The Mathematics of Games and Gambling: Planning Part 2

Saturday, August 25, 2012

Note:  This post originally appeared on my other (non-education related) blog Nachos Grande.  I'm slowly moving all the education related posts over to this blog.  Don't worry, there aren't many that need to be moved.  This particular entry was the second in what was supposed to be a fairly lengthy series.  Unfortunately, I never got around to writing a third planning post.  Thus, any future planning posts you see in relation to my Mathematics of Games and Gambling class will be new!

Image source:  http://entertainment.howstuffworks.com/how-to-play-roulette.htm  
This is the second in what will (hopefully) be an ongoing series throughout the summer as I plan my new course that I'm teaching this fall.  You might want to read the first post in order to see why I'm doing this and to catch up with me.

It's taken me over a month, but I finally have the first week of the course figured out and the lessons are (mostly) finished.  When I teach, I usually use PowerPoint so part of my course planning involves making the PowerPoint presentation to go along with the day's topic.  I also usually allow my students to print out the slides beforehand (if they so desire) so I have to add lots of animations to keep the work to problems (and the solutions) hidden from those who don't want to see the answers while they are working the problems out in class.  Obviously, it is a lot of work - but based on student feedback, it's appreciated by the majority of them.  Of course, there's always one or two students who tell me I should write on the board all the time instead of using the PowerPoint (they obviously haven't paid attention to my handwriting)!

Before I put together the first week of lessons, I sketched out the full schedule of the semester - and if time allows (and there's enough interest out there), I'll probably do a post covering each of the major topics as I complete them.  In this way, I can motivate myself to work harder all summer - and be accountable for my actions if I don't!

I've decided to start the course with a week long study of finite probabilities.  At the college where I work, the entire first week is known as Drop/Add week which means theoretically a student can add my class on Friday at 3:59 PM and expect to do well in the course (despite missing three classes already - it's a Monday, Wednesday, Friday morning class).  As a professor, I hate the drop/add week because you can't cover too much or new students have little hope of catching up, but you can't cover too little because then the students who attend the first week get bored and think the class is lame.  It's a tough life.

So, I usually do some form of compromise.  In this case, it's going to be a week of finite probabilities.  We'll look at dice and cards on Monday and Wednesday and then switch over to Roulette on Friday.  I think I can arrange to get a roulette wheel for the day so we'll probably "play" a little Roulette as a class as well.  I'm debating about giving each student a "gambling bank account" of fake cash to use over the course of the semester.  Will they listen to the odds and keep the money safely in the bank or will one student's success convince ten others to try their luck (with undoubtedly poor results)?  It should be fascinating (at least for me).

Once the second week begins, it's the proverbial pedal to the metal as we plow through a bunch of topics.  I think the second week will pick up with Roulette and mathematical expectation.  Speaking of that, did you know that the expected value of a $1 bet on ANY* type of Roulette bet on an American Roulette wheel is -0.0526?  That is, you should expect to lose on average 5 cents for every dollar you bet in Roulette.  No wonder the casinos love the game!

*There is one bet (the five-number bet) on the American wheel that gives the house an even greater edge of 7.89% or almost 8 cents per dollar.  See the image below for more details on the possible Roulette bets on an American wheel.

And for those that are wondering, there are two types of Roulette wheels in use around the world.  The "original" wheel (which is used basically everywhere except in the US) has the numbers 1 - 36 plus a green 0 spot.  The American wheel is identical except it adds a green 00 spot to go along with the single 0 spot.  In other words, the house advantage doubles on an American wheel!  It's not hard to see why Roulette isn't terribly popular in America and yet it is quite popular everywhere else around the world.
Image source:  http://www.bestamericanroulette.com/roulette-guide/american-roulette-odds/  

For the first week of my lecture, the main goal is for students to learn how to compute expected values (such as the one quoted above for Roulette).  Expected values aren't terribly difficult (but they prove to be an extremely valuable tool when analyzing some games)!  Essentially, you can compute expected values by summing up each individual event's probability multiplied by that event's respective payoff.  The one catch is that each of the events with probabilities have to be pairwise disjoint.

As with any course that discusses mathematical expectation, the section will conclude with a problem known as St. Petersburg paradox.  Essentially, the idea of the paradox is to flip a coin as many times as you can until you flip a tail (which forces you to stop).  The payoff is based on the number of heads you flipped before landing the tail.  If n represents the number of heads you landed, your payoff is 2^n.  The questions (that I won't answer here for those who want to think about it) are:

  • What is the expected payoff for this game?  
  • Is it realistic?

If all goes well, I should be able to write about games like Craps and Chuck-a-Luck and maybe even some computational rules (along with counting rules) next time as that's where the course is now headed.  I've got about two more months of summer to make this happen!

6 comments:

Carlos Salgado said...

You have me thinking about this... The question of the expected payoff is of the St. Petersburg Paradox or the Roullette?

FanOfReds said...

The question is: What is the expected value for the game described in the St. Petersburg Paradox (namely, what is the expected value for a game in which you flip a coin repeatedly until you flip a tails at which point you stop and you are paid 2^n where n is the number of consecutive heads). It's an interesting problem to be sure - and one I think will probably blow the minds of the students (assuming they understood the basic ideas of expected value seen during the first four or five days of the course)!

Carlos Salgado said...

Let me tell you something, you have me here wishing I was in college all over again. Do you use a specific textbook? I think I'm going to take this course "at large". hahaha

FanOfReds said...

I'm using "The Mathematics of Games and Gambling" by Edward Packel. It's basically the only book on the market that is appropriate for the level of course that I'm teaching. I will say that I probably won't use the book next semester if I teach the course again though, I've been disappointed in the homework problems in the text!

Carlos Salgado said...

My experience, and that is 12 years ago when I was a TA in college for the statistics professor, was that the assignments prepared by the instructor are almost always more effective than those prepared wholesale for the textbook. I'm guessing you provide them with great assignments in the powerpoint presentations, right?

FanOfReds said...

I have been TeXing my problem sets for the students and will hand them out as we cover the various sections. It's nicer that way because my problems can draw from everything that we've covered. Textbooks are sort of limited in that they can't be sure a professor covered each previous section (and so the homework tends to be much more simple computations).

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