without computing this fraction, that this average must be some constant < 50%.