This is a discussion on Bill Chen  "The Mathematics Of Poker" Study Group within the online poker forums, in the Cash Games section; Hello :) This will be the ongoing Study Group thread for Bill Chen's book: "The Mathematics Of Poker". About the book and the Study Group. For anyone who 


#1




Bill Chen  "The Mathematics Of Poker" Study Group
Hello
This will be the ongoing Study Group thread for Bill Chen's book: "The Mathematics Of Poker". About the book and the Study Group. For anyone who hasnt heard about "The Mathematics Of Poker": its a great book! Generally its about...poker math (as the title itself suggests) but it goes a bit deeper, beyond basic stuff (such as: counting outs or calculating pot odds) and presents also some interesting concepts from game theory (mostly by introducing and then solving various "toygames" which are used to model pokerrelated "scenarios"). The book consists of 5 parts. We will start from the 1st one, which generally serves as a reminder of some basic concepts such as: Probability, Expected Value, Variance, Bayes Theorem etc. Its rather boring and does not contain anything revealing but it allows to get used to the "language" of the book and authors' overall approach. Also, if anyone does not know/remember those things  thats your chance to change it The 2nd part is about "Exploitive Play", which I think is the part that most "poker math" books are all about so I originally prefered to skip it but... we dont have to. The 3rd part is IMO the most interesting one because it deals with the "Optimal Play". Thats the part that Im personally very excited about but...we will talk about it when (and if) we get there. Im aware that the book can be a bit difficult and that not everyone has a lot of spare time for poker, therefore I suggest we start with reading one chapter per week. In the meantime, throw questions (if you have any) or just start a discussion about the current chapter here. After a week, I will try to put a short summary about the crucial concepts from that chapter and we will move on to the next one. You should have access to the book > I'm not going to transcribe/copy the entire book here. Also, since the book is very theory heavy, I will try to come up with some practical examples. We can adjust the studying speed later, depending on everyone's preferences. Tbh, I have really high hopes for this thread and I think we can learn lots of interesting things together. Currently, I know of 4 people who already have the book and few others who declared an interest in the Study Group. I will try my best to keep this thread going, unless of course I end up talking to myself here. One more thing: I've never been in any "study group" before so if you have any suggestions (eg: how it should look like), let me know! Also, I created this thread but its not MY thread. You're welcomed to post whatever you like and talk with whoever you want to talk (keeping it on topic ofc ). Good luck and lets have fun! Few tips before you begin. a) Right, so the book can be sometimes a bit difficult, especially to people not used to that kind of technical, math oriented books. Dont hasitate to ask any questions about stuff you dont understand. b) Do not get discouraged if you dont get everything at the first time. There is no shame in reading a single page (or even a single sentence) few times before moving on (the same goes with going back to previous chapters). Also, sometimes its better to just leave the book and get back to it next day. c) Some parts (especially the "Optimal Play" one) may look a bit "out of touch with reality" (eg: using a [0, 1] distribution instead of hole cards) and you may not see any correlation between stuff from the book with how it could be applied to the actual game. Well, unfortunatelly (or maybe thats for the better) there is a huge gap between presented concepts and its realworld applications and there is even a lot more work to be done after finishing the book. It does not mean that the book is useless though. I firmly believe that you can learn and improve a lot even if you dont choose to follow the path presented by the book (the next step after finishing it could be: "breaking down" your game in a software such as CardrunnersEV). d) Finally: do not expect too much out of this book. The amount of work, time and patience you put into understanding it, does not necessary guarantee that you will instantly become a better player and start crushing tough games. Dont set any high goals/expectations, instead just treat it as: a learning experience, another way of looking at a game, a chance to talk with others about interesting concepts/things you dont understand, a chance to improve and learn something new. Useful resources for the book. 1) Lectures from MIT 15.S50 (aka MIT Poker Class) Lecture 3: Introduction to Game Theory Slides : http://web.mit.edu/willma/www/2013lec3.pdf Video 1: https://www.youtube.com/watch?v=BuxCNZ0RVKA Video 2: https://www.youtube.com/watch?v=xA6WWSVLitY Lecture 5: Game Theory in Practice: A Tale of Two Hands Slides : http://web.mit.edu/willma/www/2013lec5.pdf Video 1: https://www.youtube.com/watch?v=VHcrsMPQtgo Video 2: https://www.youtube.com/watch?v=dwtrcee6gNk 2) https://www.youtube.com/user/PokerDavidD/videos (31 short videos about Game Theory in Poker. They are a bit...slow (the author likes to repeat himself) but its a good start I guess.) 3) Jerrod Ankenman's (coauthor of MoP) forum thread about GT in poker (http://forumserver.twoplustwo.com/15/pokertheory/askmeanythingaboutpokergametheory1225323/) 4) http://blog.gtorangebuilder.com/ 5) Matthew Janda  "Applications of NoLimit Holdem". 
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I guess we can get started. The first 2 chapters are like...5 pages each and tbh I dont think I will have much to add to them. For sure, I will have (at least I plan to have) a "real" Holdem example for Bayes' theorem (3rd chapter). For now, I will stick to what I've said in the first post: 1 chapter per week (we can change it later). Start reading! 
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in to follow along, don't have the book...do have "the math of holdem" by collin moshman & douglas zare which is next on my reading list so don't see me buying another unless anyone knows if its perhaps lots better than the math of holdem?
I will note looks like some great folks with interest here though, so I'm sure there will be lots of knowledge to get from the discussion alone. 
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Have seen this book recommended by a lot of great players.
Sauce actually wrote an article a while back mentioning how this is one of the best (or maybe the best) reads as far as poker books go. Old article though. Excited to follow along and participate. 
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#13




