A course in gambling mathematics
I wasn't sure where to post this. My apologies if this is the wrong place, please let me know/feel free to move it.
I do a lot of gambling and tend to make good money doing it, and I would like to brush up on my math knowledge as I feel like it is beginning to hold me back. I studied quantitative degrees so I'm familiar with most of the topics I want to learn but I would like to refresh my knowledge. The topics I think are useful would be:
- Linear algebra
- Calculus
- Prob & stats
- Discrete math
Most of the textbooks I've seen for these topics tend to be either very basic, or proof-based, whereas I'm looking for an intermediate-advanced applied book. I would much appreciate if anybody could recommend some textbooks on these topics that teaches them from an applied point of view, with lots of problems, and covers the right content for a gambler.
Richard Epstein
Quote: lilredrooster"Theory of Gambling and Statistical Logic"
Richard Epstein
I own that one actually, and went through about 50-100 pages of it before deciding that I should brush up on my math first before going back to it. Which led to this topic..
https://www.unr.edu/Documents/business/gaming/practicalcasinomath.pdf
Quote: FCBLComishDoes this help?
https://www.unr.edu/Documents/business/gaming/practicalcasinomath.pdf
$88 on Amazon!
Quote: DRichBrushing up on probability and statistics would be your best bet for gaming math.
^ This.
I'm no "pro" at doing gambling math stuff like some others are, but I do by just fine most of the time. Most of figuring stuff out is just probability and statistics. Don't think I've ever used calculus, linear algebra, or anything else really when doing gambling math stuff. That isn't to say it isn't useful to know about the stuff in those subjects, because the more subjects you've studied, the easier it is to look at a problem and find a solution, since you can look at it from many different angles.
Also, depending on the type of stuff you want to calculate, you can also be more important to learn programming, although you should probably know math too, since they go hand-in-hand frequently.
Edit: Are you the same tree on Ryemo's discord?
Quote: Tree- Linear algebra
- Calculus
- Prob & stats
- Discrete math
Throw in some basic computing courses, and Matrix Algebra, and you've pretty much described what I had to take in my first two years to get into the Computer Science program at Cal.
Basic probability is probably the best place to start. Keep in mind that things like the Risk of Ruin formula are far from "basic."
Wolfram Mathworld is your friend.
Quote: RS^ This.
I'm no "pro" at doing gambling math stuff like some others are, but I do by just fine most of the time. Most of figuring stuff out is just probability and statistics. Don't think I've ever used calculus, linear algebra, or anything else really when doing gambling math stuff. That isn't to say it isn't useful to know about the stuff in those subjects, because the more subjects you've studied, the easier it is to look at a problem and find a solution, since you can look at it from many different angles.
Also, depending on the type of stuff you want to calculate, you can also be more important to learn programming, although you should probably know math too, since they go hand-in-hand frequently.
Edit: Are you the same tree on Ryemo's discord?
Thanks for the response. Yes I am the same Tree from the discord. I guess Calculus is a bit of a stretch but I had thought that some of the trickier analyses would have required linear algebra. And discrete mathematics is very directly relevant, at it's core it's the study of counting things and a lot of the problems in DM textbooks are casino game examples. Maybe I should just invest all of my time into learning Prob/Stats/DM.
I agree that programming is more important. I program quite a bit for my AP as it is. I do want to improve that a lot too but wanted to start with math as it feels more 'fundamental'.
Quote: charliepatrickAs had been said I haven't used calculus or integrals for gambling analysis. More useful is a thorough knowledge of probabilities and an understanding of statistics. Most of the rest you learn on the job or while trying to find out how to work out certain problems. I did find some interesting stuff when developing my simulations (mersenne primes, RNGs, confidence testing). The stuff I haven't yet tackled is analysing blackjack for finite decks or complicated poker games - e.g. some people talk of using bit analysis to quickly evaluate poker hands. I am assuming you are confident in writing computer programs.
Yeah I guess that calculus was overkill, hah. I agree that prob & stats seems much more important, but I also think Discrete Mathematics is very important too. I program quite a bit for my AP as it is and want to improve that anyway but wanted to brush up my math first. I understand the important topics to be honest, I was mainly looking for textbook suggestions.
Quote: ThatDonGuyThrow in some basic computing courses, and Matrix Algebra, and you've pretty much described what I had to take in my first two years to get into the Computer Science program at Cal.
Basic probability is probably the best place to start. Keep in mind that things like the Risk of Ruin formula are far from "basic."
Wolfram Mathworld is your friend.
That site looks like a good resource but not really a substitute for a good textbook, ie guided study with problem sets.
Quote: FCBLComishDoes this help?
https://www.unr.edu/Documents/business/gaming/practicalcasinomath.pdf
That looks a bit too simple to be honest although maybe that means I already know enough lol.
Wizard's web site
Try to read every page, you will learn more than any textbook alone.
After this review, certainly Richard Epstein's book should come next.
Review tiny bits of linear algebra and calculus as necessary along the way.
Quote: teliot
Review tiny bits of linear algebra and calculus as necessary along the way.
I know that you are a renowned expert on gaming math, what types of casino games would one use calculus to evaluate?
Something like this?Quote: DRichI know that you are a renowned expert on gaming math, what types of casino games would one use calculus to evaluate?
https://wizardofvegas.com/forum/gambling/slots/31945-must-hit-by-jackpot-math-question/
If you are simply asking to compute the house edge and standard deviation for a single play of the game, then I have not used calculus for this task. If that is the goal of the OP, then a couple of courses in programming in C/C++ would be useful.
Always a delight to have you weigh in on a WoV message thread. Glad to hear that being "retired" doesn't make you "too tired" to reply when the topic is right down your alley. Many thanx for posting.
Thanks. I enjoy reading *some* of the threads here and will contribute when I have something to add that others equally capable do not seem to be adding.Quote: LuckyPhowAlways a delight to have you weigh in on a WoV message thread. Glad to hear that being "retired" doesn't make you "too tired" to reply when the topic is right down your alley. Many thanx for posting.
I'm doing my best to be retired, but I get a fair number of requests. In the last week -- I was contacted by a VIP game room manager in Macau asking if players were beating his loss rebates (I declined to answer with any specifics, beyond saying "yes"), by a player with two new casino games he wanted evaluated (I sent him here) and by an attorney who needed an affidavit about a device on behalf of a player (I did that one, pro bono). My services are free, hence I decline nearly everything.
why did you not mention this?Quote: teliotI'm doing my best to be retired, but I get a fair number of requests.
Introduction to Gaming Mathematics
I have the pdf
it is very good
you could easily expand on it, if you wanted to.
some have the gift of explaining difficult math concepts so most can understand.
I find Sheldon Ross intro to probability hard to understand mostly all
I just do not get all his advanced thoughts and math
now to the OP
"I do a lot of gambling and tend to make good money doing it, and I would like to brush up on my math knowledge as I feel like it is beginning to hold me back. "
that could be enough for a book or two books.
I enjoy retirement.
added:
"By Eliot Jacobson Ph.D.
March 12, 2013
A.P Heat
I often receive E-mail from readers asking me to explain the mathematics that I present in the posts in this blog. This material can be very challenging for someone with little background in probability theory or statistics. With that in mind, I wrote a small booklet on gaming mathematics. If you want to better understand the material presented here, I offer you:"
https://www.888casino.com/blog/gaming-mathematics/casino-payout
tried to find the pdf I spoke about. Looks to be not available...
Perhaps he doesn't want to toot his own horn. But both are very pertinent to any good math evaluation of gaming. Much of the math is embedded, but I think is invaluable as applied in the field.
https://www.amazon.com/Advanced-Advantage-Play-Safeguarding-Promotions/dp/1883423392
https://www.amazon.com/Contemporary-Casino-Table-Game-Design/dp/1883423171
Care to elaborate on what methods you use?Quote: Tree
I do a lot of gambling and tend to make good money doing it
I completely forgot that I wrote it! I sold it (along with APHeat) to 888 in 2016, and they turned it (in part) into an online guide. I can't post the pdf because they now own the copyright. They didn't post the material on particular games or the long-run. This guide was a re-write of Chapter 1 of my book on table game design.Quote: 7crapswhy did you not mention this?
Introduction to Gaming Mathematics
For anyone who cares:
https://www.888casino.com/blog/gaming-mathematics/casino-payout
Hmmm ... I suppose my blackjack book is a book of a different color, so to speak. :) Thanks for the plug. They are no longer available, I de-listed my books when I retired. Any copies available on Amazon are from private sellers.Quote: beachbumbabsI own both of teliot's books
Quote: teliotHmmm ... I suppose my blackjack book is a book of a different color, so to speak. :) Thanks for the plug. They are no longer available, I de-listed my books when I retired.
Oh, heck, I didn't scroll down to see if the links were still working. I've been recommending them since you wrote them.
Can people PM you to purchase them, or is that too much trouble?
I don't accept PM's. I don't have any copies for sale anyway. But thanks for promoting my books!Quote: beachbumbabsCan people PM you to purchase them, or is that too much trouble?
I'm still stuck at basic arithmetic but I'm learning to say "maths" and "the calculus".
The only one in my collection that comes to mind is not by Mike Shackelford, Frank Scoblete or any other online gambling writers. they are only interested in how to gain the advantage over the casino game one plays or only play for entertainment, with no math, when playing against a house edge.Quote: TreeI would much appreciate if anybody could recommend some textbooks on these topics that teaches them from an applied point of view,
with lots of problems,
and covers the right content for a gambler.
S.N.Ethier comes to mind.
(Stewart N. Ethier)
lots of math applied to casino games with lots of problems and his website has an answer book also. (last time I looked)
"The Doctrine of Chances
Probabilistic Aspects of Gambling"
over 800 pages 2010 version.
http://www.math.utah.edu/~ethier/
good luck to you
Quote: teliotHmmm ... I suppose my blackjack book is a book of a different color, so to speak. :) Thanks for the plug. They are no longer available, I de-listed my books when I retired. Any copies available on Amazon are from private sellers.
I loved the BJZone. It's definitely geared towards beginners but what a great beginners book.
With that said, I must say I enjoyed the Elliot that posted on Ken Smith's forum much more than your posts here.
Enjoy retirement. It takes getting used to.
Have YOU looked at the amazon listings I posted above?
They're (third parties) scalping your books, listing from 300-1500 for Contemporary and 500-5000 for Advanced.
You should be making that money.
Quote: teliotThey are no longer available, I de-listed my books when I retired.
May I know why?
Quote: teliotHmmm ... I suppose my blackjack book is a book of a different color, so to speak. :) Thanks for the plug. They are no longer available, I de-listed my books when I retired. Any copies available on Amazon are from private sellers.
I bought several copies of Advanced Advantage Play and gave them away to people I was working with at the time. I was planning to buy several more. I think the material is awesome, but not worth $516.12 per copy, which is the current sale price.
Why are your books no longer in print?
Thanks for buying several copies!Quote: FCBLComishWhy are your books no longer in print?
My publisher came down with ALS and ceased being able to write or type (and more recently she can't talk) -- in particular she could no longer write me quarterly royalty checks. My publisher is Blue Point Books, which is really just one very nice and hard working woman. That, together with my retirement, and I decided to just pull them. I may re-list them, but I would need to get the original files to edit (I don't even know what software she used), a new publishing house and new ISBN's.
Yes, I know that there are copies for sale at outrageous prices. I seriously doubt anyone is paying that price, especially because the majority of the content is available online at 888.Quote: beachbumbabsUmmm...Elliot...
Have YOU looked at the amazon listings I posted above?
They're (third parties) scalping your books, listing from 300-1500 for Contemporary and 500-5000 for Advanced.
You should be making that money.
Quote: AxelWolfCare to elaborate on what methods you use?
Mostly online AP and some bricks and mortar machine play. I'm wary of elaborating on a public forum but if you want to ask more via private message / IM on Discord or Signal I'd be happy to.
PS - my request in this thread is not directly linked to the type of AP I currently do, I increasingly have opportunities to diversify.
Quote: teliotI recommend "Statistics for Engineers and Scientists" by William Navidi.
After this review, certainly Richard Epstein's book should come next.
Review tiny bits of linear algebra and calculus as necessary along the way.
Thank you for the response. I want to take this opportunity to say that I have great admiration for the quality and rigour of your work, I bought Advanced and had a great time reading it even though it isn't applicable to the type of AP that I currently do. Your suggestion deserves special weighting in light of this.
