A New Blog on the Mathematics of Blackjack
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Quote:I've recently published my first Blackjack blog on Wordpress. I've taken a mathematical approach to the subject and the blog itself is a conversion from a 10000+ word dissertation that I completed as a school project. It includes a factual introduction to basic strategy, the theory and purpose of the statistical processes, step by step derivation of the decision equation and, a comprehensive evaluation of the spreadsheet showing a worked example. I have written additional pages describing history of Blackjack, card counting theory e.t.c and I intend to expand this content in the future. The way that my project will be marked will mean that any incorrect information or calculation errors will not be picked up on so I decided to convert it into blog format so that the element of audience interaction would enable me to continue to expand and improve on what I have produced. If it is possible, I would love an expert's opinion on my work.
If the gaming mathematicians don't mind, especially those with some expertise in blackjack analysis, could you have a look and give your expert's opinions?
https://magiciansluck.wordpress.com/
From what I can tell from the blog, this person is an 18 year old student living in the U.K. In my opinion, she has a first rate gaming-math mind and I am hopeful that she can get some good feedback here. I recommended she join WoV to be able to engage in this discussion.
Thanks!
[Mods - link posted with prior approval.]
For example, on the "deriving the decision equation" page, it says:
Quote:The dealer is bust and the player does not. This outcome may be written as P(T>21). P(x< 21) however, since P(x< 21) =1 this reduces down just to P(T>21).
It should be (emphasis added to show the change):
Quote:The dealer is bust and the player does not. This outcome may be written as P(T>21)·P(x< 21) however, since P(x< 21) =1 this reduces down just to P(T>21).
sweet and HiQuote: alexismorgan49That's me, thanks Eliot!
now
what are you trying to add to the blackjack knowledge base that has not already been done B4?
(without looking at your Blog)
are you just trying to verify the work of others that were B4 you?
this is Ok too, btw
I mean, Dr Thorp and Don Schlesinger did lots of stuff for the game as did many others
(and they are even now very handsome men, imo)
and why you and now, except from what you mention in your blog?
good luck to you
Sally
Pretty much what will happen, LOL. Congrats on the ability to count. Hey, nudge, the same count can benefit the dealer, LOL!
I had the option at school to complete and extended project this year for extra credit. I decided to write about Blackjack because it was something that I had some knowledge about and consistent interest - I wanted to learn more about it and still do. Originally I was just going to write an informative and instructional piece on Blackjack strategies but during my research I came across a JASA pdf file on Basic Strategy. I'm currently studying physics and double maths so I was curious about what maths was involved and whether or not I would be able to comprehend it.
I decided on a whim to analyse the paper and thought I could derive the equation myself so I gave it a go. The purpose of my work on the optimum strategy could be to explain the derivation in a way that could be understood by someone who is interested in the processes but with a less advanced level of mathematics. After this, I wanted to explain how the formula could be used to find the minimum standing values but the method described in the pdf file was very brief. I had little idea of how Baldwin's team actually produced their calculations so I tried to reproduce some of their values quoted in the paper with my own method using Microsoft Excel just to see if it could be done. Surprisingly I wasn't a million miles from some of the figures they quoted in the paper but I couldn't actually check any of my results because I have been unable to find any references that contain raw data from Baldwin or Thorp.
Since converting my project to blog format, I am hoping that the opportunity for interaction with like-minded individuals might enable me to improve my methodology and locate the source of error in my work to which I can currently only speculate.
Surprisingly, I enjoyed producing this project and want to continue analysing other gambling strategies (taking a similar approach) in the future.
Apologies for the bad English.
Quote: countblackjackhttps://www.youtube.com/watch?v=LgD8rkIxj7o
Pretty much what will happen, LOL. Congrats on the ability to count. Hey, nudge, the same count can benefit the dealer, LOL!
Interesting, this will happen to me? Your concern for my mental welfare is touching indeed, but I'm afraid that I'm not old enough to gamble in US casinos let alone become addicted to gambling.
Now i really wish you good luck there.Quote: alexismorgan49I'm currently studying physics and double maths so I was curious about what maths was involved and whether or not I would be able to comprehend it.
I had
to take math (i had no choice)
to go with my Biology studies.
I married a math (and video game) genius (and a cute guy) but they (the school - UCI) did not want to accept that.
what? that is funnyQuote: alexismorgan49Apologies for the bad English.
UK English or US English (yes, i like Aussie English too)
Quote: harvson3Learn LaTex (use a nice GUI to start out) instead of bothering with Word Equations Editor. It'll pay off if you go into a field that requires technical writing.
Okay, thanks I'll check it out
UK English or US English (yes, i like Aussie English too)
UK English I guess haha I have poor sentence structure - writing is hard
Overall, on the blog, it looks like there's the framework for a lot of information to be put up there (i.e. an examination and discussion of the various blackjack variants, including more modern ones) and the opportunity for a little more research on your part. For instance, if you actually research Spanish 21, you will see that it's low house edge (0.44% with H17 and redouble) actually makes it one of the best blackjack games left on the planet.
First, this spreadsheet does not take into account the composition of the player's hand. I don't know if that was your intent, though. If you did intend to take into account the player's hand, then you would need to adjust the probability of the dealer getting any card that appears in the player's hand.
Second, a handful of calculations are incorrect in column K, due to the denominator. If you draw one card, you expect to see 1/51, two cards 1/51*1/50, three cards 1/51*1/50*1/49, etc. I know you understand this from reading the blog, but still some formulas have errors. For instance, cell K35 has denominators of 52, 51, and 50. Another example, in cell K298, there are six cards drawn, so the denominators should be 51, 50, 49, 48, 47, 46.
Quote: CRMousseauI will agree that the formatting makes the blog look difficult, but I think that with the formatting cleaned up a bit, that you will definitely have a useful internet resource for the "hard math" on the game. In particular, my eyes lit up a bit when I saw all those inequalities from that famous first paper on Basic Strategy, and having someone that worked through the derivation of the equation is certainly of much interest.
Overall, on the blog, it looks like there's the framework for a lot of information to be put up there (i.e. an examination and discussion of the various blackjack variants, including more modern ones) and the opportunity for a little more research on your part. For instance, if you actually research Spanish 21, you will see that it's low house edge (0.44% with H17 and redouble) actually makes it one of the best blackjack games left on the planet.
Thanks, the maths content was really fairly simple in the end I just felt that the difficulty came in requiring a lot of lateral thinking in order to cancel expressions. Initially the expectations worried me but upon examination I understood how they worked and that was really all that I needed in the end. I really felt that the optimum strategy paper was over-brief especially in the sense of explanations. There was one instance where they quoted two seemingly different lines of formula that were linked in-between by the sentence 'subtracting off Es,x and and performing some simple algebraic manipulation' that actually took a couple of sides of working before I could come up with the same result.
Really? wow, I never knew that! I was quoting Blackbelt In Blackjack but I've really not done much research into game variations and the effect of side rules yet. Its definitely a topic I'd want to study in further detail in the future, thanks for the feedback.
Quote: CrystalMathI haven't looked through all of what you have, but I did take a peek at the Basic Strategy Final Spreadsheet D10 x19, and I have some comments for you.
First, this spreadsheet does not take into account the composition of the player's hand. I don't know if that was your intent, though. If you did intend to take into account the player's hand, then you would need to adjust the probability of the dealer getting any card that appears in the player's hand.
Second, a handful of calculations are incorrect in column K, due to the denominator. If you draw one card, you expect to see 1/51, two cards 1/51*1/50, three cards 1/51*1/50*1/49, etc. I know you understand this from reading the blog, but still some formulas have errors. For instance, cell K35 has denominators of 52, 51, and 50. Another example, in cell K298, there are six cards drawn, so the denominators should be 51, 50, 49, 48, 47, 46.
Its hard to explain but the fact that the denominators of my respective probability calculations do not reduce further that 50 is intentional and this is supposed to be a way of taking the cards the player draws into account.
When I was experimenting with my second draft, my first thought was to decrease the denominators as you said and I wasn't sure if they just ignored the player's affect on the deck either - I did actually try using decreasing denominators (I think I still have a version of this spreadsheet if you would like to have a look, but my column totals were still pretty far out.) When I consulted the optimum strategy paper, this is what I found:
Basically, I interpreted this to meant that their game rules make the assumption that the first three cards are always dealt to the dealer, and this allows us to calculate exact probabilities for the first three cards dealt - 'three card probabilities' e.g. like you said a/51*b/50, but after this, cards are dealt to players with each remaining card having equal probability of being dealt to them. I think that, because it would be too difficult to take into account each possible outcome of cards being drawn and also the possible number of players from 1 to 6, they decided to use a method of 'sampling with replacement' to accommodate this. I'm not sure why they used 1/52 - surely they should use multiples of 1/49 and just keep multiplying like: a/51*b/50*c/49*d/49*e/49*f/49 e.t.c but the paper stated they used 1/52 for some reason - I might post a version using the equiprobability denominator of 49 for hands of 4+cards to see how it compares but I tried to follow Baldwin's method as closely as possible in order to reproduce his column totals. Because of the use of equiprobability, my J values were also calculated using multiples of 1/52 for each additional card drawn and I'm also assuming my initial total/ x value of 19 was drawn after the dealers first three cards i.e. this would also be composed of multiples of 1/52 making it independant in the case of respective probabilities for the outcomes of D, but since this value is outlined for a specific situation when x is 19 then then P(x=19) is just 1 anyway so we wouldnt have to consider it.
I did find this concept confusing (still do) and my interpretation could be completely wrong, but I found that my results were closest to those quoted in the JASA article when I used this method. :)
When I analyze blackjack games, I use a recursive routine that will choose the optimal decision for each unique hand. To speed things up, I save the optimal ev and player action for each unique hand. That way, I don't need to calculate the same hand twice. For instance, if the player hand has a 2 and a 5 and then they hit and got a 6, this hand will be identical to having a 6 and 2 then drawing a 5. Once I calculate the first one, I don't need to calculate the second one. This is similar to the method you investigated, except it uses exact probabilities instead of infinite deck estimates.
Quote: CrystalMathI see. They were simplifying the calculations after the first three cards. Of course, with modern computers, we can calculate the exact probabilities.
When I analyze blackjack games, I use a recursive routine that will choose the optimal decision for each unique hand. To speed things up, I save the optimal ev and player action for each unique hand. That way, I don't need to calculate the same hand twice. For instance, if the player hand has a 2 and a 5 and then they hit and got a 6, this hand will be identical to having a 6 and 2 then drawing a 5. Once I calculate the first one, I don't need to calculate the second one. This is similar to the method you investigated, except it uses exact probabilities instead of infinite deck estimates.
so, by calculating the exact probablilities, does this take a long time? would I be able to run simulations on excel or would this be inefficient ?
Quote: beachbumbabsWelcome to the forum, alexis! Really nice to see your interest on the topic.
Thanks, I'm enjoying the discussiong - it really helps!
If you want to enhance your understanding of the BJ math, I suggest reading The Theory of Blackjacj by Peter Grifin.
Even though this book was writen ages ago, it is the book that formalised the Math behind BJ and is primarily about Math. Almost all other BJ books are primarily about how playing BJ and learning to Count.
Especially about the concept of Effect of Removal (EOR) which all counting systems are based on. In the book there are formulas of how EOR, Betting Correlation etc are calculated and mathematical proofs for all these.
Regarding calculating Basic strategy, I understand that you used Infinite deck approximations to calculate them in Excel.
In Excel you can also use Visual Basic macros to write a recursive algoritm to calculate exactly Basic strategy. The most difficult part of such algoritm are the Split decisions.
