Expected Value and a real game, (for kids)
My Combinatorics class needed a final unit that would tie together a bunch of what they had learned in some sort of real world application.
The disappointing part of this is that we never reached Deal or No Deal in class. We needed to have the kids watch episodes, and analyze them, and the time just wasn't there.
(For the record, we would have done very simple expected value calculations, discovered that based on expected value alone that the last few offers were the only ones that were ever worth taking, but we would have explored other rational decision-making factors other than expected value.)
(Further, we would have attacked the banker's math, but I don't think the kiddies would have cracked his plan. I haven't, though I haven't tried too hard. The median and mean seem involved, but there is something inversely proportional to the number of remaining cases, as well.)
Instead, we learned to play Rouxletxte. (excuse the x's, I am hoping they defeat the search engines). We learned what the wheel looked like, where the chips go, what the payouts are. And I didn't need to tell them anything.
I taught them to calculate expected value, and they found that (on the american wheel, at least) that each wager has an equal, negative expected value.
It works so easy, I will never truly understand why so few people get it. Our calculations were a shade more complicated, but if you covered each number, $1 each (though we talked about 'points'), you would have 37 losers (-1 each or -37 altogether) and one winner (+35, afaiu). You lose 2 on 38 wagered, or a bit over 5%. Same math holds for black/red: you get an even money payoff even though slightly less than half the numbers are red (0 and double 0 are neither red nor black). Turns out to be the same 5+% (lose 20, win 18, out of 38).
What else? They ran all the wagers, all had the same negative expected values. They read some telegrams from Dostoevsky to his wife. (Dostoevsky was addicted and the kids got to read how excited he got when he won, and how convinced he was by his wins that his losses were aberrations).
They learned a little about progressions, but also about gambler's ruin, and then they got to try playing some progressions. Bingo, half the players ran out of funds.
I didn't actually have to tell them – by the end of last week they were trying to explain why people would play this game at all (they found some reasonable explanations, plus they had studied gambling in Health earlier this year). And they know how to make money at this game: open a casino.
JD2718 
This is way cool. What are the prerequisites for your combinatorics class? Does NY have state standards just for combinatorics? In California there are “Probability and Statistics” standards randomly mixed into most of our math classes. They’re also called out on their own, but it doesn’t seem like anyone does non-AP versions, and the AP version is very stats-heavy.
New York State does not have combinatorics standards.
Instead (rant alert), probability and statistics strands are littered throughout the K-12 curriculum. Now, I think they are important for kids to learn. But basic stats in science class. Chart reading in social studies. You get the idea. In mathematics P & S displace the core skills that are so important. They do not readily connect to other parts of the curriculum. But science is the natural subject for applications of mathematics, and it has been de-mathified.
One of my favorite blogs is a science teacher who insists on doing math with her kiddies. It is Ms Frizzle, and here are a few examples I quickly found: here and here and here and there really is much more. Tons more. Fantastic. But Frizzle is the exception.
So, combinatorics in my school. It is an elective for juniors and seniors. Some kids want an extra math. Some kids want an extra ‘fun course’ which it might be. Also, we require four years of math (NYS only requires three), and some kids need an alternative after trig or precalc because they do not feel ready to go forward.
Instead of doing stats (which none of our teachers wanted to teach) or discrete (which I feel is usually too ‘grab-baggy’) I developed two one-term electives (one for each term).
Logic was the Fall course. It barely qualified as a math course. I followed a text (Hurley) that was designed for a logic course in a philosophy department. We did get a decent dose of mathematical logic in there, though. Certainly it was far more than the dribs and drabs that are presented over four years of high school. And even the non-math stuff counts: I was working with a kid, struggling to study for a state exam, when he noticed ‘wait, this indirect proof, you assume it’s false and show that it leads to a contradiction? that’s just a validity test, right?”
Combinatorics was the Spring elective. I used Niven’s classic, Mathematics of Choice, but freely diverged from it. The last few weeks we worked with coins and cards and even dice. This kept the interest level high (graduating seniors, remember) and brought them to do the analysis above as a sort of culminating unit.
So, we invented non-AP electives so kids could have fun, have extra math, and meet our school’s requirements. We created them because we thought it was right, not because we know of many other schools doing it.
Jonathan — a couple of questions. What school do you teach at, or is that private? Also, does your school have a math team and if so are you involved in coaching it?
I’d rather not post my school’s name; it’s not really secret, but I don’t want to hit the search engines. Feel free to e-mail at [blogname] at gmail dot com and I’ll tell you.
Yes, we do have a math team, and a math club. The team has been around for two years, but only really got off the ground this year. I am the coach.