The statistics book looks great, exactly what I was after. Maybe I should try searching for '...for engineers and scientists' in the future. I understand why simply reviewing LA and Calc would be enough. However I am surprised that you do not think a book for Discrete Mathematics is necessary. I never studied this subject in my degrees (whereas I did study prob/stats/LA/calc) but it seems to be to be at the heart of a lot of AP problems. Perhaps that is only true for basic material in the field of Discrete Math (counting problems) but the more advanced material is no longer relevant?
Quote: teliotYes, I know that there are copies for sale at outrageous prices. I seriously doubt anyone is paying that price, especially because the majority of the content is available online at 888.
I have to say that I probably would pay the £500 price that is listed on Amazon.co.uk if I did not already own a copy.
Thank you again for the reply and I hope you are enjoying retirement!
Quote: 7crapsThe only one in my collection that comes to mind is not by Mike Shackelford, Frank Scoblete or any other online gambling writers. they are only interested in how to gain the advantage over the casino game one plays or only play for entertainment, with no math, when playing against a house edge.Quote: TreeI would much appreciate if anybody could recommend some textbooks on these topics that teaches them from an applied point of view,
with lots of problems,
and covers the right content for a gambler.
S.N.Ethier comes to mind.
(Stewart N. Ethier)
lots of math applied to casino games with lots of problems and his website has an answer book also. (last time I looked)
"The Doctrine of Chances
Probabilistic Aspects of Gambling"
over 800 pages 2010 version.
http://www.math.utah.edu/~ethier/
good luck to you
This looks like a great book, I had not heard of it before. Thanks for the recommendation.
Quote: teliotThanks for buying several copies!
My publisher came down with ALS and ceased being able to write or type (and more recently she can't talk) -- in particular she could no longer write me quarterly royalty checks. My publisher is Blue Point Books, which is really just one very nice and hard working woman. That, together with my retirement, and I decided to just pull them. I may re-list them, but I would need to get the original files to edit (I don't even know what software she used), a new publishing house and new ISBN's.
That's really too bad. The content of the book is excellent, a must read for anyone attempting to protect these games. Some of the content is on 888, but if I recall correctly, you took a bunch of it down when you released the book. I could be wrong about that though.
Have you considered self-publishing on E-book formats?
I was one of your proofreaders of the very original document. I may be able to come up with it if it was on my personal e-mail, but I am afraid it was sent to my work e-mail at the time, and being that I am no longer there, it is not possible for me to retrieve it.
Quote: teliotThanks for buying several copies!
My publisher came down with ALS and ceased being able to write or type (and more recently she can't talk) -- in particular she could no longer write me quarterly royalty checks. My publisher is Blue Point Books, which is really just one very nice and hard working woman. That, together with my retirement, and I decided to just pull them. I may re-list them, but I would need to get the original files to edit (I don't even know what software she used), a new publishing house and new ISBN's.
That's a bummer - I imagine you must not be hurting for retirement income? If so, it would seem that the nominal work to get the book back out there and published by a different publisher would be worth it, since it's clearly in demand.
But I know nothing about authoring and publishing and the potential income stream vs work requried. Absolutely, if you are happy in retirement and don't want to do whatever work would be required, that's your prerogative! I got my copy, and won't be selling it any time soon. Of course, would help to actually read it...
Thank you! Glad to help. I'm always on the side of anyone who wants to learn.Quote: TreeThank you for the response. I want to take this opportunity to say that I have great admiration for the quality and rigour of your work, I bought Advanced and had a great time reading it even though it isn't applicable to the type of AP that I currently do. Your suggestion deserves special weighting in light of this.
The statistics book looks great, exactly what I was after. Maybe I should try searching for '...for engineers and scientists' in the future. I understand why simply reviewing LA and Calc would be enough. However I am surprised that you do not think a book for Discrete Mathematics is necessary. I never studied this subject in my degrees (whereas I did study prob/stats/LA/calc) but it seems to be to be at the heart of a lot of AP problems. Perhaps that is only true for basic material in the field of Discrete Math (counting problems) but the more advanced material is no longer relevant?
I have to say that I probably would pay the £500 price that is listed on Amazon.co.uk if I did not already own a copy.
Thank you again for the reply and I hope you are enjoying retirement!
As per "discrete math", if you look through the table of contents of a book on the topic you will find some combinatorics, but that is already covered by your probability/stats book. There is a fair amount of set theory and logic, some recurrence relations stuff (i.e. Fibonacci sequences), I taught this course at UCSB for our C.S. majors. The purpose of the course was to teach the kids everything they needed to know as undergraduates about "finite" mathematics in a single course.
My wife is going to see her today (she is the durable power of attorney for health care) -- with a portable hard drive in hand. I am going to get all the files. Apparently they are in some old version of Quark Express. This way I will preserve my options.Quote: FCBLComishThat's really too bad. The content of the book is excellent, a must read for anyone attempting to protect these games. Some of the content is on 888, but if I recall correctly, you took a bunch of it down when you released the book. I could be wrong about that though.
Have you considered self-publishing on E-book formats?
I was one of your proofreaders of the very original document. I may be able to come up with it if it was on my personal e-mail, but I am afraid it was sent to my work e-mail at the time, and being that I am no longer there, it is not possible for me to retrieve it.
As for 888, I had over 300 articles on APHeat when I shut it down. I think I sold about 170 of them to 888 and took the rest down. Not sure what the exact overlap with AAP is.
As for an E-book, they don't work well with large tables of data. I've tried.
Quote: teliotThank you! Glad to help. I'm always on the side of anyone who wants to learn.
As per "discrete math", if you look through the table of contents of a book on the topic you will find some combinatorics, but that is already covered by your probability/stats book. There is a fair amount of set theory and logic, some recurrence relations stuff (i.e. Fibonacci sequences), I taught this course at UCSB for our C.S. majors. The purpose of the course was to teach the kids everything they needed to know as undergraduates about "finite" mathematics in a single course.
Thank you. I take it that you don't think any of this (apart from the combinatorics stuff which as you said would be covered anyway) is really worth putting dedicated effort into learning? I don't see it really coming up in AP material.
Inclusion/exclusion sometimes is used in game analysis, that topic would be in Discrete Math and may not be covered in a stats book. Aside from that topic, I can't think of anything else such a book might uniquely contain that would be useful to learn.Quote: TreeThank you. I take it that you don't think any of this (apart from the combinatorics stuff which as you said would be covered anyway) is really worth putting dedicated effort into learning? I don't see it really coming up in AP material.
Quote: teliotInclusion/exclusion sometimes is used in game analysis, that topic would be in Discrete Math and may not be covered in a stats book. Aside from that topic, I can't think of anything else such a book might uniquely contain that would be useful to learn.
Thanks for all the help, I wish you and yours a merry christmas/happy holidays!
Quote: beachbumbabsUmmm...Elliot...
Have YOU looked at the amazon listings I posted above?
They're (third parties) scalping your books, listing from 300-1500 for Contemporary and 500-5000 for Advanced.
You should be making that money.
0_0
If anyone wants to trade a me copy of Beyond Counting for Advanced Advantage Play shoot me a PM ;)
Quote: teliotFor example, the "Normal Distribution" is continuous and the area under it is an integral that involves exponential functions. Many questions about the long-term behavior of games require at least this much. Understanding the Central Limit Theorem is huge. So I would say all types of casino games require calculus to evaluate.
Teliot knows a lot more math than I do, but I think discrete math is making a comeback. The Normal Distribution was invented as an approximation to the Binomial Distribution and works best when p is near 1/2. If p is very small or very large the approximation is not so good. In the days of mainframes there was a book published Tables of the Cumulative Binomial Distribution for Small Values of p.
It is easy to write a program to calculate the Cumulative Binomial Distribution and it will run very fast on today's home computers. You can even get results online and the calculator won't balk at a large number of tries.
(I tried to include a link to the Stattrek site on the word "online" but the site editor wouldn't allow it.)
Textbooks are slow to catch up, however, and they still use the Normal Distribution quite a bit.
The Wizard recently posted an excellent problem "The Three Waitresses Problem" and there was a discussion of using a triple integral to solve it. MY GOD! Triple integrals are used to model situations in three dimensions. What if there were four or five waitresses? I have never seen an integral of order greater than three in print. A reader submitted a short elegant solution using only discrete math.
The trouble with calculus is that it is based on shaky logic. This was much debated at the time it was invented but is not mentioned in today's textbooks. I think it should be. I didn't begin to understand it until I read Sir Isaac Newton's The Method of Fluxions and Infinite Series (1736).
