Extra Rules Affecting Blackjack Odds European No-Hole-Card Rule. Some blackjack variations are played with a hole card that is dealt to the dealer only after all the players have played their hands. This rule affects player strategy when playing against dealer up 10 or an Ace. The fascinating game of Blackjack is peppered with amazing statistical probabilities. Enjoy this guide to Blackjack statistics at 777.

1. Wizard Of Odds Blackjack
2. Odds Of Getting Blackjack

The Only True Bible of Blackjack Mathematics and Probability

By Ion Saliu, Founder of True Mathematics of Blackjack

Verse One: History or How the Blackjack Odds Were Calculated — John Scarne
Two: True Mathematics, Combinatorics, Probability Applied to Blackjack
Three: Breakthrough Software to Generate All Blackjack Hands and Calculate the Odds Accurately
Four: History in the Making — Shocking but True New Odds of Blackjack Dealer Bust
Five: Possible Bust Situations; Comparative Analysis of New Bust Probability Versus Old 28% Bust Odds
Six: Conclusions Regarding the Greatest Breakthrough in Blackjack History — New Strategy
Seven: Resources in Blackjack, Mathematics, Probability, Odds, Software, Algorithms

1. Being dealt with certain cards in blackjack can significantly affect the odds of the dealer on obtaining a high or low bust percentage which can affect the outcome of their dealer's game. The card value of 5 gives the blackjack dealer the highest odds of busting at 43 percentage rate.
2. Dealer's Bust-Out Rate If you've ever been undecided about whether to hit or stand in a game of blackjack when the dealer's upcard is a 'bust card,' learning about the dealer's bust-out rates can greatly improve your chances.

I. History or How the Blackjack Odds Were Calculated — John Scarne

First capture by the WayBack Machine (web.archive.org) October 3, 2009.

• I do have a strong interest in blackjack. It is well documented on this site. Matter of fact, I consider myself the best blackjack player ever. As Muhammad Ali, the greatest of them all, put: 'It ain't bragging if you back it!' So, I put the money where my mouth is: I issued a casino gambling challenge, especially at the blackjack tables. So far, nobody has dared to honor my challenge. The real casino challenge is open to any gambler, gambling author, or gambling system developer.
• 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):

• '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) = (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!

• '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.
  ... Two-card counts of 17, 18, 19, 20, and 21 will show up 462 times of 1,326... The Black Jack rules demand that the dealer must stay on all those hands, so he cannot bust on any of them.
  ... Two-card counts of 12, 13, 14, 15, and 16 will be held by the dealer 514 times out of 1,326... Since the rules demand that the dealer must draw to a count of 16 or less, he will bust... [or stand]...'

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!)

On the other hand, I commend John Scarne for taking on such daunting of a challenge. He did it without the help of computers — and he came pretty close to what good computer programming achieves.

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!

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 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.

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 BjBustOddsOrder. 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. 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.

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! 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:

The arrangements scenario regarding Dealer's bust probabilities for the game of blackjack reads:

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):

Or, stand on 14 or greater (as arrangements):

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!

• All figures above regarding the bust odds figures apply only to one full 52-card deck. Programming BJ software for more decks is severely limited by today's computers I have access to. For one, the size limit of an object (e.g. file) is limited to 2 GB by the Windows operating system. The speed of today's computers will also make virtually impossible to generate all possible blackjack hands in lexicographic order for multiple decks. John Scarne tried to figure out the Dealer's bust odds by just using 13 blackjack cards from 2 to Ace — one suit only!

  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 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 house advantage (HA) at blackjack is virtually a coin toss: from 0.5% to 1%.'

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:

V. Possible Bust Situations; Comparative Analysis of New Bust Probability Versus Old 28% Bust Odds

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.

Wizard Of Odds Blackjack

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.

• It is one more compelling reason why I consider the 28% bust probability to be farther away from casino reality than my new odds figure. The 28% bust odds figure leads to the misconception that blackjack is, in the worst-case scenario for Player, a 50–50 game. Blackjack is NOT better than coin-tossing... it is a FAR-CRY from that!

VI. 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. 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. 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:

• My blackjack software to calculate accurately the odds, including the source code, was available to buy for a limited time. The offer was withdrawn for patenting reasons — the source code was outrageously cheap.
• Having said that, I created a watered-down version of the software that generates only 2-card blackjack hands:Blackjack2Cards. You can get the programme for free from the aforementioned Web page. It shows every 2-card hand perfectly, plus 100% accurate statistics, such as PAT hands, STIFF hands, PAIRS, DOUBLE DOWNS, BJ NATURALS. Don't mention it, axiomatic one! I am one of the very best humans who ever was, despite rough edges… especially when I harangue down aggressors…

VII. 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.
• Reality Blackjack: Real, Fake Odds, House Advantage, Edge.
• Run Blackjack, 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 –
• Probability Software to Analyze Blackjack Streaks: Wins (W+), Losses (L-), Busts, Pushes.
• Casino Winning Percentages In Atlantic City Beat The Odds And House Advantage By Wide Margins.
• Lawsuits Against Casinos By Gamblers, Blackjack Card Counters, 21 Film.
• The following is an application of gambling streaks — My Mental Blackjack System:
• The Best Casino Gambling Systems: Blackjack, Roulette, Limited Martingale Betting, Progressions.
• No, the casinos couldn't resist reacting… once again! They should know by now that I don't get intimidated. Yet, they keep reacting every time I publish something of great importance. No doubt, my new research in calculating the blackjack odds of busting is ground-breaking. The new figures will surely have an impact on the game of blackjack. They even make me a criminal by association! They compare me to the inmates who become their own lawyers!! Please read a hostile email I suspect originated in a highly paid office of a casino:
• Hostile Reaction by Casinos to the New Blackjack Dealer Bust Odds, Probability Calculations.
• Download Blackjack, Casino GamblingSoftware.

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If players’ priority is to win and to improve with every single blackjack game, then they need to put effort and time into achieving it. There are many other things that gamblers need to take into account and focus on prior to playing and during the game. To make the best possible decisions every time, players should prepare themselves beforehand.

For instance, it is recommended for them to have a look at every possible hand they can get and what their best choice will be in such cases. Knowing how to proceed in the difficult situations when players have a hand 14, 15, or 16 is crucial as the total value of any of the hands is significantly high which means that the chance of players busting increases.

Whenever players’ hand total value is above 14, they are already in an unfavourable situation. Such cases require a good strategy and it is necessary for gamblers to think them through beforehand. Preparation and knowledge are compulsory if one wants to make the best possible decision.

Of course, as already mentioned, several other factors play a crucial role such as money management, discipline, decent bankroll and not to forget a bit of luck. Whenever players have the chance to turn the tables in their favour, they should take full advantage of the situation.

Important Things To Consider When Having Hand 14, 15 or 16

When players are unfortunate enough to get a hand 14, 15, or 16, they need to be very careful and stick to the strategy they have chosen. These are situations in which players are already one step from going bust regardless of the dealer’s upcard. However, this doesn’t mean that they should ignore its value, players need to always bear in mind that it is necessary for them to make their moves based on the dealer’s upcard.

Also, how players proceed in such situations shows their level of competence and knowledge to the rest of the players on the table. Since gamblers are already in an unfavourable situation, winning the hand shouldn’t be their priority. Instead, they should try to make the best move which is the move that will leave them with the smallest possible amount of money loss.

These are some of the trickiest hands in blackjack and as such, they are frequently misplayed. Part of the confusion results from the discrepancies in strategy charts. The optimal playing decisions vary depending on many different factors. These include how many decks are in play at the table, whether or not the dealer must hit soft seventeen, and whether players have the chance to late surrender.

All of these factors must be taken into account before you grab a basic strategy chart to take with you at the blackjack tables. What is optimal for hard totals 14 through 16 in a single-deck game may no longer be correct in multiple-deck variations.

What are Breaking Hands

The term breaking hands is commonly used in blackjack which places it among the important phrases players need to get familiar with. Gamblers have a breaking hand when they get their first two cards and they total 12 or above. The reason for this is that almost every single card will cause players to go bust (i.e. break their hand) as their total value will go over 21. Also, it can be referred to the dealer’s position as ‘breaking’ if their upcard is a 2 through 6.

The trouble with the so-called breaking hands 12 through hard 17 is that they invariably lose in the long term against all upcards of the dealer, including those that have higher bust rates for the house. Hard 14, 15, and 16 are particularly tough to approach because they lose more frequently than they win over the long haul.

