Ugh -- so I've sent mysteriously coded messages twice in one day. No more cut-and-paste for me. And if that wasn't enough, I earlier wrote:
That's just the probability that you're surrounded by squares whose color is not your own, so (1/2)^4 = 1/16, right?
Likewise for a domino, there are four neighbors you could domin with (now *there's* a non-sactioned noun-to-verb conversion), and six border squares that would have to be of the opposite color. So the odds that a random square is in a domino is 4 * (1/2)^7 = 1/32. And picking random squares makes dominos twice as findable as monominos, so both come up one sixteeth of the time if you pick a random square and look at its connected component. Right?
Wrong on the factor of two at the end. The probability that a random square is in a domino is indeed 1/32, so half the probability of a monomino. But if you pick random polyominos, instead of random squares therefrom, the domino becomes harder to pick, not easier -- you'll get a monomino four times as often as a domino in that case. J.D. Williams's _Compleat Strategyst_ introduced the notion of an "oddment" -- the number 4 in the statement "the ratio of monominos to dominos is 4:1", a numerator of a probability in which you do not yet know the denominator. For picking a random polyomino, we can easily get the oddments of any particular shape, but I don't see how to get the denominator (hence the probability) without doing all infinitely many of them. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.