I. History or How the Blackjack Odds Were Calculated — John Scarne
This book is about the true mathematics of blackjack, insofar as precise probability calculations are concerned. You might be shocked, but the mathematical truth is that your knowledge of blackjack probabilities or odds is wrong. Everything you had known was based on guesswork, albeit it educated guesswork.
There is a chance I'll make available the new, accurate software, plus its source code…for a fee, of course! The software generates every possible blackjack hand, based on the BJ Dealer's rule (draw to 16, stand on 17). The algorithms do not skip any configuration, such as four consecutive Aces. As rare as such a configuration might be, it must be taken into consideration as it is an element of the total number of possibilities. The probability is founded on number of favorable cases and number of total possible cases.
To this date, the blackjack odds are the same as John Scarne calculated in the 1950's. The computers were not the commodity they are today. And John Scarne was NOT a computer programmer! A far cry from that! The way he calculated the odds made sense for the first two and three blackjack cards in a round. I quote from his "Scarne's New Complete Guide to Gambling" (pg. 363):
"We find that the dealer's first two cards can produce the counts from 2 to 21 in 1,326 ways."
Indeed, Combinations C (52, 2) = 1,326 two-card blackjack hands (combinations of 52 cards taken 2 at a time). That is the only thing … half way mathematically correct! The truly correct method applies the mathematics of combinatorics alright. But instead of the numerical sets known as combinations, we must apply the mathematics of arrangements. The combinations represent boxed arrangements. In this case, C (52, 2) = (52 * 51) / 2 = 1,326. Arrangements A (52, 2) = (52 * 51) = 2,652 — or double the amount of combinations. Hence, we played cute and said half-true for the blackjack combinations case!
"We'll discover that we need to know, however, and avoid most of the fractions, if we multiply 1,326 x 169 to get a common multiple of 224,094."
Now, that's a big mystery! How did he come up with that 169 factor??? Well, that's what they call an educated guess, or guestimation! John Scarne, requiescat in pace, didn't have a clue, mathematically speaking. He does not explain how he came up with that 169, kind of a new number of the beast! Yes, some notified me that 169 is 13 to the power of 2! Indeed, there are 13 cards in the blackjack deck, from 2 to Ace, counting only one suit. But how do you incorporate 13 ^ 2 in the house edge calculations?!
II. True Mathematics, Combinatorics, Probability Applied to Blackjack
In order to calculate the probability PRECISELY, we must generate all the elements (blackjack hands) in LEXICOGRAPHICAL ORDER! Nobody even knows how many hands are possible, as their size varies widely: From two cards to 10 cards (for one deck)! When two or more decks are employed, the blackjack hands can go from two cards to 11 cards.
` Of course, there is a lot of blackjack software out there! But all that software belongs to the SIMULATOR category! That is, the blackjack hands are dealt RANDOMLY! Based on the well-known-by-now Ion Saliu's Paradox, random generation does NOT generate all possible combinations, as some elements repeat!!! So, we can never calculate the probability precisely based on random generation! If there are 334,490,044 total possible complete hands in blackjack, only 63% will be unique and 37% will be repeats — IF we randomly generate 334,490,044 hands.
I rolled up my sleeves again. I had started years ago a blackjack project to generate all possible hands. It was very difficult. I put it aside and forgot about it, as other projects felt more compelling to me. I found the project and also the code to generate sets from a list. In this case, the list is 52-line text file with the values of the blackjack cards, from the four 2's to the 16 Tens, to the four Aces. That's a stringent mathematical requirement. The deck of cards must be also ordered lexicographically, if we want to correctly generate all qualified sets in lexicographical order.
Let's make this analogy to lotto combinations. Generating lotto combinations in lexicographic order is far easier than generating blackjack hands. The lotto combinations have a fixed length. For example: A lotto 6/49 game consists of 6 numbers per combination, from 1 to 49. There are 13983816 total possible elements for this type of lotto game.
Incidentally, I released my lotto software in 1988 (Wow! How time flies! Two decades ago!) At that time, I was the only one who knew how to generate lotto combinations in lexicographical order. A few others were able to generate lotto combinations only in random manner. It took other programmers several years before they discovered the trick of lexicographical lotto generation. History always keeps accurate and unbiased records!
The lotto combination generation in lexicographical order is very simple today.
FOR A = 1 TO 49 FOR B = A+1 TO 49 … FOR F = E+1 TO 49
The lotto generation in lexicographical order as arrangements is also very simple … to me!