The result is that the player ultimately ends up in the red with these totals no matter what playing decision they make. You probably think that’s bad. It is, but it gets worse since you will end up receiving horrible stiff underdogs approximately 40% of the time you spend at the blackjack tables.

What is a player to do with these hard totals then? The answer is simple – trust in basic strategy and approach such underdog hands courageously by making the correct playing decisions. And by ‘correct’ we do not necessarily mean the ones that would secure a winning outcome.

The moves basic strategy recommends for these three hard hands are considered optimal because they reduce your negative expectation, i.e. you will end up losing less money with these stiffs over the long haul. In other words, you are on the defence rather than adopting an offensive approach. In some cases, the strategy plays help you escape a highly disadvantageous situation and increase your winning chances.

Breaking Hands’ Situations

As the name hints, breaking hands is the situation when either the player or the dealer is in a very weak position. Such cases are extremely hard to cope with and the only way to have a chance of not going bust is to follow the basic strategy. Breaking hands are the reason why so many people lose in the long run as well. Thus, if players manage to learn how to deal with them, they will significantly improve their performance at the table.

The bottom line is that once gamblers have such breaking hands, the chances that they will go bust and lose are very high. However, if they use basic strategy, they will be able to improve their hand once in a while.

The move hitting here plays a crucial role and players should take their time to observe when they should hit and when this task should be left to the dealer. In cases where players have a breaking hand and the dealer’s position is standing, then they should hit. Otherwise, they risk losing the hand.

When Players Have Hand 14, 15 or 16

It will be best to have a look at all of these cases at once where players have a hand totalling 14, 15, and 16 as the strategy that needs to be used is the same. It is worth mentioning that the same goes for the situations when players have a hard 13 against different combinations of the dealer’s upcard.

Let’s first elaborate on the recommended strategy plays for hard 14. These are the easiest to remember since there are no discrepancies in the strategy based on decks and dealer rules. Hard 14 is always a stand when the dealer exposes small cards 2 through 6. The player must hit their hard 14 against all other upcards, namely 7 through ace.

When dealt A-3, you have a soft 14. This is a much better situation to be in since it is impossible to break this hand with a one-card draw. You have an advantage against a dealer who starts with weak small cards 4 through 6.

In single-deck S17 blackjack, you must double on soft 14 versus the dealer’s 4, 5, and 6. When playing with two to eight decks, you should double on A-3 only when the dealer has a 5 or a 6 provided that the S17 dealer rule applies.

A pair of 7-7 also adds up to a total of 14. In shoe games, this pair must be split when the dealer has 2 through 7 and hit versus upcards 8 through ace. At single-deck tables, the pair should be surrendered against the dealer’s 10. The rest of the moves coincide with those for multi-deck blackjack.

The player is also in a tough spot when holding a hard 15. This is a bad hand, to begin with, no matter what value the dealer’s upcard is. However, things get trickier since the best plays are influenced by the dealer’s drawing rules and the number of decks.

We shall tackle the strategy differences in more depth in the surrender section of the article. Provided that late surrender is unavailable, you should stand on hard 15 against low-value cards 2 through 6 and hit versus 7 through ace. Undoubtedly, hitting a hard total of 15 is not the easiest decision to make at the blackjack table, especially against the dealer’s 10.

Nevertheless, it has to be done because it results in the lowest possible negative EV for the player. The main idea behind hitting 15 against a 10 is that it gives you a shot at improving your situation. The differences might appear negligible but in the long run, they are not.

Soft 15 (A-4) requires a different approach due to the flexibility the ace gives you. You should double down versus low cards 4 through 6 and hit against all other cards the dealer starts with.

EV of Hitting and Standing on Hard 15 vs. High Upcards 10 and Ace in Multiple-Deck S17 Blackjack
EV of Hard 15 vs. 10		EV of Hard 15 vs. Ace
Hitting	-0.504428	Hitting	-0.480006
Standing	-0.540430	Standing	-0.666951

The situation of the player worsens even more when they are dealt a hard 16, the worst total one could possibly obtain in blackjack. There are several ways to get this terrible hand including 8-8, Q-6, 9-7, and 3-5-8. The pair of 8-8 should be split against all dealer upcards in the vast majority of blackjack variations. We expand on the particulars of this pair further on in this guide.

The correct strategy moves for hard 16, assuming you cannot surrender are relatively easy to remember. You stand on 16 against 2 through 6 and hit versus 7 through ace. The same applies to multi-card totals that add up to 16, or at least if you follow total-dependent basic strategy. In the absence of late surrender, hitting is again more optimal because it gives you the chance to improve your total against the strong dealer.

Hands 14, 15, and 16 Basic Strategy without the Late Surrender Option
Players’ Hand	Dealer’s Upcards
	2	3	4	5	6	7	8	9	10	Ace
14	S	S	S	S	S	H	H	H	H	H
15	S	S	S	S	S	H	H	H	H	H
16	S	S	S	S	S	H	H	H	H	H

Again, all three hands will end up losing over the course of thousands of rounds played. The optimal decision is the one that cuts down your long-term losses the most. The above moves are recommended under total-dependent strategy. If you want to take your game to the next level, you can switch to composition-dependent strategy.

The latter takes into account the exact composition of the cards your hand consists of. Composition-dependent strategy recommends you to stand on hard 16 versus a 10 when your hand contains three or more cards like K-3-3. Also, when the 16 results from splitting a pair, you should stand rather than hit under composition-dependent strategy.

The Option to Surrender

There is one alternative move players can make which leads to the least money losses. However, due to this fact many landbased casinos do not offer the option to surrender as they know that if players know when to take advantage of it, they can greatly benefit from it.

Many professional gamblers won’t play in a casino which doesn’t provide the option to surrender when playing blackjack. However, if players find themselves in such a situation and there is no surrender option, all they can do is act according to the above-mentioned strategy and hope for the best.

When this option is available, most casinos tend to offer late surrender. This allows players to give up on poor hands like hard 15 and hard 16 in exchange for half of their original wager. With late surrender, you can forfeit a bad hand after the dealer has checked for blackjack when starting with an ace or a ten-value card.

Late surrender is beneficial to players because it takes away around 0.07% from the house advantage. Most novice players are averse to surrendering as the name of the move itself evokes negative connotations. However, surrendering is a smart move when you are dealt negative-expectation hands like 15 or 16 versus strong dealer upcards like 10s and aces.

Odds Of Getting Blackjack

As a general rule of thumb, surrendering is recommended whenever you receive a hand whose expectation of winning is less than 50%. The main advantage of surrendering is that it saves you money when you find yourself at a disadvantage. Late surrender is a defensive play which also allows advantage players to temper the effect variance has on their blackjack bankrolls.

The correct late surrender plays depend on deck number and the dealer’s fixed standing rules. In single-deck blackjack, surrender is recommended when you have hard 16 versus the dealer’s ace or 10.

Provided that the dealer hits soft 17, it is also recommended to surrender hard 15 against an ace. You should forfeit paired 7s against the dealer’s 10 in single-deck S17 blackjack. If one deck is in play and the dealer must hit soft 17, surrendering is advisable against tens and aces when you have 7-7.

In double-deck H17 games, surrender is advisable when you hold hard 15 and hard 16 against the dealer’s ace and 10. Paired 8s should also be surrendered against the ace in double-deck H17 blackjack.

As for shoe-dealt games, basic strategy suggests surrendering hard 16 against strong upcards such as 9, 10, and ace. The approach toward hard 15 in multi-deck variations depends on the dealer’s fixed rules for drawing and standing. You surrender the 15 against the dealer’s 10 in S17 variations and against the 10 and the ace in H17 multi-deck blackjack.

Hard 16 Consisting of Paired 8s

If you have read CasinoGuardian’s blackjack guide carefully so far, you probably remember that the rule of thumb of basic strategists is to always split pairs of 8-8 rather than forfeiting them despite the fact this is still a hard total of 16. Some gambling authors recommend surrendering the pair of 8s versus high dealer upcards like 10, K, Q, and J.

Regrettably, this is a major mistake, and here is why. The dealer undoubtedly has an advantage over you when you hold 8-8 versus a ten-value card. He or she would arrive at standing totals 17 through 21 77% of the time on average.

