A standard mattress can be positioned on a bed frame in any of four orientations. Suppose that four times a year the mattress is rotated into one of the three possible new orientations, chosen at random. According to an article in The American Scientist, a computer simulation suggests the rather surprising result that over a period of 10 years, the most frequently occurring orientation occurs 31 percent of the time, while the least frequently occurring orientation occurs just 19 percent of the time. This paper contains an investigation of this phenomenon, which supports the claim of the simulation. Along the way, it considers a similar but more easily explored problem in coin flipping, deriving the distribution functions for the more frequent and less frequent outcomes of heads and tails for a sequence of n independent flips of a fair coin.