FOR A = 1 TO 49 FOR B = 1 TO 49 IF B = A THEN GOTO 222 … FOR F = 1 TO 49 IF F = E OR F = D OR F = C OR F = B OR F = A THEN GOTO 666 … 666 NEXT F … 222 NEXT B 111 NEXT A
Blackjack lexicographic hands can be only generated from a list, such as a deck of cards. Lotto combinations can be generated from a list as well. The list, however, must be ordered lexicographically; e.g from 1 to 49, 49 elements, one element per line. Blackjack hand generating face a tremendous obstacle of its own. The number of elements per combination (or per arrangement) varies widely: From two cards to 10 or 11 cards per hand. That was the challenge that kept gambling programmers paralyzed up until now. It is so much easier to say simulation! That is, generate blackjack hands randomly! Many kids can do that, like many adults. But I reserve the right to doubt everybody's accuracy in generating blackjack hands! That's me! I've seen so much falsity out there, including pathological lying and deception. Card counting is the most blatant example.
The year of grace 2009 has proven to be augural to me. For I was able to successfully finalize the software to generate ALL POSSIBLE BLACKJACK HANDS IN LEXICOGRAPHICAL ORDER. It is not easy and it wasn't easy, ever. I went through various methods and algorithms. Verification is the hardest part, as there are NO mathematical formulas regarding blackjack combinatorics and probabilities.
I played blackjack in Detroit in the year of grace 2009. I visited with my daughter after many painful years of separation. She is an adult now, although they always check her ID before entering a casino. The first time, in June 2009, I and she too were shocked how easy it is to lose playing those CSM blackjack tables. CSM stands for continuous shuffling machines. I make the same statement I made about the blackjack slot machines. They are deceptive devices: The devices are programmed to beat the player far worse than the situation of games with normal blackjack shoes. I stand by my statement even if corrupt judges might be attracted by the smell of my blood (I always wear nice deodorants!)
I visited my daughter and Detroit again in August. We only checked the so-called high-limit gambling room. The maximum-to-minimum ratio of the table limits was worse than on the casino floor. We also did an experiment playing slot blackjack. I left the area after a couple of hours by winning just one dollar! Nobody is supposed to win the blackjack slots! Nobody can beat the devilishly programmed chips! My strongest impression was related to betrayal of the basic strategy (BS it is!) on some occasions. My daughter almost always advised me to stand on 15 or 16 (even 14) against dealer's high cards. That made the difference against the masked bandit inside the programmed chip.
III. Breakthrough Software to Generate All Blackjack Hands and Calculate the Odds Accurately
As I said, the hardest part of my blackjack software development was the verification phase. I generated blackjack hands as combinations or arrangements. Then, I opened the output files (text format) and checked as many hands as possible. Yes, computing things are so much better today than just 5 years ago. The generating process is significantly faster. Also, opening large files is much easier today. My text editor of choice is my own MDIEditor And Lotto WE. It opens reasonably fast text files of several megabytes in size. The editor also uses a fixed-width font, which makes reading blackjack hands easier.
I started by creating one deck of cards. That is, write a simple text file with 52 lines, consisting of one card (number) per line, from the four 2's to the four Aces (written as 11). The file name is BjDeck1.TXT and is absolutely free to download. From that pivotal layout file (deck of cards), I deleted 4 of the 10's to get the deck used in the Double Attack Blackjack game. The new text file has only 48 entries (12 Ten-value cards, instead of 16). The file name is BjDeck2.TXT and is absolutely free to download. The output files for the arrangements generation are gigantic! Thus I needed to create smaller deck files. The reduced deck files helped me tremendously with the verification process. One deck, BjDeck1-11.TXT had only one suit of the 13 cards (one 2, one 3, …, to 4 Tens, one Ace). The fourth deck, BjDeck4-11.TXT had only one suit of the 13 cards (one 2, one 3, …, to 4 Tens), except for Aces. All four suites of the Ace are in that layout file (16 entries in the file).
I created two more deck layout files (also completely free to download). One file was the result of shuffling (randomizing) the regular 52-line BjDeck1.TXT: BJDeck1Shuf.TXT. The other file was the result of reversing the order in the deck. The composition is from the four Aces to the four deuces: BjDeck1Rev.TXT. These two BJ deck files prove that the arrangements method of generating hands is the most precise. All three layout files generate the same amount of hands and the same bust percentage: BjDeck1, BjDeck1Shuf, BjDeck1Rev. By contrast, the combinations method of generating hands leads to three different result files.