What adds insult to injury is that the dealer’s probability of busting when starting with a ten-value card is rather small at 23%. A pair of 8-8 will cost you money no matter how you approach it but splitting is recommended because it reduces your losses the most. With 8-8, you have a great opportunity to turn a terrible stiff 16 into two brand new hands starting with an 8 each. This improves your chances of forming a good hand and beating the dealer.

Assuming you play standard six-deck S17 blackjack where you can resplit to up to four hands and double down after you split, your 8-8 will lose against the dealer’s ten-value card 77 hands out of every one hundred hands and win 23 times out of every one hundred hands. However, your win rate improves to 38 hands out of every hundred rounds when you split your 8-8.

Let’s suppose you are flat betting £1 per hand for simplicity’s sake. If you merely hit the pair, you will lose £77 and win £23 every one hundred hands on average. This makes for an average net loss of £54 in the long run. Meanwhile, if you consistently split your 8-8 versus the dealer’s ten-value card, you will lose £62 and earn £38.

Your net losses drop to 2 x £24 = £48 per every one hundred hands in this case. This may not sound like a significant improvement but you are still losing £6 less compared to drawing or staying on your pair of 8-8. Meanwhile, if you choose to surrender this pair, as some gambling authors advocate, you will net losses of £50 per every one hundred rounds on average (you lose only half your original wager when surrendering).

Therefore, consistently surrendering your pair of 8s versus the dealer’s 10 turns out to be £2 more expensive than splitting over the long haul. At this point, it is pretty much obvious you are in a losing spot when holding 8-8 no matter how you decide to play it out.

The bottom line is you will save more money by splitting in the long run compared to surrendering or the other possible plays. The only consolation of blackjack players in such instances is that they have made the mathematically optimal decision when dealt a long-term loser like stiff 16.

You can compare the expectation of each basic strategy play with paired 8s and 7s against the dealer’s 10 upcard below. The calculations are courtesy of mathematician and gambling expert Michael Shackleford, known as the Wizard of Odds.

EV of Paired 8-8 and 7-7 vs. the Dealer’s 10 in Multiple-Deck Blackjack
EV of 8-8 vs. 10		EV of 7-7 vs. 10
Doubling	-1.079653	Doubling	-0.938247
Standing	-0.540430	Standing	-0.540430
Hitting	-0.539826	Hitting	-0.466307
Splitting	-0.480686	Splitting	-0.657268

As with all rules, there are always exceptions, however. Paired 8s should always be split unless one is playing shoe games and double-deck blackjack where the dealer must hit soft 17. Under these playing conditions, the odds favour surrendering the pair of 8s rather than hitting when the dealer shows an ace.

Exceptions are also made for paired 7-7 in single-deck blackjack. When the dealer must stand on all 17s, you surrender the 7-7 against a 10. Provided that the dealer must hit soft 17, basic strategy recommends you to surrender this pair whenever you are up against a 10 and an ace.

Odds and Probabilities

Players need to know what their odds are in case they happen to have a total value of 14, 15, or 16 in their hand as this will help them make better decisions. The following odds are accurate proving that the game is played with more than one deck and that gamblers decide to hit. If they have a 14 hand, the chance that they will go bust is 46%, and if the total of their hand is 15 – 54%.

In cases when they get unfortunate and get a total of 16, their chances of going bust are 62%. The worse-case scenario is when players get 19 or 20 as this leaves them with more than 85% chance of going bust. The good news is basic strategists never draw to hard 19 and hard 20 so their chances of busting with these totals are practically nil. Logically, the higher the total of the cards in the players’ hand, the more they are likely to bust by taking a hit.

Blackjack Odds and Probabilities
Players’ Hand	Chances to bust on a Hit
11 or less	0%
12	31%
13	38%
14	46%
15	54%
16	62%
17	69%
18	77%
19	85%
20	92%

Conclusion

Having to deal with a 14, 15, or 16 hand is a big challenge which requires a lot of preparation and a good strategy and even then, players’ success is not guaranteed and they can only hope that the odds will be in their favour.

However, if they follow a strategy or choose the option to surrender, any of these choices will lead to the least amount of money they will lose. Once players find themselves in a bad situation, they should consider how to get out of it with minimal money losses instead of how to win the hand as in many cases this only pushes them to certain doom.