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 BC325 at 11:30-1pm
email: mylastname @ mathstat then the usual yorku and ca ending

The course will be meeting on the following dates:
September 10, 17, 24, October 1, 8, 22, 29, November 5, 12, 19, 26, December 3
January 7, 14, 21, 28, February 4, 11, 25, March 4, 11, 18, 25, April 1.

The grade will be based on the following components:
weekly homework
participation (incl attendance)
Review assignments
30% (15% x 2 at end of each FW term)

The homework will usually just be a couple of problems related to the subject we happen to be studying.  The review assignments will be more in depth and more like a take home exam or at least more involved than the homework assignments.

Sept 10, '09
Roulette -  about the game, winning strategy/gambler's fallacy
HW #0 given
Sept 17, '09
basics of probability, roulette
HW #0 due, HW #1 given
Sept 24, '09
beating roulette, expected value, odds
HW #1 due, HW #2 given
Oct 1, '09
probability of dice events, the game of craps
HW #2 due, HW #3 given
Oct 8, '09
play craps, conditional probability
HW #3 due, HW #4 given
Oct 15, '09
reading week

Oct 22, '09
conditional probability, analyze craps
HW #4 due, HW #5 given
Oct 29, '09
probability paradoxes
HW #5 due, HW #6 given
Nov 5, '09
train paradox, lottery MAX v. Super 7
HW #6 due, send me your Lotto MAX picks
Nov 12, '09
lottery and combinatorics
HW #7 given
Nov 19, '09
lottery and combinatorics
HW #7 due
Nov 26, '09
birthday paradox
HW #9 given
Dec 3, '09
Winning the lottery twice
HW #9 due
Jan 7, '10
Card shuffling

Jan 14, '10
Perfect shuffles, probability and cards

Jan 21, '10
Poker (play)
HW #10 given
Jan 28, '10
Jacks or Better strategy
HW #10 due, HW #11 given
Feb 4, '10
Texas Hold'Em (play)

Feb 11, '10
Monte Carlo methods with Texas Hold'Em

Feb 18, '10
Reading week

Feb 25, '10
Spent day working on Jacks or better hw
HW #11 due
March 4, '10
Blackjack, basic strategy, card counting

March 11, '10
50-50 games, betting strategies
HW #12 given
March 17, '10
More 50-50 games, betting strategies
HW #12 due
March 24, '10
Weak law of large numbers
HW #13 given
April 1, '10
Weak law of large numbers & hw13

April 9, '10

HW #13 due

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


(April 1, 2010) Thanks everyone. I had fun and learned a lot. If you have any questions on the last assignment let me know and I will try to answer them by email. The best place to hand in your assignment is at my office TEL 2028. I'll accept scanned/typed versions too if you want to email them (but please no MS Word documents, I really can't read those!). Last (and worst option) is to drop it off at the Math department Ross S525, but then it has to pass through campus mail.

(February 11, 2010) I forgot to announce here and change the date for HW 11 on Jacks or Better (I did in class). It is a challenging problem and since I don't want to give a new assignment over reading week anyway, I will delay the due date until February 25. I want you to work together and compare answers.

(February 4, 2010) It turns out that the university network went down just when I wanted to use YouTube in class. I think that for that short time it was possible to access the network within the university but not websites outside of university.

(January 29, 2010) So the example that I did in class yesterday of the game Jacks or Better was start with the hand 4H, 3D, 3C, 9H, KH. Two possible ways we can play this hand is to pursue the pair of 3s or the flush with 4H, 9H, KH. Here are the photos of the calculation that I did 9, 4, K of hearts and pair of 3s (sorry, but one of those photos is barely readable). I calculated that the expected value if I keep the 4H, 9H and KH is 0.3441 and the expected value if I keep the pair of 3s is 0.8236. Looks like I should go for the two pair or better with the pair of 3's!

(November 19, 2009) We have to understand lottery better before we can move onto card games because poker hands is much more difficult than lottery but uses the same ideas. Even when we get to simple games like rock-paper-scissors next term we will use these counting ideas.

Here is something to think about until the next time I see you: Say we have 4 distinct dice, the largest value is twice the smallest. So either the smallest was 1 and the largest was 2 (we said this could happen in 14=4+6+4 ways) or the smallest was 2 and largest was 4 (this could happen in 50=4+6+12+12+4 ways) or the smallest was 3 and the largest was 6 (this could happen in 110=6+4+12+12+12+24+12+12+4+12 ways). Another way of counting this is 14 = 2^4 - 1^4 - 1^4, 50 = 3^4 - 2^4 - 2^4 + 1^4 and 110 = 4^4 - 3^3 - 3^3 + 2^4. Exercise (since I didn't give you the Powerball assignment this week): figure out why this is true. Use the same reasoning to figure out the problem for 5 dice.

(November 14, 2009) Someone pointed out to me a weird phrasing in problem 3 and 4 of the homework. They didn't make sense as written. They should read "the largest value is twice the smallest value" and not the reverse as it was written. I have posted a corrected version here.

(November 14, 2009) I got a number of people who sent me Lotto MAX picks before last night's draw. I will analyze the picks to see if we won anything as soon as I get a few minutes. I did buy a ticket on Thursday (with ENCORE^{TM}) just to see what it is like to play. Watch this website to see if class is cancelled because I am celebrating that I won $15 million dollars. Oh, yea. Don't forget that tonight is the World Rock, Paper, Scissors competition at the Steam Whistle Brewery (just near the CN tower). Sadly, I am not going to be able to go tonight.

(November 7, 2009) A lot of good sending the email about the Lotto picks on Thursday did me. I got one response before Friday at 9pm and he sent me 20 picks. I took those picks plus 27 of my own (including the 2 I had in the email). Between the 47 picks we won $0 + 2 free plays. Only twice did we actually get 3 correct and never 3 with the bonus. We also matched 3 on the extra MAX picks a few times but in order to win you need to get all 7 right for the bonus draws. BTW, I didn't mention that it costs $5 to play each pick so if I had actually played all 47 draws it would have cost me $235. Because the jackpot was over $50 million the lottery triggers more draws. They actually picked tons of winning numbers.
Draws for November 6, 2009.
2 7 8 11 16 17 18 bonus: 12
extra bonus draws because the jackpot was over $50 million. You must get all 7 correct to win.
1 3 10 24 31 32 42
9 14 24 35 37 38 40
24 28 30 32 43 44 47
2 19 30 38 40 46 47
6 8 11 21 23 28 48
7 11 14 16 34 44 46
10 13 17 18 26 28 40
1 4 10 21 29 36 40
6 9 10 24 29 47 49
2 14 16 18 23 36 44

(November 5, 2009) Someone left their umbrella in the classroom. I took it with me back to my office. If you want it, drop me an email and let me know how to get it back to you. I am in TEL 2028 if you want to pick it up or if you send me a reminder I can bring it to class next week.

(November 5, 2009) We didn't get to some of the new stuff I wanted to talk about so there isn't a homework assignment this week except to get me your Lotto MAX picks before Friday 9pm. Think of a reason why OLG changed from Super 7 to Lotto MAX.

(October 29, 2009) Mainly because I've been giving a lot of homework, but partly because I forgot to bring a printed copy, I decided to make the 6th homework assignment optional. If you want to use it to replace one of your other assignments you can do it and bring it next week, but I won't count it unless it is a better grade than one of your others.

(Sept 25, 2009) I forgot to mention, there was a typo on the homework sheet that I handed out (it is corrected on the electronic version here). An American roulette table has 38 values, a European table has 37.

(Sept 24, 2009) We got a new room! It isn't fantastic because it isn't set up the way that a seminar room should be, but at least it 10 times better than BC325. Starting October 1 we will be Ross S174.

(Sept 24, 2009) Register now for the World Rock Paper Scissors championship. It will be held here in our lovely city on November 14, 2009 at the Steam Whistle Brewery. I went last year and didn't win.

(Sept 8, 2009) Here is a list of some of the topics I hope to spend time on in this seminar. The discussion that we have in class will guide the topics we cover so please bring your own ideas to the class discussion:
Games - roulette, craps, blackjack (basic strategy and card counting), Poker, Texas Hold'em, lottery
Finance and risk
gambler's fallacy
the weak law of large numbers
probability paradoxes
Kelly Criterion and information theory
fair games and betting strategies