1) The name of the first program is BjDealerOrder.EXE. It generates all possible blackjack hands as arrangements. That is, the order of the cards is of the essence. Total number of qualified (completed) BJ hands is staggering for all one-deck regular files: 334,490,044. As of the Double Attack Blackjack game: 302,394,480 total possible completed hands. I only calculated the hands, but did not generate them. I simply commented out the statements for printing to an output file. I did generate, however, the arrangements for the BjDeck1-11.TXT and BjDeck4-11.TXT source files of incomplete decks uv cards. The output files are also available absolutely freely as downloads: BjAllHands1-11Ord.TXT and BjAllHands4-11Ord.TXT.
Those two output files for incomplete decks played a role of biblical importance, as it were. They helped me discover the most subtle errors of generating blackjack hands of variable size (length). Generating hard-count hands was as easy as a breeze. The big problem came from the Aces, as they are counted either 1 or 11. Blackjack hands such as 2+A+2+A must be hit by the dealer; hands such as 2+A+3+A are mandatory stand. I perfected my blackjack probability software due in great part to those two output files. One can easily check the error-free files in MDIEditor And Lotto WE. I have checked several times: Each and every hand is absolutely correct and no hand is missing. Of course, we follow the rules for the blackjack casino dealer. Yes, some casinos rule now that hit soft 17 (e.g. A+6) is mandatory for the dealer. First off, that's a bad rule for the blackjack player; avoid such tables at all costs. Secondly, my software can be easily adapted to accommodate the hit soft 17 rule.
2) The name of the second program is BjDealerCombos.EXE. It generates all possible blackjack hands as combinations. That is, the order of the cards is not important. Hands such as 2+A+2+A or A+A+2+2 are always written as 2+2+A+A. Total number of qualified (completed) BJ hands is far-cry lower for all one-deck regular files: 297,615 (for 52 cards). As of the Double Attack Blackjack game: 257,877 total possible completed hands. I calculated the hands and also generated them. The output files are not available absolutely freely as downloads. They require the reasonable Permanent Membership: BjAllHands1Combos.TXT and BjAllHands2Combos.TXT.
IV. History in the Making — Shocking But True New Odds of Blackjack Dealer Bust
NO LONGER APPLICABLE AD LITERAM! You, Player, try to avoid busting, first and foremost! Let them Dealers bust more than You, especially when FEW or NO Players before You and Dealer busted! Mathematical tip: If a majority of Players before You busted, expect a lower bust DC (degree of certainty) for Dealer — You gotta play more aggressively!
And now, the shocking results! If you go all the way down to the bottom of BjAllHands1Combos.TXT, you see that the bust percentage is 41.97%. WOW! We all believed John Scarne and his "biblical" figure of 28%! Keep this new figure in mind: The odds for a blackjack dealer's bust are at least 41.97%. The bust probability is calculated by dividing the number of dealer's busted hands to the total possible blackjack actions. Blackjack actions is a parameter that counts everything: Busted hands, pat hands (17 to 21), blackjack hands, and draws or hits (incomplete hands). The software does NOT print the incomplete hands.
The best-case scenario regarding dealer's bust probabilities for the game of blackjack reads:
Total BJ Actions: 476246 Total Draws (Hits): 178631 Total Non-Bust Hands: 97735 Total Dealer BUSTS: 199880 (41.97% = 199880/476246) Total Complete BJ Hands: 297615 Total BlackJacks (10+A): 64 (4.83%)
The worst-case scenario is represented by the arrangements method of generating blackjack hands. The worst-case scenario regarding dealer's bust probabilities for the game of blackjack reads:
Total BJ Actions: 345948532 Total Draws (Hits): 11458488 Total Non-Bust Hands: 110020276 Total Dealer BUSTS: 224469768 (64.89%) Total Complete BJ Hands: 334490044 Total BlackJacks (10+A): 128 (4.83%)
The bullboochah subspecies will bullbookate their eyes in anger and hatred. They would fight vociferously against this very bible. They would not accept these truly mathematical figures for a number of reasons. For one, the angry beasts are strongly involved in the business of card counting training for the game of blackjack. Some sell packages for upwards of dozens of thousands of bucks. There are also the casinos. It is in the financially selfish interest of the casinos to advocate the application of blackjack basic strategy (BS) and, especially, card counting (BCC). The game of blackjack is highly profitable for the casinos mostly because of the BS and BCC.