Geez... I'm wishing I would've payed more attention in math class.. Or maybe I did and just can't remember most of it?
I can understand some of the equations but I'm getting stumped on simple stuff(or just don't undetstand how we got there). I don't want to post stupid questions, but if I'm having troubles understanding the math in chapter 1, I don't know how much I'll be able to contribute( if any at all). Do you mind explaining certain math equations if we don't understand them, Martin? Or would this be too much hassel throughout our study group? Tks in advance! 
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re: Poker & Bill Chen  "The Mathematics Of Poker" Study Group
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I will post an example for Chapter 1 tomorrow btw. 
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Im just going to continue reading and skip through anything I don't understand, for now. Tks 
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Example: Probability of flopping a flush draw, holding 2 suited cards: There are 3 possible flops for your flush draw (S  your suit; O  not your suit) A: [S S O] (1st card has to be your suit (11 out of 50 left), 2nd also your suit (10 out of 49 left), 3rd not your suit (9 left of your suit, you do not want them so you take remaining (48  9) out of 48 left)). B: [S O S] C: [O S S] P(A) = 11/50 * 10/49 * 39/48 = 0.036 P(B) = 11/50 * 39/49 * 10/48 = 0.036 P(C) = 39/50 * 11/49 * 10/48 = 0.036 (for mutually exclusive events) P(A or B or C) = P(A) + P(B) + P(C) = 0.108 = 10.8% (I hope I got it right. I've never been good in probability problems but Flopzilla says its 10.9% so I guess its just a rounding error. If you do this in Excel you will get 10.94%) Same goes with eg: flopping a set (you have to remember about not flopping quads AND not pairing the board (FH)) or any other stuff. I wouldnt be paying too much attention to that kind of calculations, you can always use software such as Flopzilla or ProPokerTools for them (or look here (http://www.flopturnriver.com/pokerstrategy/pyroxenescommonflopodds19147)). 
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I'll sum up the first chapter now, because I wont have much time to post during this week. Three important things to remember from this chapter:
1. EV is additive. If you are desperate enough to do those various hand vs. range EV calculations (eg: in Excel), what you can do is calculate EVs for each hand in that range and than just sum them up, keeping in mind number of combinations (that last equation for <A, B> on page 20). Additionally, if you really dont know what to do with your spare time and you do some kind of multistreets EV calculations (comparing various lines etc), you have to account for EVs from previous streets (+ changes in pot size, card removal effect etc. If you also operate on ranges with assigned various weights > it becomes even more f**ked up) as well. 2. The mathematical approach to poker is concerned with the maximization of EV. There is no metagame/feel/reads/historybased play involved. Its all just about making the highest EV decisions (making only +EV decision is not enough. It has to be the highest possible +EV one). 3. Gamblers Fallacy. Dont treat independent events as if they were dependent. Hitting heads 3 times in a row in a coin flip does not mean tails is more likely to come in a 4th attempt. Same goes with: "I havent hit a set last 8 times, so this time Im due to hit it!" type of thinking. Also, bringing a duck to the table or jumping 3 times on one leg with closed eyes ("for luck") before a session does not guarantee that you will hit flush draws more often. I saw a few people had a problem with understanding that (mostly in the famous "R..." thread) so its good to set this straight before we move on. Quote:

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I'll just post an example that I have for Chapter 1, mostly concerning the maximization of EV. I came across this toygame long time ago (I dont know if its in MoP or not) and I was going to share it anyway so instead of posting lame, standard and boring "coin flip/call a shove with a flush draw/dice roll" type EV calculations, I'll post this one: The Game. * Its a halfstreet type of game, in which there are only 13 cards used: 2T, J, Q, K, A * Two players: Villain and Hero. They both get dealt a single random card. Rules. * There is already $100 in the pot. Hero and Villain have $100 in stacks left. * Villain is first to act (OOP) and he always checks. * Hero can either a) shove or b) check behind. a) If Hero checked, there is a showdown and the player with the higher rank card wins the pot. b) If Hero shoved, Villain:  Calls with top of his range: T, J, Q, K, A  Folds with 29 So even though this is just a fixed game/scenario and rules for Villain's actions are predefined, his strategy does not seem that "bad" (given that he knows nothing about Hero's betting patterns). There is no point for him in calling with 2 or 3 because ... he's not beating much, so he might as well just only call with some % of his strongest hands. Card Removal. There is of course a card removal effect here: if I have a card from Villains check/folding range, lets say: 3, he will fold not 8 cards (29) but only 7 (2, 49). Same goes with his check/calling range. Maximizing EV between lines for every hand in our range. As stated in MoP, the goal is to play each card in our range as best as possible and since we only have to make a decision between Checking or Betting we only need to calculate EVs for those decisions. Then, we will pick the one with the higher EV. EV of Checking: <Check> = [(amount_of_hands_we_beat/12) * $100] + [(amount_of_hands_that_beat_us/12) * $0] EV of Betting: Two possible outcomes when we bet. a) We win the pot without the showdown. b) We go to the showdown and a player with the higher rank card wins. <Bet> = (F/12) * $100 + (C/12) * [(W/C) * $200] + (C/12) * [(L/C) * $100] where: C  amount of hands villain calls with F  amount of hands villain folds W  amount of hands we beat L  amount of hands we lose to And obviously: C + F = 12 W + L = C Exemplary calculations. * We have an Ace. <A, check> = [(12/12) * $100] + [(0/12) * $0] = $100 <A, bet> = (8/12) * $100 + (4/12) * [(4/4) * $200] + (4/12) * [(0/4) * $100] = $133.3 * We have a 9. <9, check> = [(7/12) * $100] + [(5/12) * $0] = $58.3 <9, bet> = (7/12) * $100 + (5/12) * [(0/5) * $200)] + (5/12) * [(5/5) * $100] = $16.6 EVs for our entire range. So here is how our entire range looks like with all those EVs calculated. I also marked a decision with the higher EV. RangeTable.jpg20150216 19_33_25Chapter1example.ods  OpenOffice CalcNew.jpg So this is kinda interesting, right? Our betting range is polarized, meaning it contains both strong hands and weak ones and nothing in between A few conclusions. I know it might be obvious for some people but for the sake of formalities: * There is nothing suprising with betting the top of our range. The EV of betting is higher than EV of checking therefore when we have strong hands, we should definitely bet (for value). * When we are at the bottom of our range, the EV of checking is 0 or very close to it > we have no chance/very small chance to win at the showdown and thats why its often correct to bet (as a bluff) with that part of range (EV(bet) > EV(check). * There is also a region in between containing medium strength hands, which have some showdown value. This the area where, when we bet  we only fold out worse hands and get called by better. There is a big value in checking those hands and keeping all the other (weak) ones in Villain's range (EV(check) > EV(bet). * There are also 2 hands, Q and 4 which are indifferent between both lines (EV(check) == EV(bet)). The catch. Thare is a little catch though Villain has to possess a fold button. In this example, we are getting 1:1 on a bluff (we're risking $100 to win $100) so it has to work at least 50% of the time. The reason it does work is because when we are at the bottom of our range, Villain is folding 7 (card removal!) out of his 12 possible cards which is: 58%  he folds too much. 
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we should probably not our overall EV in this game is very very high. EV of bluffing seems small compared to EV of having higher cards. but a 17% edge on a pot is substantial in poker

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re: Poker & Bill Chen  "The Mathematics Of Poker" Study Group
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I assume Villain would have to pick some kind of a mixed strategy which would make Hero indifferent to bluffing: defending 50% (given the pot odds) of his entire checking range. He could defend top X% of his hands and add some % of his bluffcatchers => he can accomplish this in several ways. Then we pick the best counter strategy for Hero and...if Villain cannot improve his EV (pick another strategy for defending)...we found the equilibria? Does it make any sense? 
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11/50 * 10/49 * 9/48 = 0.008418367 = 0.84% _ 
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To me sometimes some of the maths stuff is like reading a sentence in a foreign language, you know some of the words and try and guess what they are on about 
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re: Poker & Bill Chen  "The Mathematics Of Poker" Study Group
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Naah, I was always that "average guy" as far as university math related classes went. I dont even like math that much, tbh. I just find some things interesting and worth getting to know, thats all. Quote:
