## Monday, March 09, 2009

I don't know if it's illegal or not, but it's certainly taboo in some corners to teach about gambling in schools. I looked but cannot find the words or phrases "gamble", "gambling", or "games of chance" anywhere in California education code (search here), but I'm not sure.

This is remarkable to me. Our state has a lottery system, some of the proceeds of which fund public education. The odds of winning are printed on each ticket!

The entire field of probability was jump-started when some enterprising individuals tried to determine probabilities for winning at games of chance! I can't tell you how many students ask me about Texas Hold'em, tournaments of which they watch on television but about which I know nothing.

Just as it's against the law in California to teach about communism with the intent to instill in the minds of students a preference for communism over our representative form of government, I decided that if I teach about gambling I must do so without glorifying it or instilling a preference for it. I assume that this is how sex, drugs, and alcohol are also addressed in the appropriate classes.

As my Algebra II students have been learning about probability, permutations, and combinations recently, games of chance are very relevant to the subject matter. Many of the problems in our textbook have dealt with probabilities of drawing certain cards from a deck, so I've extended that to other probabilities--for instance, the probability of being dealt a royal flush in 5-card stud (or 5-card Darren, same thing).

And then there's Keno.

Keno is similar to the state lottery in that balls are drawn from a hopper, but more interesting. Players mark numbers from 1-80 on a Keno ticket, and then 20 numbers are drawn from the hopper. If a certain number of the players numbers are drawn, the player wins! Keeping with the "don't glorify it" theme, we calculated the probabilities of the player's choosing only 1 number and having that number chosen. Turns out there's a 1/4 chance, meaning that if the player chooses only one number (say, 25), that number will be drawn on average only once in four games. Since Keno is generally \$1 a game, the player would have to play 4 games at a cost of \$4 in order to win one. According to my Circus Circus Reno Keno payout sheet, a player would win \$3 for getting one number drawn out of one.

Did you get that? Mathematically, you pay \$4 in order to win \$3. Those casinos weren't built because all the players were winning, you know!

But no one picks only one number--birthdays are a favorite. So we calculated the probability of choosing two numbers and having both of them drawn. Skipping all the math here, the probability is about 1/17. In other words, you'd have to play 17 games (costing \$17) to win one--and the payout would be only \$14 according to the Circus Circus booklet.

We did the same for 3 numbers--probability of 1/72, for a payout of only \$42.

The students seemed to enjoy the "practical" lesson and I taught much about counting principles and probability. Additionally, no one could say I was glorifying gambling at all. Sounds like a win-win to me.

What's the probability of that?

Steve USMA '85 said...

One I used before teaching probably was the odds on the craps table. Problem nowadays is that few people play craps so the lesson isn't as relevant. Still, I find this kind of lesson fun because I watch the lightbulbs go off in people's head as they begin to realize that casinos and lotteries are in business to take money, not give it away.
Another exercise is calculating the odds of the mega-millions type lotteries. First, calculating how infinitesimal the odds are to win, then showing how buying a second ticket really doesn't increase your odds in any meaningful way.
Still, all you need is one parent who thinks any discussion of gambling is one more step along the highway to hell to ruin your day.

Joshua Sasmor said...

My favorite example is the roulette wheel, with and expected payoff of -5.26% on every bet available on the Vegas table :) I then talk about independence of trials, and the "history board" that the Vegas tables installed about 10 years ago to push people to make bets that are "due"... Definitely not glorifying the game, but certainly instructive.

As for the history, go back to the original discussions between the Chevalier du Mer and Pascal and Fermat about how to determine the pot split on an interrupted game :) That's the foundation of expected value and modern probability theory.

Superdestroyer said...

You may want to look at your assmptions on winning kemo. If you have a 1/4 chance of winning, the the chance of not winning is 3/4. If you play twice, the chance of winning becomes 1-(3/4)^2. If you sent p=0.5, then the formula for calculating a 50% chance of winning is 0.5 = 1-(3/4)^N. thus you have a 50% chance of winning with three plays and a 90% chance of winnng in 8 plays.

Or ln(0.5)= n * ln (

Doug said...

My school district has a "Disciplinary Chart" that outlines the punishment for certain student behaviors and which section of the ed code it relates to. Gambling is on there, and the code they use is 48900 (k), which says : (k) Disrupted school activities or otherwise willfully defied the
valid authority of supervisors, teachers, administrators, school
officials, or other school personnel engaged in the performance of
their duties.
I think you are safe in a math class dicussing probability in games of chance.

Anonymous said...

Would it be okay for you to *run* a "pick one" game in class every day and have the kids keep track of how they were doing? My thinking here is that 180 school days is enough so that "win \$3 for every \$4 you put in, on average" would tend towards everyone losing. Which is the point.

One wouldn't play for money, but could/would keep a chart of how everyone was doing.

I expect that the end of the year when *EVERYONE* in the class was behind would have a powerful impact on how the odds really worked.

You'd continue to do the calculation work, too, of course.

-Mark Roulo

PeggyU said...

Superdestroyer - Let me see if I follow. Wouldn't that mean that the chance of winning at least once on those 3 attempts would be around .5? But most of those "winning at least once" are just "winning once" or breaking even. The odds of winning on 2 out of 3 or on 3 out of 3 would be less.

PeggyU said...

I have an 11-year-old son who is keenly interested in poker of all kinds, as well as other games of chance and strategy.

He thinks it is not only NOT wrong to teach probability as it relates to gambling, but imperative to do so since it will lead people to better decision making.

PeggyU said...

He asked me to share this quote from the foreword of one of his favorite books, An Introduction to Poker by Stewart Reuben:

"Poker has long been a most popular card game in the U.S. Of course it has a somewhat sleazy reputation, some of it deserved. Of course, it's a gambling game, but gambling is part of our everyday life. If you define gambling as wagering, then investing in the stock market is gambling. Deciding to go to school, instead of straight to work after high school graduation is a gamble that you will eventually make more money that way."

"If you define gambling as playing against the odds, I have just one word you should take to heart: DON'T! I have never been tempted to buy a sweepstakes ticket. Only about 50% of the money staked is returned as prizes, with massive odds against winning. The worst odds in poker are slighty better than even money." He goes on to point out that games of strategy (played well) improve chances of winning, and the point is to analyze the information at hand and never play with money you can't afford to lose. Obviously, that would apply to other financial decisions outside of the poker room as well.

Anonymous said...

I have a few favorite sayings, and one that I frequently trot out in math classes is that the lottery is a special tax on people who are bad at mathematics. Recently there has been a show on a cable channel called "The Curse of the Lottery" that makes this saying resonate a little more.

As far as teaching about gambling, I think it is reasonable to consider it under the general goal of enhanced numeracy. While I freely concede that I have a weakness for craps, at the same time I haven't been in a casino since 2001, so I'm not exactly a hypocrite when I suggest to students that gambling is generally a waste of money. Even low-level students can understand the concept of expected value as a "long-term average." If you explain to them, for example, that a payout ratio (expected value) of 0.85 for a slot machine means they will lose, on the average, \$0.15 of every dollar bet, they can appreciate and understand what that means, I think.

Too, when I collected paper and pencil homework, back in the day (I collect whatever math homework that I collect using online ancillaries now), I used to play a round of craps with the students to see whether I'd collect a particular assignment or not. Nothing fancy... just the student playing the role of a (right) pass-line bettor and me playing the role of the house. If they won, they could decide whether the class would submit the assignment or not; if I won, I could decide. Admittedly, I had to explain the rules, briefly, to every player just about every time, but I would also suggest they consider how many times they lost while playing.

As an aside, about ten years ago I had a guy in one class who worked as a programmer for a slot machine manufacturer, and we had a very interesting conversation regarding how the machines of today are programmed. I think it is fair to say that gaming machine manufacturers are well aware of certain aspects of human psychology, and how to employ them to maximize revenue for their customers. In the old days of purely mechanical slots, each outcome of the wheels was equally likely. Today, the machines are electromechanical, and the outcome of each wheel is no longer equally likely. The machines use a lookup table, in which small jackpots (intermittent reinforcement) are assigned relatively high probabilities, and huge jackpots are assigned miniscule probabilities. The result is an expected value in line with what the customer wants/needs, namely a machine that will separate the player from as much of their money as possible in a reasonable amount of time.

rightwingprof said...

I once had a really dense class. I asked them: "A container contains 100 balls. 60% are red, and 40% are blue. What is the probability that you will draw a red ball out of the container?"

Nobody knew.

Mrs. C said...

Darren, in case you don't check back, I have amended my post to include the important information you mentioned!! :]

The MAN Fan Club said...

LOVE the post. Not many guidelines and many gray areas in teaching for sure. Only a problem if someone complains.

My keno story. I took \$40 to play and was playing 25 cent games all night. My friend blew through his \$200 so when I had \$5 left I bumped my keno play to \$1.25 per game and on my last push I hit all 5 numbers and it payed about \$245. You want to win when you are leaving for sure. My friend drove and I tipped him \$50.