Statistically Unlikely

I’ve been playing solitaire Mahjong on my laptop a bit lately, and I’ve noticed something odd about the way the computer sets up the board. In the solitaire “turtle” layout, the middle is four columns of tiles, with a tile centered on them pyramid style; you can’t move any of the four tiles on top of those columns until that fifth tile goes away.  In the last 30 or so games I’ve played¹, there has been one in which one of the four tiles under the centered one isn’t one of its matches–that is to say, if the centered tile is 6 of spots, so is one of the four directly under it.

There are 136 tiles in a Mahjong set; any given tile has 3 others that it can match with.  So you’ve got the top, centered tile, which has three possible matches; for each of the four spots under it, there’s a 2.2% chance that they’ll be one of those matches–just over one out of 50 times.  I have forgotten precisely what transformation to apply to get the chance one of them will come up given four tries, but I’m fairly sure it should be less often than 29 games out of 30.

I have no idea whether this is a deliberate feature of the way the computer makes its layouts, and if so what purpose it serves other than to taunt the player with the knowledge that one of the most vital tiles on the board is blocking one of its own matches…

1: I haven’t kept track precisely.