And then there is that sub-human group of the afflicted. You surely know of UFO afflicts who are medically afflicted by irrationally believing in UFOs. You also know of religious afflicts who are medically afflicted by irrationally believing in gods (like "God speaks to me.") The same is true about the milder and far smaller group of card-counting-afflicted blackjack players. Some of them are medically afflicted by irrationally believing in the true effectiveness of card counting. I can only feel for such individuals, humans (Homo sapiens). But I am truthfully harsh against them pathologically liars who use lying and deception for financial gains.
Pathologically-deceiving liars who make statements on TV that they were abducted by aliens and tortured inside UFOs — and thus trying to cash in. Pathologically-deceiving liars who broadcast TV religious programs, shamelessly claiming that God speaks to them constantly — and thus trying to cash in. Pathologically-deceiving liars who shamelessly claim that they make big money by counting cards at blackjack — and thus trying to cash in. But none of those authors, developers, vendors of card counting systems and training materials apply their systems at the blackjack table — ever! Virtually none of them plays blackjack at all! The latter three groups do it knowingly — hence the pathological attribute I mercilessly but truthfully attached to their characters.
There will be also probability purists who will fight the blackjack figures my new blackjack software reveals. They will argue against a number of blackjack hands generated by software: 2 2 2 2 3 3 3 or 3 3 3 2 2 2 2 or 2 2 2 2 3 3 11 11 11, or 2 2 2 2 3 11 11 11 11 3, etc. Such hands will never come out, they scream! But, hey, who decides what hands come out and what hands will never come out? Is there a god of blackjack who makes such decisions? NOT!
The purist argument resembles the older heated debate regarding the lotto combination 1 2 3 4 5 6. Indeed, lotto combination 1 2 3 4 5 6 does come out so rarely that it has not been drawn in our lifetime! The standard deviation plays a hostility-causing crucial role. The same should be true about another random phenomenon as the game of blackjack. There are some issues here that we must address. Calculating the standard deviation for lotto is as easy as it can be (just use my SUMS.EXE standard deviation software). It is extremely hard to calculate standard deviation for blackjack output files. Think of those huge 10 GB files! We must have a 64-bit operating system and 64-bit compilers to create adequate software to handle that size. Here is the most important issue. We know exactly how to calculate the probability of any lotto combination. Par exemple, we can generate all 13983816 lotto 6/49 combinations in lexicographic order and see exactly one 1-2-3-4-5-6 combination. We must do the same thing in blackjack: Calculate the probability precisely p = n / N.
V. Conclusions Regarding the Greatest Breakthrough in Blackjack History — New Strategy
I got my conclusions, of course. And I do have a new blackjack strategy. For starters, figure out the BJ dealer bust probabilities this way. 41.97% bust chance for the first hand at the table. Getting a second bust hand at the table is .4197 ^ 2 = 17.6%. This is the new and most accurate figure representing the blackjack house edge or advantage. According to John Scarne, the biblical figure accepted today regarding BJ house edge is .28 * .28 = 7.8%. The 10% difference in house advantage is significant. That's the reason why the generic blackjack player loses faster and more than the generic roulette player. The so-called gambling gurus or experts will never admit that to that … truth!
How about three busted hands in the same round? .4197 ^ 3 = 7.4%. Four busted hands? 41.97% ^ 4 = 3.1%... And so on… Granted, the busted hands may not be exactly in consecutive fashion. We will be in a probability situation known as the odds of M successes in N trials. You can perform such calculations easily with my free mathematical software SuperFormula.Exe. But please be always mindful that the blackjack players get their bust hands before the dealer; the busted players will lose their bets immediately, before the dealer even plays her hand!
See how well placed the blackjack dealer is? I know … I may not place myself as well as the BJ dealer. I can only place myself as well as the next position before the casino dealer. I don't know why they call it the third base ... it should be called the premier base, for it is the real premier position at the BJ table for a player! How many players busted before me? The more of them busted, the happier I would be! I might play more aggressively the hit/stand situations… and vice versa…
It took me a while to reach this moment in my activity. I am highly conscious of validity and validation. There is no doubt in my mind now that this new blackjack theory of mine is valid. I verified many times. I finally decided that my theory was validated beyond reasonable doubt. I made several verification/validation files available for free to everybody. The registered members of my website have access to additional files that validate my algorithms and theory. How about the software?
VI. Additional Resources in Blackjack
Blackjack: Software, Content, Resources, Systems, Basic Strategy and Card Counting.
See a comprehensive directory of the pages and materials on the subject of blackjack, baccarat, software, systems, and basic strategy.
Blackjack: Basic Strategy, Card Counting; Charts & Tables, Probability, Odds, Software.
Remember to check the specific software downloads site: