Revisiting “SET® Probabilities Revisited”

In my last post, I discussed a ‘quasi-thrifty’ algorithm to select cards during the play of SET® in order to take as many as possible by the end of the game, thereby increasing the chances of ending the game with a cleared table.

In fact, the algorithm enjoys another consequence: after a few rounds of play, it starts to affect the odds a Set is present among the displayed cards on the table, a statistic detailed by Henrik Warne in his interesting post “SET® Probabilities Revisited.”

Henrik created a simulator to model human play of the game, collected large amounts of data, and then used that data to analyze statistics during play. He found that the game begins with 30:1 odds a Set is present in the first round, but then the odds sharply decline, and after about the fourth round, they are 14:1, slowly declining to about 13:1 by the final round.

I asked my simulator to perform the same analysis, using its random selection setting to model human play, and I found similar statistics.

However, my simulator also enjoys a ‘quasi-thrifty’ selection setting, which uses an algorithm that intends to clear the table at the game’s end. The algorithm works by removing those cards which eliminate the fewest number of Sets from the list of all of those that could possibly be formed using the remaining cards, thereby increasing the chances that more Sets will remain to be collected later in the game.

When using this selection method, the results were as expected: the odds of a Set being present among 12 cards were, on average, higher. More precisely, the odds still decrease sharply at the start, but after about the 6th round, the odds start to increase, gaining momentum until the odds are nearly 300:1 that a Set is present by the time the deck runs out.

It’s possible to run the same type of analysis for the odds when 15 cards are present, and the results are similar: the ‘quasi-thrifty’ algorithm enjoys, on average, better odds than random selection, and these odds actually increase as the game progresses.

These curves would probably smooth out a bit—and we would obtain more accurate statistics—if I had the patience to run more simulations. In these graphs, I’ve only accumulated data from 100 thousand games (whereas Henrik has data from 10 million!).

Nevertheless, these analyses cast some doubt on the statistics listed in the instruction manual of the game, which states that the odds of a Set among 12 cards are about 33:1, and a whopping 2500:1 for 15 cards. This doubt is not particularly new: it was first discussed, as far as I can tell, some time ago by Peter Norvig in his interesting post “The Odds of Finding a SET in the Card Game SET®,” which actually served as some source of inspiration for Henrik’s post.