Bethune 1800: Mathematics of Gambling, a seminar series on probability, combinatorics, statistics and information theory and their relation to gambling. Professor Mike Zabrocki, Office: TEL 2028, web page:, Course meets Thursdays in BC214 at 11:30-1pm

The course will be meeting on the following dates: Sept 4, 11, 18, 25, Oct. 2, 16, 23, 30, Nov 6, Feb 5, 12, 19. March 5, 12, 19, 26, Apr 2, 9, 16, 23, 30, May 7, 14

The grade will be based on the following components:
weekly homework
participation (incl attendance)
project and presentation

The homework will usually just be a couple of problems related to the subject we happen to be studying.  More details on the project and presentation will be available at the beginning of the Winter term.

Sept 4, '08
Roulette -  about the game, winning strategy/gambler's fallacy

Sept 11, '08
randomness, card shuffling, a magic trick, introduction to probability
hw1 due
Sept 18, '08
more on probability, expected value, odds, craps
hw2 due
Sept 25, '08
roulette probability, dice probability, craps
hw3 due
Oct 2, '08
analyzing craps
hw4 due
Oct 16, '08
birthday paradox, more analyzing craps

Oct 23, '08
Monty Hall Paradox, RPS
hw5 due
Oct 30, '08
AIDS test paradox, addition and multiplication principle, nPr, nCr

Feb 5, '09
fair games, weak law of large numbers, more combinatorics
hw6 due
Feb 12, '09
fair games with a limited bankroll, the weak law of large numbers
hw7 due
Feb 19, '09
weak law of large numbers (slower this time), lottery

March 5, '09
random variables, lottery 6/49

March 12, '09
More lottery, discussion about final projects
hw 8 due
March 19, '09
Cards, Poker, rankings of poker hands

March 26, '09
class cancelled

April 2, '09
hw 9 in class, "Jacks or better" strategy

April 9, '09
more "Jacks or better" strategy
project proposals due
April 16, '09

April 23, '09
basic strategy
hw 10 due
April 30, '09
card counting and the Kelly criterion

May 7, '09
Texas hold 'em

May 14, '09
Texas hold 'em

May 22, '09
Gambling Fair

Bibliography - I will add books to this list that I think are of interest to this course


(Sept 26, 2008) Don't bother with the the last 5 questions on the homework. We didn't have time last time to do conditional probability as I thought. The formula for these probabilities is in the notes and we will do it next time. The homework that I posted on the web does not include these 5 questions.

(October 2, 2008) We will go over the expected values of the bets that we did in groups today next week. I decided against giving you homework but homework next week will be to fill in more of that table of expected values of bets. REMEMBER: No class October 9. I will see you again on October 16.

(October 18, 2008) In the last class I asked you to fill out the table of bets in the craps table as homework 5. You might want to print out a fresh copy since I corrected a few errors on those pages.

(October 23, 2008) I forgot the article on Rock-Paper-Scissors that I wrote yesterday and I have posted it in the list above. Wish me luck this Saturday. Next week we will talk a little more about RPS, the contest and then start some other topics.

(October 23, 2008) For the horn and world bets on the betting sheet turned out to be wrong I have corrected them. They may look wierd because the payoff says (for instance) that on the world bet the payoff is 26:5 on 2 or 12, but this is right. This is because if you bet $1 on each of 2, 3, 7, 11, 12 and 12 comes up you will win $30 and lose $4 that was on 2,3,7,11 and you will keep your $1 bet that was on 12. At the end you have won $26 more than your $5 bet so we say the odds are 26:5.

(October 30, 2008) In the event of a strike class will be cancelled and the assignment will be due another day.

(October 30, 2008) Be sure to check out my notes about rock-paper-scissors. I will probably continue to update them but the current version was last updated just after the contest and has a picture of the scissors drawn on the back of my head.

(October 30, 2008) I posted links about the addition and multiplication principle from wikipedia in the list of links above. They are called "rule of sum" and "rule of product" there (I think those names are a lot less common than the ones I use).

(November 5, 2008) I didn't want to acknowledge it last week but it now looks very likely that class will be cancelled tomorrow due to the strike. If negotiations come through and things work out then I will see you tomorrow. If CUPE 3903 goes on strike, then class will be cancelled and I will try to keep you informed about what I know here. The homework of course will not be due until we get back.

(November 6, 2008) Bad news. The university will not be holding classes due to the strike. I have been following updates at the York University website. What is particularly depressing is that the vote wasn't very close.

(January 25, 2008) There is a possibility of the strike ending soon. Please pay attention to the notifications about the strike at York U's website and I will post any relevant information that I have here.

(January 30, 2009) Welcome back all! I'm looking forward to continuing this class. Our next meeting for the 'Fall' term is February 5. We will also have class Feb 12 and Feb 19 and then we have 'Fall exam period' and we are back March 5.
We have a lot to cover and we can get into more interesting topics next term. I want to start out by continuing building some techniques of combinatorics and probability. Those of you that have asked, we will get to Poker/Texas Hold'em...eventually. This next class we will be playing a new game which I hope will demonstrate one of the main theorems of probability, the weak law of large numbers. I am asking that the homework that I gave you for November 6 be done for February 5.

(February 5, 2009) If you didn't hand in your homework last week get it to me next week. You will also have homework 7 due.

(February 12, 2009) I postponed homework 8 to be due the first week of the winter term for two reasons. First, I noticed that there was an error on the paper version of homework 8 (which is relevant for problem 4, but it is still ok for problems 1, 2, 3). I will fix it before next week. Second, I want to spend a little more time on the weak law of large numbers again and we will try to understand its consequences. I am starting to write up what I did in class today and last week but it may take a while.

(February 19, 2009) I fixed the homework assignment that I am posting today. It is pretty short if you understand how to use the theorem. I will be posting my writeup about the two experiments the we did with rock/paper/scissors and the mathematics of the probability behind them when I get some time, but I am not feeling rushed because you will all be busy with this compressed schedule. We will start up again in early March and start to talk about lottery and start working our way into card games early next term. Good luck on finals.

(March 6, 2009) We are going to need to discuss final projects next week but I don't want to write up the instructions until after we have had this discussion. Start forming an idea of what you want to do for your final projects. This way when we talk about it in class everyone can get an idea of what others are doing and I can figure out what guidance I can provide. They can be done with a partner (or two, but I encourage keeping the groups small). The projects will be due during the finals period and we will hold a 'gambling fair' to show off what you have done.

(March 6, 2009) I had a homework assignment from the last class but I decided to wait on it since I asked you to re-redo Homework 8 and hand it in next time. I posted it above and will hand it out next week.

(March 6, 2009) I posted an explanation that a friend of mine received by writing or calling the Ontario lottery board and asking them for an explanation of how to calculate the odds posted on the website. The person who wrote it obviously thought that the formulas for the probabilities were some mysterious magic thing that when you plug in the right numbers you get the answers out ("Our Statistician has advised us that to calculate this ..."). Hopefully the discussion we had in class shows that there is nothing surprising about the probabilities of winning the lottery. I think that there are a couple of errors with that formula too which I pose as a challenge problem: find the mistakes.

(March 17, 2009) I didn't give you a homework assignment last week so we will worry about it for the week after (it has been posted for a while). I just posted the project description that we talked about last week. If you read this in advance of the class and have questions let me know so that I can correct it.

(March 19, 2009) Do assignment 9, but don't worry about handing it in. We will talk about it on April 2 in class. I won't be here on March 26. I offered to play the movie "21" with all the DVD extras and the characters explaining how to count cards but most people have already seen it. If you would like to see it, I have a copy and I will lend it to you for a week.
Project proposals are due on April 2. If you have any questions please email me. Start thinking through the math on your project. It is easy to pick a game where the math becomes too difficult to figure out really fast and you don't want to find out 2 weeks before the project is due that your project is too hard to complete. You should try to pick a game where there is a choice to be made with only a very few options and you want to critically analyze the those options...which one has the better probability of leading you to winning the game? Maybe you don't want to worry about winning the game, but just winning the round or trick (make the assumption that winning as many tricks in a card game as possible will win the game for you).

(March 31, 2009) I looked in the York library and strangely there aren't any books on card games there. However, I looked at the Toronto public library and there are lots. If you need references for your project consider taking a trip to your PUBLIC library (and not the university library).

(March 31, 2009) Someone in the class wrote and asked the following question. I am posting both what they wrote and my response because I think it is information that will help everyone come up with a better proposal:
- What exactly is necessary in our proposal?
- We were thinking of a card game called hand or 14, it seems to be a combination of rummy and contract rummy, since there isn't much background about this game called "hand" should we stick to one type of rummy or continue with our original game?
- How detailed would you like the proposal to be?
- Should calculations be present in the proposal? or should we simply give a description of the kind of calculations we will be doing?
- Is there anything else we need?

My response: So your proposal should be a description of what you intend to do with your game. Roughly this is what I imagine an ideal proposal:

First paragraph you can tell me what the game is about and (briefly) how to play.

The second paragraph you are going to tell me what part of the game you are going to analyze. So this should be a statement that says
(a) what point in the game a decision needs to be made (e.g. in a board game you might have a place where you pick a card vs. roll again, in a card game you might need to bid the number of contracts you think you can make)
(b) how you are going to decide which option you are going to take (e.g. I am going to make a table of all possible hands and for each type hand possible we will calculate the probability of winning and bid accordingly)

Basically in (b) you will need to summarize the information in the game and break it into a (hopefully small) number of cases that you can handle and tell me what strategy you should use for each of these cases and why. You don't need to do any calculations (yet) but you do need to convince me that you are going to be able to come up with a strategy for your game.

(April 10, 2009) On the homework I have broken down the cases of each question by what are the possible hands you can get after you redeal the cards. I am going to break these down further in a writeup for problem 1, further in a writeup for problem 2, further in a writeup for problem 3, further in a writeup for problem 4 and you can check your approach against those. Note that your method for counting these might be different but your final numerical answer should be the same.

(April 10, 2009) If you want to use the computer to help you keep track of calculations try using SAGE. It really works a lot like a calculator does but the advantage is you can cut/paste/edit the way you do with a word processor. This might be very helpful if your project requires that you to do lots of calculations. For instance if you type:
then the computer will respond:
If you get annoyed with typing the word binomial every time you can enter once "C = binomial" and from then on you can use C(4,2) as shorthand notation for binomial(4,2). If you want a decimal answer, then you might try:

(April 11, 2009) We will have a special guest coming on April 16. The author of The worlds greatest blackjack book, Igor Kusyszyn, is a professor in the psychology department at York. He has agreed to come to class and teach us blackjack.

(April 19, 2009) So I delayed the homework 10 yet again so that it would be due on April 23. It is long but with the hints that I posted above it should be easier.

(April 19, 2009) I received a message that the gambling fair would be May 22, 2009 from 8:30-11:30am. The location is still to be determined.

(April 21, 2009) I just found out yesterday that there is a poker tournament on campus today and tomorrow. Registration starts at 6pm (except that they had online registration and other registration dates earlier) and the tournament starts at 7pm.

(April 23, 2009) I received a message that the gambling fair would be Friday, May 22, 2009 from 8:30-11:30am and it will be held in BC202.

(May 12, 2009) I forgot to post the homework earlier. The second problem is is tough, but not as hard as the Jacks or Better homework.

(May 12, 2009) Course evaluations are online (and have been since May 1). I would like feedback especially written comments about how to improve the course. Are there topics that you would like to see more/less of? What do you think of the projects? I will be holding this seminar next year so any ideas are welcome! Course evalations close on May 21: