I'm gearing up for teaching my graduate course on randomized algorithms and probabilistic analysis next semester. It's been a while since I've taught it, and I'm somewhat uncertain what to do with the course, precisely since I wrote the textbook. Me lecturing from the textbook is boring, both for me and for them, but of course the textbook contains exactly what I think is important.
Somehow I'll have to have them read the textbook offline and try to do more online learning in class. I'm working through creating some programming exercises based on the game Set. (Amazon picture below.)
Set is a great card came (wikipedia entry, or the Set web site). As it says in Wikipedia, "The deck consists of 81 cards varying in four features: number (one, two, or three); symbol (diamond, squiggle, oval); shading (solid, striped, or open); and color (red, green, or purple)." You turn over 12 cards, and look for a set, which is three cards so that for each feature, EITHER all cards are the same in that feature, or they are different. So below is an example of a set. (different in each feature). If they cards were all green, it would still be a set -- they can be all the same in some features, and all the same in others, and still be a set.
The first player to find a set picks it up, and replacement cards are dealt in to get you back to 12 cards. The player with the most cards at the end of the game wins. My eldest had already seen the game in school, and after a few months, she and I now are pretty competitively matched. I'd put it up there with Mastermind as a good mental exercise game for kids.
So the reason I thought about Set for my class is that the game instructions say that the odds of not getting a set with 12 cards dealt out is 33 to 1. (When this happens, or you can't find one, you can deal another card out; with 15 cards, the claim is more than 2500:1.) But when you play the game, it seems you don't find a set much more often. While my daughters and I are probably missing some sets some of the time, it's also clear that conditional probability is coming into play here. At any point in the game, there aren't 12 random cards on the table. Most clearly, suppose I deal 12 cards, find a set, and replace the 3 cards. What's left isn't 12 random cards; it's 9 cards left after a set was removed, and 3 cards remaining from the deck.**
This seems like a nice way to introduce conditional probability in a concrete but perhaps subtle way. And it seems like there are plenty of other related questions one can ask as well. (What's the probability of not finding a set on the kth turn, given a set has been found in the first k-1 turns...) Feel free to suggest exercises. (Apparently the largest number of cards without a set is 20. I wonder if there's a short proof of that without just doing exhaustive search.)
At the very least, it will be a good excuse to get a couple of boxes of Set for the office.
Have to go. My daughter just asked to play a game of Set...
** Of course I'm not the first to have noticed this. Peter Norvig posted on it as well.