February 28th, 2009
Set is the classic pattern match card game in which players compete to find 3-card “sets” out of 12 (or more) cards on the table. I played occasionally while growing up, and I started playing a little bit again recently with Cho.
The instruction booklet states that with 12 cards out, the odds are 33:1 that a set exists. In other words, “don’t give up — there is probably a set, but you just don’t see it”. Despite this, it seems like we get stuck on 12 cards more often than only once every 34 times.
To test whether we’re actually just bad at Set, I wrote a simple Java app that simulates Set gameplay. First, I simulated 100,000 random 12-card deals. The 12 cards were “setless” 3.1% of the time, which agrees with the predicted 3.0% (≈1/34 or 33:1). Next, I simulated 10,000 actual games: for each game, the computer played through the entire deck, selecting sets randomly if more than one existed, and adding 3 cards when there were no sets. During gameplay, 7.3% of all 12-card configurations were “setless” — more than double that of completely random configurations. This is likely because when sets are removed and replaced, the remaining cards are no longer random.
The instruction booklet’s claim of 33:1 odds aims to encourage players to look hard for sets. But the number doesn’t apply to actual in-game scenarios, in which 13:1 is more accurate.