## Saturday, April 23, 2011

### Imperfect Spider wins and losses

kw: games, statistical distributions

During the last year that I had a computer running Windows XP I kept statistics for 670 Spider Cell games. I won 169 of the games, or 25.2%. I have not yet kept any statistics for the Windows 7 version, but the program reports that I have won 29%. Perhaps this version is easier, or perhaps I am just temporarily ahead of the curve. Also, it may be that keeping the statistics interrupted the flow of play enough that I didn't play as well.

I gathered these statistics to see what the probabilities are for games of various length. The shortest possible winning game is 96 moves. Though there is no longest possible game, because you can use useless moves to inflate the numbers, I used rational rules of play to avoid making extra moves.

Of the games I won, the final tally ranged from 112 to 165, with a mean value of 140. This chart shows that the tallies are normally distributed, with a standard deviation of 11. That means that the intercept at a tally of 96 is at a standard deviation of -4.0. Thus I would expect a game in the 96 range about once per 31,600 winning games. At the rate I win, I might see such a score if I played about 120,000 games. The lowest tally I've seen is 108. That is at 2.9 sigmas, or once per 536 wins.

The statistics on losing are equally interesting. The most likely circumstance is just moving cards about and getting no suits to "complete". Generally, if you can get four suits completed, you will win, and I consider getting five completed means winning is assured, but I've had two games that had five completed suits, yet no win was possible. A statistical chart of losing games, charted by completed suits, is no surprise:

The more suits you complete, the more likely you are to be able to play longer, because typically more cards get uncovered. Note that my shortest game tallies 25 moves. There were two deals of six for which no move was possible. The shortest possible game is 0 moves, but that would require all the deals to be stonewalls, with no possible moves.

Though there is a little curvature to these distributions in line-normal space, we can estimate how likely such a situation is. Zero-deck games are nearly normally distributed with an average of 54 and a standard deviation of 13. This intersects zero at -4.15, meaning once each 60,000 zero-deck games. Such games make up about 45% of all games, so again, it would take playing about 120,000 games to have much chance of seeing a total tally near zero. The negative curvature of the line hints that this estimate may be very optimistic!

Well, I've certainly spent a lot of time gathering these data. Analyzing them has been fun. At the moment I don't expect to gather more statistics. I got different kinds of irons in the fire at present.