Wednesday, June 20, 2012

The Mathematics of Games and Gambling: Part 2

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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.
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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!


Josh D. said...

I think you mean -.0526. :-)

Brad's Blog said...

Can you let me know if Let it ride has any decent odds at all, i just dont get why anyone spends money on that game

FanOfReds said...

Josh: Yes, (and fixed). I don't think many people would play if they lost 50 cents per dollar (of course, some still would).

Brad: I haven't decided if I want to include Let it Ride or not yet, but I'll look into it!

Robert said...'s an interesting article you may be able to use about probability

Sharpe said...

Very cool stuff.

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