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 them in the 1950s. The computers were not the commodity they are today. And John Scarne was not a computer programmer! 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):
Indeed, Combinations C (52, 2) = (52 * 51) / 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!
Now, that's a big mystery! How did Scarne come up with that 169 factor??? Well, that's what they call an educated guess, or guesstimation! John Scarne didn't have a clue, mathematically speaking. He never explained how he came up with that 169, kind of a new number of the beast! ((Some people emailed me with the explanation that 169 = 13 ^ 2. But that relation has absolutely no mathematical rationale in Scarne's calculations… except for the 13 cards of one suit in a blackjack deck!)
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!
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 SIMULATION 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. 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.
The lotto generation in lexicographical order as arrangements is also very simple... to me!
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!
Random-generating (or so-called simulation) does not satisfy the condition of accuracy. The undeniable Ion Saliu's Paradox proves that only around 63% of all possible sets will be generated. 63.2% is a mathematical limit that does apply to blackjack as well. In roulette, it is around 65%. In 38 American roulette spins, only 24-25 numbers will be unique.
The year of grace 2009 has proven to be augural to me. 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.
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 BjBustOddsOrder.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 of cards. The output files are also available absolutely freely as downloads: BjAllHands-1-11-16.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 WEor Notepad++ (potent free editor). 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 BjBustOddsCombos.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. BjAllHands1Combos.TXT and BjAllHands2Combos.TXT.
If you click on a text file, it opens directly in your browser; if you right-click, you can choose to download (Save as) to your computer.
NO LONGER APPLICABLE AD LITERAM! You, Player, try to avoid busting! 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!
If you go all the way down to the bottom of BjAllHands1Combos.TXT, you see that the bust percentage is 33.55% (as arrangements) or 33.61% (as combinations). Keep this new figure in mind: The odds for a blackjack Dealer's bust are at least 33%. 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 to the first 2-card hands(incomplete hands). The software does NOT print the incomplete hands.
The combinations scenario regarding Dealer's bust probabilities for the game of blackjack reads:
Total BJ Actions: 594768 Hits to 1st 2-Cards: 297153 Total Non-Bust Hands: 97735 Total Dealer Bust Hands (*): 199880 Percentage Dealer Bust: 199880 / 594768 = 33.61% Natural Blackjacks (10+A): 64 / 1326 = 4.83% Total Complete BJ Hands: 297615
The arrangements scenario regarding Dealer's bust probabilities for the game of blackjack reads:
Total BJ Actions: 668979164 Hits to 1st 2-Cards: 334489120 Total Non-Bust Hands: 110020276 Total Dealer Bust Hands (*): 224469768 Percentage Dealer Bust: 224469768 / 668979164 = 33.55% Natural Blackjacks (10+A): 128 / 2652 = 4.83% Total Complete BJ Hands: 334490044
How can we apply the new programming to determine the bust odds for the blackjack Player? After heated debates in forums in 2014, I simply modified my software. The hit-stand limits can be set by the user. Initially, it was fixed — the ubiquitous hit all 16s and under, stand on all 17s or greater.
The software user can set the hit-limit to any value. The choices are, obviously, from 12 to 16. I tried, for example, the hit limit to 11 — that is, hit anything 11 or under, stand on anything 12 or higher. Evidently, there is no bust in such situations. That's another proof that my programming is 100% correct.
I believe that setting the hit limit to 14 or 13 reflects pretty closely the bust odds for the Player. That is, stand on 15 or greater (as arrangements):
Percentage Player Bust: 61656 / 274254 = 22.48%
Or, stand on 14 or greater (as arrangements):
Percentage Player Bust: 570 / 3702 = 15.40%
Now, the house edge goes between something like .3355 * .2248 = 8.3% and something like .3355 * .1978 = 6.6%. It averages out to 7.5%. It is a far cry from the intentionally false house advantage (HA) of 1%, or even .5%! The overwhelming majority of blackjack players lose their bankrolls quickly, because this is NOT a 50-50 game or so much close to that margin! And always be mindful that blackjack is strongly sequential: The Dealer always plays the last hand. Otherwise, the casinos would go bankrupt!
Using various partial decks, I noticed that the bust odds grow with an increase in the number of cards, therefore the bust odds grow with an increase in the number of decks of cards. That is valid for both Dealer and Player. It also means that the house edge also increases (better for the casinos, worse for the gamblers).
Just about everything I write about generates strong reaction. I do not talk about the positives regarding my ideas — they are clearly in the majority. I am referring here only to the negative reactions.
There is honest criticism, especially rooted in less knowledge on the subject. There are also objections in the manner of common sense. There are probability purists who will fight the blackjack figures my new blackjack software reveals. They will argue against a number of blackjack hands generated by the BJ odds programs: 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 as p = n / N or Favorable_cases / Total_cases. There is no other way in mathematics - only mathematics counts here!
The most vociferous reaction against my ideas comes from a minority who deeply hates me, no matter what I do and what I say. Most of them are jealous and resentful authors of gambling systems. There are also casino representatives (executives, agents, or moles as I also call them). The casinos have a vested interest in aggressively fighting my gambling theories and systems. There is deception and conspiracy to commit deception as regards the game of blackjack. They preach:
The casino executives and their "mathematical" consultants know it all too well. They've heard continuously players complaining that they lose too much and too fast at blackjack. Some players believe the BJ dealers cheat (sleight of hand). I remember, a few years ago, a player shot dead a floor manager in Atlantic City. The player claimed, under arrest, that he committed the crime because he was cheated!
My opinion is mathematical-based. The blackjack HA is higher than that unreal 0.5%. Let's look again at one of my examples: $100 bankroll, $10 minimum bet, 1% house edge. Has anyone seen a situation like this in 100 bets? 51 losses for the Player against 49 wins; total loss in 100 bets: 2 * 10 = $20. NOT!
The overwhelming majority of blackjack players lose their initial $100 bankroll long before the 100-bet mark. (Actually, I was generous and set HA to 2%, as you won't have 50.5 hands v. 49.5 hands in 100). As per my example above, I challenged self-proclaimed blackjack players to a challenge. Let's see if they can resist 1000 hands playing strict basic strategy! Their only response to my challenge was in the form of cursing me!
My blackjack challenge to prove the house edge is detailed further on Facebook by applying the test of the binomial distribution and the normal probability rule:
Axiomatic ones, "The style is the man," said that witty writer Molière. I do have my own style. Sometimes it brings out the worst in the creatures of smallness! This very topic attracted a lot of virulent reactions from half-dozen or so lowlifes (should I "forgive" them? NOT!) Basically, they started a war against my new figures for the Dealer bust at blackjack!
As coolheadedly as only me can be (kidding!), let's analyze my new bust probability figures versus the old and entrenched 28% odds for a BJ Dealer's bust.
We'll analyze the easiest case here, although it is NOT common by any stretch. Expect the situation to be worse for Players in the most common casino situations: multiple decks and several players at the same table.
There are 2 (two) elements: Dealer and Player. There are 2 (two) bust situations: well, Bust and No-bust. Total possible situations: 2 x 2 = 4 (four). (That's the way I explained in a gambling forum, as my way to "lighten up" virulent lowlifes… innocuous fun!)
1) Player_No-bust AND Dealer_No-bust
2) Player_No-bust AND Dealer_Bust
3) Player_Bust AND Dealer_No-bust
4) Player_Bust AND Dealer_Bust.
A) First, let's analyze the game for my new odds figure: Dealer bust = 0.335 or 33.5%. Therefore, Dealer No-bust = 0.665 or 66.5%. We count blackjack hands only, not bets.
1) Player_No-bust AND Dealer_No-bust: 0.665 * 0.665 = 44% of hands;
2) Player_No-bust AND Dealer_Bust: 0.665 * 0.335 = 22%
3) Player_Bust AND Dealer_No-bust: 0.335 * 0.665 = 22%
4) Player_Bust AND Dealer_Bust: 0.335 * 0.335 = 12%.
In 3) and 4) we must make adjustments, as the Player can bust less if applying basic strategy. We deduct 4% from the previous figures for 3) and 4):
1) Player_No-bust AND Dealer_No-bust: 0.665 * 0.665 = 44% of hands (half favorable to Dealer, half favorable to Player);
2) Player_No-bust AND Dealer_Bust: 0.665 * 0.335 = 22% (all favorable to Player);
3) Player_Bust AND Dealer_No-bust: <0.335 * 0.665 => 18% (all favorable to Dealer);
4) Player_Bust AND Dealer_Bust: <0.335 * 0.335 => 8% (all favorable to Dealer).
In situation 1), Dealer and Player have an equal opportunity to win, lose, or tie. Let's divide the 44 out of 100 hands equally: 22 favorable to Dealer, 22 in favor of Player. Thusly, Player wins 22 + 22 = 44 hands; Dealer wins 22 + 18 + 8 = 48 hands. We notice now only 92 hands out of 100. Mystery? NO! The 8 missing hands are those 8 cases when the Dealer does NOT even play her hands out — the simultaneous bust cases!
B) Second, let's analyze the game for the old odds figure: Dealer bust = 0.28 or 28%. Therefore, Dealer No-bust = 0.72 or 72%. We count blackjack hands only, not bets.
1) Player_No-bust AND Dealer_No-bust: 0.72 * 0.72 = 52% of hands;
2) Player_No-bust AND Dealer_Bust: 0.72 * 0.28 = 20%
3) Player_Bust AND Dealer_No-bust: 0.28 * 0.72 = 20%
4) Player_Bust AND Dealer_Bust: 0.28 * 0.28 = 8%.
In 3) and 4) we must make adjustments, as the Player can bust less if applying basic strategy. We deduct 4% from the previous figures for 3) and 4):
1) Player_No-bust AND Dealer_No-bust: 0.72 * 0.72 = 52% of hands (half favorable to Dealer, half favorable to Player);
2) Player_No-bust AND Dealer_Bust: 0.72 * 0.28 = 20% (all favorable to Player);
3) Player_Bust AND Dealer_No-bust: <028. * 0.72 => 16% (all favorable to Dealer);
4) Player_Bust AND Dealer_Bust: <0.28 * 0.28 => 4% (all favorable to Dealer).
In situation 1), Dealer and Player have an equal opportunity to win, lose, or tie. Let's divide the 52 out of 100 hands equally: 26 favorable to Dealer, 26 in favor of Player. Thusly, Player wins 26 + 20 = 46 hands; Dealer wins 26 + 16 + 4 = 46 hands. We notice now only 92 hands out of 100. Now, this is a mystery! The 8 missing blackjack hands are NOT covered by those 4 cases when the Dealer does NOT even play her hands out — the simultaneous bust cases!
The most disturbing fact in B), however, is the equality of number of hands won by Dealer and Player! That can never — ever — be true! Blackjack is far, far from being a 50–50 game! (Remember, I am talking here only about number of hands, not bets! The Player gets a 2% advantage if the BJ natural is paid 3 to 2.) That is an affront to the intellect and sanity. It is also crass casino deception — they know that 9 out of 10 blackjack players lose their bankrolls quickly. IF the game was truly 50–50, then an equal number of blackjack players will win an amount of money double to their bankrolls.
Again, that never — ever — happens. The casinos know that reality very, very well. Casino reports show intakes of 20% – 25% for the blackjack games. The casinos justify there are many idiotic blackjack players. Yet, the truth is the vast majority of the players know basic strategy. In fact, even the casino dealers offer free advice to less knowledgeable players. Something like: "The book says to take that action in your situation." Other players at the table who are knowledgeable also offer advice based on basic 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. 22% bust chance for the first hand at the table. Getting a second bust hand, with only one Player at the table, is 0.22 * 0.336 = 7.5% = simultaneous bust for one Player and Dealer. The simultaneous bust is what creates the house advantage in the game of blackjack. This is the new and most accurate figure representing the blackjack house edge or advantage.
How about three busted hands in the same round (2 Players, one Dealer)? 0.22 * 0.22 * .336 = 1.6%. 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 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! The more Players at the table, the lower the bust chance for the Dealer goes.
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? You can read more details and see screenshots, plus download lots of text files showing all possible blackjack hands for various decks of cards — see link below:
Run Blackjack.exe, probability and statistical software to analyze thousands of blackjack hands from the perspective of a strict blackjack basic strategy player. The program calculates the number of streaks of losing exactly N consecutive blackjack hands (from the Player's perspective). We symbolize a Dealer's win by L and a player's win by W. We can calculate the number of the streaks consisting of exactly, say, 4 consecutive losses for the Player: WLLLLW –
The following is an application of gambling streaks — My Mental Blackjack System:
Check out the software site: