Bill Chen - "The Mathematics Of Poker" Study Group

[dealio96]

Chapter 2 Summary.

1. Variance and Standard Deviation.

Variance describes how far from the EV, you can expect your results to be. It is:

a) Always positive.
b) Additive across n trials:

(just like EV).
c) Loose, wild games produce more variance than tight, passive ones.

Standard Deviation is just a square root of Variance:

So for a given n number of hands:

whereas

Few Standard Deviations for most popular games (you can actually find those numbers in HM2, probably also in PT4):

View attachment 76365
2. Central Limit Theorem and Normal Distribution.

Central Limit Theorem - the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed.

(So this actually applies perfectly to poker - we have a large number of independent events (hands) and for each one we can calculate EV and V (both are finite).)

Normal Distribution is given by a formula:

(Btw, that is a probability density function so: )

Spoiler

a) It doesn't contain negative values (because...probability is always >= 0):

b) The area under the curve is always equal to 1:

Its peak is located at the mean and Standard Deviation influences the width of the curve (well...because by definition Std.Dev describes how far results are from the mean).

Spoiler

- mean (EV)

- standard deviation

- Variance

Here is how it looks like:

To calculate the probability of a certain event falling between a and b, we need to calculate the area under the curve for that region. Unfortunately, the integral cannot be solved analytically so we need to do it numerically. Fortunately, it has already been done: formula 2.7 in MoP

I dont want to get into this stuff too much but there is this CDF function which basically answers the question: whats the probability of a certain event a being less or equal to it (so its calculating the probability of: (- infinity; a> ). So if you have 2 events: a and b, what you can do is calculate CDF(a) and CDF(b) and then substract each other and in result you will get the area (which is probability) between both events (thats what the 2.8 formula in MoP really does). In reality, you will usually just calculate a
Z-score (more on this later) and then use a website such as this one:HERE to get what you need

One more thing, we could play a bit with the formula and transform it like this:

Now, the:

is the distance from a given x to the mean (EV). If we divide it by

, we get that distance in terms of standard deviations. This is actually called a Z-score:

Z-score – indicates how many standard deviations an observation is above or below the mean (EV).

Example: You are winning at $25/100 hands over a decent sample with a $350/100 Standard Deviation. Whats the Z-score of you breaking even for the next 20k hands?

<EV> = $25/100 * 20000 = $5000
Std. Dev = $350 * sqrt(200) = $4950
Z = (0 – 5000$) / (4950$) ~= -1

This basically means that you are 1 Std.Dev below expectation (EV). You can enter that Z-score into the website, I just posted a link to, or look at the normal distribution graph to get the probability associated with -1 Std.Dev. Either way, its about 15.9%. So if you had 100 samples of 20k hands, you can expect to be break even in 16 of them

(There are some examples in MoP with those type of calculations so I dont know if I have to post more of them. Anyway, they are not really that important, so if you dont understand them (or you dont even want to) just skip them.)

3. Final notes about Variance.

a) Variance is a bi*ch.
b) Variance does not justify bad play.
c) Variance does not only relate to downswings. It also accounts for those upswings/heaters which recreational players love so much.
d) The goal should not be to decrease Variance (???) but to increase Winrate!

(Here is an example in 10NL NLHE showing the effect of increasing WR)
View attachment 76362

View attachment 76363

View attachment 76364

For more, you can check out this website (it has a Variance Simulator): http://pokerdope.com/poker-variance-calculator/

Nice recap, Martin! I've yet to finish chap 2 but feel I'm understanding it for the most part. It's taking me a bit longer than most bc I find myself rereading certain areas that I just can't comprehend. I'm taking your advice though, and making sure I understand how we got there opposed to try and solve every single equation.

This seems like a good chapter for players who have a problem with understanding the expectation of variance throughout large sample sizes.

I'm also wondering If you could break this down and apply it to your own sample and see if you could predict positive/negative variance swings? I mean... It may be a bit more exciting if we can do it in "real life" opposed to looking at it over 200 die rolls. IDK... maybe I'm missing something here... just figured I'd ask.

Just realized you put up a few examples with a variance simulator, Will look into this after awhile. TY sir for all of the hard work you're putting into this!

 

[deuceswild]

Ok I for one do recognize the importance of covering this topic, what it basically means to me at least is that even if you flop top set and the only threat is a flush draw, the time will cone when the flush draw hits and you lose to 9-10 suited. Also it means that this may happen 7 times in a row BUT that does not mean that this is not a case where you do not want all your money in. Long term its a profitable solution. Conversely, if you have a flush draw and are going heads up against two pair or a set even though your flush may hit it does not justify calling bets ( I know pot/bet size) you will long term lose money even though short term results may state otherwise. Pretty basic I think to everybody, I am greatly looking forward to the breakdowns of the minigames, so everybody got this?

 

[Fknife]

Nice recap, Martin! I've yet to finish chap 2 but feel I'm understanding it for the most part. It's taking me a bit longer than most bc I find myself rereading certain areas that I just can't comprehend. I'm taking your advice though, and making sure I understand how we got there opposed to try and solve every single equation.

I've read the first part of the book about 15 times already.

I'm also wondering If you could break this down and apply it to your own sample and see if you could predict positive/negative variance swings? I mean... It may be a bit more exciting if we can do it in "real life" opposed to looking at it over 200 die rolls. IDK... maybe I'm missing something here... just figured I'd ask.
Just realized you put up a few examples with a variance simulator, Will look into this after awhile. TY sir for all of the hard work you're putting into this!

Yea, I did something like this in those tables (btw: Low(1) means: 1 sigma below EV, High(2): 2 sigmas above EV etc). You can do this pretty easily in Excel and see for yourself how different parameters (Standard Deviation, Winrate) affect possible swings.

On a side note, I think there is an option in PT4 to see the Normal Distribution Curve with various types of hands marked on it. Like: how much below/above the EV you are with hitting flush draws or sets etc.

Looking back at that post, I'm not sure if I explained/covered everything well (on the other hand I dont want to just copy everything thats in the book - it doesnt make sense) so...if anything is not clear, you know what to do.

 

[Fknife]

Ok I for one do recognize the importance of covering this topic, what it basically means to me at least is that even if you flop top set and the only threat is a flush draw, the time will cone when the flush draw hits and you lose to 9-10 suited. Also it means that this may happen 7 times in a row BUT that does not mean that this is not a case where you do not want all your money in. Long term its a profitable solution. Conversely, if you have a flush draw and are going heads up against two pair or a set even though your flush may hit it does not justify calling bets ( I know pot/bet size) you will long term lose money even though short term results may state otherwise.

Well, that pretty much sums it all up I mean, everybody "knows" about short time variance or the importance of long term when looking at the results but...lots of people dont realize how big of a deal Variance really is and how bad/hot you can run playing good (or even bad and in result thinking you're a good player when in reality, you're just on a heater).

Pretty basic I think to everybody, I am greatly looking forward to the breakdowns of the minigames, so everybody got this?

I appreciate your enthusiasm but I hope you've read the first post -> we're doing only 1 Chapter per week for now so...it will take some time till we get to those "minigames" (especially if we are also going to cover Part 2 of the book). Btw, I'm also pumped up for those "toy" games

 

[duggs]

Iv done lots of that so will skip that chapter, onto 3!!

 

[deuceswild]

Hah ya forgot about the 1 chapter a week thing, npnp didn't mean to jump ahead of anybody just have been looking for somebody to pour over this book with since I finished it. Even my most die hard poker buddy for some reason won't give this book a chance. Neither here nor there BUT am really looking forward to next week and hope this thread stays alive so we can actually study the whole book.

 

[Fknife]

KK if everything is clear so far, I guess we should be moving on to the Chapter 3. This is the last chapter of Part 1 of the book and I think we should come to some interesting (and practical) conclusions after studying this one (or not).

 

[Figaroo2]

Martin
Over the years I have done a reasonable amount of work using the z score to assess whether certain of my employers strategic and short term tactical plans have impacted on the the business in a way that is statistically significant. (Usually they were not which is never what they want to hear of course).
However in terms of poker no matter how you are running if you make any big tactical adjustments to your poker game you should be able to use this methodology to assess the impact on your results. For example next month I intend to double my 3bet percentag from its usual 4-4.5% up to 8-9% and see what happens

 

[Fknife]

Martin
Over the years I have done a reasonable amount of work using the z score to assess whether certain of my employers strategic and short term tactical plans have impacted on the the business in a way that is statistically significant. (Usually they were not which is never what they want to hear of course).

Thats interesting I've never used Z-score beside those few assignments, which I had to do for my uni classes...

However in terms of poker no matter how you are running if you make any big tactical adjustments to your poker game you should be able to use this methodology to assess the impact on your results. For example next month I intend to double my 3bet percentag from its usual 4-4.5% up to 8-9% and see what happens

Well, Stars seems to run as usual (although I heard some people had problems logging in) so I guess nothing that terrible has happened yet (I don't know why, but it sounded like a threat to me ). Just in case, I won't even run the Stars client in this month and I'll just leave you alone on your quest to 3betting the sh*t out of those poor FR nits

(On the other hand, I was looking recently at some stuff regarding Blinds vs BTN/CO play and the 3bet ranges, which I came up, were much wider than 8-9% )

 

[Fknife]

First example for Chapter 3.

So, Bayes Theorem basically allows to answer various questions about certain events, providing that we know something about previous ones (we make some assumptions/observations). It's really that awesome!

There is a very popular "medical" problem, which demonstrates the use of this theorem. It goes like this: there is a mediacal test, with some accuracy parameters. The goal is to solve for probabilities of for example: a healthy person getting a positive result.

Because, as always, I want to be original I won't go into this kind of lame examples - besides there is already a similar one described in the book. Instead, here is a poker related one (this one is a classic but you can do a similar one on your own):

Problem: A new guy (unknown) just sat down at a table and he 3bet x out of his first n hands. How likely is that he could be 3betting light?

So first of all, we have to make some assumptions about the average player in that population. I just exported my HM2 database (about 305k hands, excluding my own) into Excel, did some magic tricks there and came up with this:



(I can go more into how to do this, if anyone is interested)

So basically, I treat everyone whose 3Bet is less than 3% to be 3Betting only premium hands - 44.02% of the entire population does that. Then, I have a more reasonable interval (3 - 7%), which is very common among TAGs. Last, there are guys who 3Bet more than 7% - I treat them as light 3Bettors. Oh and btw, I have at least 200 hands on every player in that population (so a donk who just sat down at a table and went broke in 5 hands by 3Bet shoving his entire stack preflop and then left, won't be taken into account).

Now, it's just a matter of asking various questions and applying Bayes Theorem.

Note: Because we have more than 2 possibilities (3 different player types) we can't just use that Bayes form directly from the book - it "applies" only to 2 complementary possibilities (like: someone can either have a car or not). Instead, we have to use the 'Extended form' of the formula (wikipedia.org -> Bayes' theorem -> Forms -> Extended form).

Anyway, lets say that the guy hasn't 3Bet a single hand out of all his 2 hands. How likely is that he's a Value NIT?

We have two independent events (reminder: P(A and B) = P(A) * P(B)). We know that NITs in this population, 3bet on average 1.35% of time so:

P(3Bet 0/2) = P(NIT hasn't 3Bet his 1st hand) * P(NIT hasn't 3Bet his 2nd hand) = (1 - 0.0135) * (1 - .0135) = 97.33%

The entire population consists of 44.02% NITs who will 3Bet 0/2 hands with a probability of P(3Bet 0/2) so multiplying those two together (part of the Bayes Theorem denominator) gives: 42.84% (I refer to this as Weighted %). We have to do the same for each remaining player type in that population and then sum up the results to get the denominator of Bayes Theorem formula (Weighted % Population? )

Next, we just need to see what part of Weighted % Population, each of those players make up for: (Weighted % of each type) / (Weighted % Population).

The results are below:



So yeah, if a guy hasn't 3Bet a single hand out of 2 opportunities that he had, he's more likely to be a NIT than a Light 3Bettor (well, nothing surprising here...)

Edit: Looking back, I should have probably picked a more interesting situation... but hopefully you got the idea behind all of this though.

I did few more situations. Results are for your own interpretation:

1. Never 3Bets (1..4 hands):

Spoiler

2. Always 3Bets (1..4 hands; when he 3Bets 3 out of 3 opportunities, he's very likely to be 3Betting Light than to be a NIT (0.58% only )):

Spoiler

3. Mixed 3Bets. (notice P(3Bets 5/7)).

Spoiler

I'll try to post one more example of Bayes involving hand ranges/hand-reading because I think it will be quite important later in the book!

 

[Panamajoe]

Ordered the book from Amazon today. Looks like a very useful thread to improve my game. Just kinda wish I'd seen it earlier in order to get the book in time to follow in real time but I'll try to catch up.

Thanks for the work you are doing Fknife!

 

[rhombus]

Thats interesting I've never used Z-score beside those few assignments, which I had to do for my uni classes...

What did you study at Uni, was it a degree in Mathematics or more specific like Statistics??

 

[Fknife]

Ordered the book from Amazon today. Looks like a very useful thread to improve my game. Just kinda wish I'd seen it earlier in order to get the book in time to follow in real time but I'll try to catch up.

Thanks for the work you are doing Fknife!

You can join anytime you want to, just start from the beginning of this thread and move forward as you continue reading the book. Thanks

Btw, you're not the first person, who bought MoP to follow this thread. I hope, you won't be disappointed (with the book).

What did you study at Uni, was it a degree in Mathematics or more specific like Statistics??

Electronics engineering

 

[Fknife]

Chapter 3 Summary.

1. Two ways of estimating probabilities of our preditions.

Classical Statistics (frequentist): just observe (eg: collecting hand histories), no assumptions, unbiased. It takes a long time to converge.

Bayesian statistics: assume likely starting distribution (a priori) and go from there - how likely is something given that assumption (assumptions are logical). Allows to quickly recognise rather improbable events (eg: a winrate of 20bb/100 with a 60% VPIP), which elimination would take a lot of time (and sample) when using frequentist method.

2. Bayes Theorem.

Allows us to answer a lot of questions about various events given previous observations/assumptions.

The general formula:

...but for two complementary events (this one is in the book):

Example: An unknown guy sits down at a live poker table and during the first orbit, he only played one hand (1/9). How likely is that his VPIP is >= 50 given that about 8% of players at that limit play > 50 VPIP and the remaining ones (92%) have an average VPIP of 20?

P(plays 1/9 with a 50% VPIP) = (0.5) * (1 - 0.5)^8 = 0.19% <- P(A | B)
P(plays 1/9 with a 20% VPIP) = (0.2) * (1 - 0.2)^8 = 3.35% <- P(A | ~B)

Denominator of Bayes Theorem (aka Weighted % in my tables from my previous post):

P(A | B) * p(B) = 0.19% * 8% = 0.016%
P(A | ~B) * p(~B) = 3.087%

Sum it all up: 0.016% + 3.087% = 3.10%

P(in top 8% of > 50 VPIP players given that he played 1/9 hands) = P(B | A) = 0.016% / 3.10% = 0.5%



3. Apply Bayesian Inference!

Use all the available information to get reads/assumptions: HUD stats, stack sizes, betting patterns, previous actions/showdowns, known cards etc.


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Sorry guys, didn't have time to prepare a Holdem example for Bayes. It should be presented in the book somewhere though (things like: card removal, weighting actions/ranges). If not, I will prepare something later.

Anyway, now that we finished Part I of the book...what do we do next? (I need at least 4 "votes")

a) Move on to Part II - Exploitive Play
b) Move on to Part III - Optimal Play
c) Take a one/two/? weeks break an then move on to a) or b)
d) Leave the book alone because it sucks (I didn't say this)

 

[deuceswild]

I'd go with A, but since we are crunching numbers not entirely on topic but.... What hands does only 3 betting 3% include exactly, also what is the range for 3-7%, and what hand are we assuming the loose players are playing with greater than 8%?, I'm a live player so haven't had the benefit of a hud to illuminate me as to what these percentages mean

 

[Fknife]

I'd go with A, but since we are crunching numbers not entirely on topic but.... What hands does only 3 betting 3% include exactly, also what is the range for 3-7%, and what hand are we assuming the loose players are playing with greater than 8%?, I'm a live player so haven't had the benefit of a hud to illuminate me as to what these percentages mean

Top 3% is exactly: JJ+, AK (= 40 combos). If you additionally include TT, AQ and KQs then it's about 5% (= 66 combos). Add AJ and you have 6.2% (= 82 combos). Thats a one way of looking at it. You can also have a polarized range with a clear value component (hands which you plan on 5bet-shoving; usually those are like: x% of the strongest hands and some bluffs: low PPs or A2-A5s etc) and a bluff component (hands which you fold to a 4bet). For instance a range: [JJ+, AK, AQs, 44-55, 54s-JTs, 97s-QTs, KTs] (= 104 combos) is about 7.8%.

I'm not making any assumptions about actual hands in ranges in that example, its just a number that I picked to differentiate between a Light 3-bettor and a Regular player to show how Bayes works. I even saw maniacs 3betting any Ace or King preflop so... you can imagine how those > 8% ranges might look like

 

[duggs]

i choose A, although I'm open to a week off as i have a hand in. thread is great tho keep it alive man

 

[Fknife]

Wow, lots of votes! Thx! (...) Ok, in that case we will take a 1 week off and then move on to Part 2 of the book.

In the meantime, a little question/assignment regarding pot odds for you, guys! It's not a tricky question or anything like that - just basic pot-odds. They will be heavily used throughout the rest of the book so I just want to make sure we are on the same page with this concept.

So here is the situation: we are on the flop. Hero is IP, Villain is OOP. Pot is $10 and both players have $130 in stacks. Villain always bets pot on each street and Hero always calls -> it looks like this: OTF (pot: $10) Villain bets $10, Hero calls; OTT (pot: $30) V bets $30, H calls; same goes OTR (pot: $90). Now, suppose Hero always has, let's say: 40% equity (it's a really static board run-out and Vallain barrels 3 streets with the same range). Is it profitable for Hero to make such a call-down?

(hint1: every time Villian bets, Hero's getting 2:1 on a call)
(hint2: if you really, really want to construct some ranges for this scenario, here is a one possibility:

board: AhAc2d2c3s
Hero: [88, 55]
Villain: [TT, 99, 44, 65s]

(@duggs is exceptionally not allowed to post his answer to this one this time because he knows all my "tricks" - unless of course a month passes and nobody still will have any clue what the correct answer is...)

 

[deuceswild]

Ty much for that breakdown, clears a lot of mystery up when I'm looking at these HUD generated stats. Also I'm a little disappointed there isn't a few more votes, all the people who didn't have the gusto to read the whole book have already dropped off lol.

 

[deuceswild]

As far as the quiz, hmm I suppose it depends on if you are ahead of 33% of all possible hands, I'm going to guess without getting my calculator out and doing some rough off the cuff estimations, if the villain is always betting any 2 cards that yes this would be profitable. 55-88 seems ahead of more than 2/3rds of any two cards.

 

[Fknife]

Ty much for that breakdown, clears a lot of mystery up when I'm looking at these HUD generated stats. Also I'm a little disappointed there isn't a few more votes, all the people who didn't have the gusto to read the whole book have already dropped off lol.

Well, it's not an easy book and it's not like you can just read 1-2 pages regarding some concept, go back to the tables, apply it and see the result immediately. There is still a lot of work to be done beside just reading the book. On top of that, not everyone has a lot of time or enought patience etc. If this thread actually works out and we manage to cover most of the book's chapters + have some interesting discussions, I would love to see a mod stick this thread so that others who decide to give this book a try in the future, could find some really good information/help here. Thats why I put so much work into this thread. This will be my CC legacy :cheers:

Anyway, back to the stats. You can also look at them like this:

There are 1326 (52 * 51 / 2) possible starting hand combinations. One combo is "worth": 100% / 1326 = 0.075%. So:

A Pocket Pair (6 combos): 0.45%
A Suited Hand (4 combos): 0.30%
An Offsuited Hand (12 combos) : 0.90%
An Unpaired Hand (Suited H + Offsuited H): 0.30% + 0.90% = 1.20%

When looking at stats, I used to have various mental shortcuts such as:

There are about 2 PPs/3 Suited/1 Offsuited hands in 1%.
One Unpaired Hand (eg: AK) makes up a little more than 1% of all starting hands.
A 4% Cold Calling range consists mostly of about 8/9 pocket pairs. (because its hard to find that many suited combos to fill that entire range whereas most offsuited hands are "too big" for such a narrow range)
If someone has a rather low Cold Calling range, he can't have that many Offsuited hands

etc

Btw: there are also applications such as PokerStove(free), Equilab(free) or Flopzilla (trial) which help with visualizing ranges.

 

[Fknife]

As far as the quiz, hmm I suppose it depends on if you are ahead of 33% of all possible hands, I'm going to guess without getting my calculator out and doing some rough off the cuff estimations, if the villain is always betting any 2 cards that yes this would be profitable. 55-88 seems ahead of more than 2/3rds of any two cards.

The exact hands/ranges or the board do not really matter, all that matters is that you always have 40% equity against whatever Villain is barreling with. If you want, you can either PM me or just post your thought process here so that we all can learn something

So far I got 1 PM with the correct answer (@Ambur) and there are few others who are on the right track but not "there" yet.

 

[SillySayLoR]

do not tell me where to be found in the public domain this book?

 

[deuceswild]

Ah so if you are assuming 40% equity then absolutely call every bet. But by the river do we not have either 100% or 0% equity? If you think you are ahead of 59% of the hands villain will play then I think its still a profit making move. Is this correct?

 

[deuceswild]

Wait no you have to be ahead of only 39% of hands villain would play?