Blackjack Probability Table
2021年7月6日Register here: http://gg.gg/va9f9
Blackjack is the name of the game. So what are the odds of getting dealt a blackjack? The probability depends on how many decks are played. Of course the essential ingredients to blackjack are an ace and a ten-value card (10, jack, queen, or king). In a single deck game there are 4 aces and 16 ten-value cards. You can get blackjack in one of. Blackjack is the name of the game. So what are the odds of getting dealt a blackjack? The probability depends on how many decks are played. Of course the essential ingredients to blackjack are an ace and a ten-value card (10, jack, queen, or king). In a single deck game there are 4 aces and 16 ten-value cards. You can get blackjack in one of. Blackjack Dealer Outcome Probabilities The dealer outcome probability charts on this page can be helpful in determining which of two tournament strategies is preferable. The values in these charts were calculated using a combinatorial analysis from a full deck after removing only the dealer’s upcard.
*Blackjack Probability Worksheet
May 24, 2017
Plan ahead before going on a casino tour: download our 6 Deck Blackjack Strategy Chart in PDF or image format and print it before taking a seat at any table!
Click to download PDF!
Sometimes it’s just more convenient to use a printed Blackjack Strategy Chart, instead of a digital one. For example, when you want to play online blackjack in full screen mode or on mobile. Therefore, we created a 6 Deck Blackjack Strategy Chart PDF and JPG to make your life easier. Just download the 6 Deck Blackjack Strategy Chart PDF to your computer, or open it in your browser and click on print. You can resize the documents based on your liking. If you plan to use it at a land-based casino, you probably want them to be smaller, but if you play from home it probably doesn’t matter.
Tags: 6 deck blackjack strategy, 6 deck blackjack strategy chart, 6 deck blackjack strategy chart download, 6 deck blackjack strategy chart print
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By Ion Saliu, Founder of Blackjack MathematicsI. Fundamental Probability Issue: True Odds at Blackjack, Software Calculator
II. Fundamental Myth of Blackjack Gambling: Counting Cards
III. Theory of Streaks: Foundation of Blackjack Gambling Strategy, Systems
IV. Blackjack Resources, Software, Systems, Basic Strategy Cards (Color Charts)1. The Fundamental Probability Issue: True Blackjack Odds - New Software CalculatorFirst captured by the WayBack Machine (web.archive.org) on November 21, 2002.
*They say roulette is the queen of casino games. Then, blackjack is the king of the casino. Many believe that Blackjack, or 21, or twenty-one is the most popular casino game in the world. Blackjack is also the most researched game ever. It is also the only casino game with fluctuating odds (or probability). The winning chance changes with the composition of the deck. This is more about blackjack mathematics than anything else.
* For gambling is such a mathematical phenomenon that the casinos would do .. you know what to you .. if you knew well about it! This casino game is so easy to win .. but it is NOT about card counting! In truth, card counting at blackjack lopsidedly serves the greedy interests of the casinos.
* The blackjack player is honestly served only by the Fundamental Formula of Gambling (FFG). And thus the casino henchmen will threaten you if you simply write down in a notebook what you lost and what you won. Hey, that’s a tax requirement in any jurisdiction!
Let me start by saying that the game of blackjack has caused me the most serious problems with casinos and gambling developers/authors/system vendors. Blackjack or Twenty-one (seen the movie 21?) is the most popular casino game and the most researched one. There are plenty of books dedicated to the so-called mathematics of blackjack.
There is worthiness in a few of such books or eBooks. For the most part, however, there isn’t much mathematics in all those blackjack studies. The heart of the matter is a worthless concept known as card counting.
I do have a strong interest in blackjack. It is well documented at my website. As a matter of fact, I consider myself the best blackjack player ever. As Muhammad Ali 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 — card-counting or not.
I wrote a book about the true mathematics of blackjack, insofar as precise probability calculations are concerned. You might be shocked to hear, but the mathematical truth is that your knowledge of blackjack probabilities or odds is dead wrong. Everything you had known was based on guesswork, albeit it educated guesswork.
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) = 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!
In the case of the first 2-card hands, the combinations-generating method greatly simplifies the problem at no additional cost. 10-7 is equivalent to 7-10. The problem comes to life beginning with 3-card hands. A hand like 10-6-7 is not equivalent to 10-7-6: Dealer must stop at 10-7.
*’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 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 has never explained how he came up with that 169, kind of a new number of the beast! (There are 13 cards in each of the 4 suits in blackjack; 13 to the power of 2 equals 169.. what’s the relation?!)
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 found the project in the year of grace 2009 and also the code to generate sets from a list (last update: 2014). In this case, the list is a 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.
I generated blackjack hands as both combinations and 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 wrote a special Web page dedicated to the topic of calculating precisely mathematically the bust-odds at blackjack following the Dealer’s rules. There are lots of details, plus screenshots of the probability programs:
*Blackjack Dealer Bust: Software to Calculate Probability, Odds, House Edge, Advantage HA.
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 bj hands.
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!
I have seen lots of search strings in the statistics of my website related to the probability to get a blackjack (natural). This time the request was personal and directed to me:
*“In the game of blackjack determine the probability of dealing yourself a blackjack (ace face-card or ten) from a single deck. Show how you arrived at your answer. If you are not sure post an idea to get us started!”
* Oh, yes, I am very sure! As specified in this eBook, the blackjack hands can be viewed as combinations or arrangements (the order of the elements counts; like in horse racing trifectas).
1) Let’s take first the combinations. There are 52 cards in one deck. There are 4 Aces and 16 face-cards and 10s. The blackjack (or natural) can occur only in the first 2 cards. We calculate first all combinations of 52 elements taken 2 at a time: C(52, 2) = (52 * 51) / 2 = 1326.
We combine now each of the 4 Aces with each of the 16 ten-valued cards: 4 * 16 = 64.
The probability to get a blackjack (natural): 64 / 1326 = .0483 = 4.83%.
2) Let’s do now the calculations for arrangements. (The combinations are also considered boxed arrangements; i.e. the order of the elements does not count).
We calculate total arrangements for 52 cards taken 2 at a time: A(52, 2) = 52 * 51 = 2652.
In arrangements, the order of the cards is essential: King + Ace is distinct from Ace + King. Thus, total arrangements of 4 Aces and 16 ten-valued cards: 4 * 16 * 2 = 128.
The odds to get a blackjack (natural) as arrangements: 128 / 2652 = .0483 = 4.83%.
The generalized formula is:
Probability of a natural blackjack = (A * T) / C(R, 2)
* A = number of Aces remaining in the deck;
* T = number of 10-valued cards remaining in the deck;
* C = combination formula;
* R = total cards Remaining in the deck.
* Read a whole lot deeper analysis:
*Calculate Probability (Odds) for a Blackjack or Natural.2. The Fundamental Myth of Blackjack Gambling: Card CountingYou might have seen that movie 21. It had absolutely no success in theaters. A DVD was released in 2008 with much more success. The 21 DVD reopened the huge gambling appetite for the so-called sure-fire strategy of counting cards at blackjack. The movie also introduced a powerfully symbolic ghost: The MIT Blackjack Team.
If you watch all the features of the DVD, you will see the author of the original book that inspired the 21 movie. In his interview, the book author and the screenplay writer admitted that his book was the result of a rumor! How could people with the heads on their shoulders believe that an MIT blackjack team was even possible?! MIT (Massachusetts Institute of Technology), such a prestigious institution, would even accept the rumor of a gambling team on the premises? Let alone a real gambling team consisting of faculty and student body?! But adding MIT to a blackjack team did wonders!
The legend of card counting started with a well-written book: ’Beat the Dealer!’ The author, Edward O. Thorp, was a mathematician working for IBM. He also learned computer programming in order to prove his theory on blackjack card counting.
If the player keeps track of the cards that were dealt, there will be variable situations for the player. Thorp speculated that the situations were favorable to the player when ten-valued cards and Aces (high cards) were predominant in the remainder of the card deck. Reversely, the situation was unfavorable to the player when there were more small cards (2 to 6) compared to high cards. The 7, 8, and 9-valued cards were considered neutral.
The same John Scarne we talked about before puts jokingly the advantage of card counting. Suppose there is a one-deck blackjack game with 100% penetration (i.e. all cards are dealt). The player tracked the entire deck absolutely precisely. There are 5 cards remaining in the deck: 3 eights and 2 sevens. The player would bet the maximum immediately (actually, millions if it were possible!) There is NO way the player can lose! The player would always stay on two cards (it doesn’t matter if it is 7+7, or 7+8) or 8+8)! On the other hand, the dealer would always bust. It doesn’t matter: 7+7 (under 17); draws an 8 and busts. Or, 8+8 (still under 17); draws a 7 and busts. Or, 7+8 = 15 (under 17); either 7 or 8 as the third card would bust the dealer’s hand!
A situation like that would have occurred, but extremely rarely. To have 2 sevens and 3 eights at the bottom of a 52-card deck has a degree of certainty in the same category as the moon colliding with the earth! Keep in mind, total possible permutations of 52 cards is calculated by factorial of 52 (52! = 1 x 2 x 3 x 4 x 5 … x 50 x 51 x 52). Who can say that number?!
As a matter of fact, John Scarne challenged Edward O. Thorp to a real blackjack game in a casino. I quote from Scarne’s New Complete Guide to Gambling (pg. 361):“In 1964, in an effort to test Professor Thorp’s ’winning Black Jack’ statements I challenged him to a $100,000 contest to be staged in Las Vegas. Thorp’s reply was a big ’No’.”
This excerpt is from page 348:
*’… if he [Thorp] would like to team up with me and my partner to beat the Nevada Black Jack tables by making use of his unbeatable system. Thorp agreed and after the first three days of play in Reno, Nevada, we realized that Thorp knew nothing about the science of Black Jack play, and his countdown system never seemed to work… Thorp later admitted to us that he never really gambled…”
There is one more notable name to be mentioned here: Ken Uston. He became really famous when he appeared on CBS’s newsmagazine 60 Minutes. The curiosity of the TV news division was triggered by a successful lawsuit that Ken Uston had won. He charged that he was barred from playing blackjack because of his skills as a blackjack card counter. The court system decided that no player may be discriminated against based on the skills of the player.
The curious thing is that Ken Uston became a spokesperson for the casino he had filed lawsuit against: Resorts. He appeared in TV commercials aired in New York and other big cities of the eastern board of the U.S. Even more curiously, the 60 Minutes Ken Uston segment was shot inside the same casino! Moreover, Ken Uston was allowed to win big money, as if were a fictional movie! Meanwhile, if a regular casino patron tries to take a benign photo on the premises, he/she might as well be thrown in jail! Not to mention that doing card counting can result in harassment and eviction by force, sometimes!
Card counting, as devised by Edward E. Thorp is a footnote to gambling history now. It offered a slight advantage in one-deck games, and especially towards the end of the deck. Ideally, a player could destroy the blackjack game if knowing the composition of the deck and the sequence of the remaining cards in the deck. The latter part is the real problem: Nobody will ever be able to know the sequence of the remaining cards in the deck. The count may be +5, but often the sequence is Low card, High card, Low, Low, High, High, Low, Low, High, etc. The dealer has the same probability to get the high cards. It is even more complicated when one considers that there are several blackjack players at the table.
The cards, Low and High, will be distributed randomly among them. What makes a particular player believe he/she would be the one to get the High cards? In one-deck games, and only playing head-to-head against the dealer, there is a slightly higher chance for the player to get a blackjack. The dealer also has an equally higher chance to get a blackjack. The difference is th
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Blackjack is the name of the game. So what are the odds of getting dealt a blackjack? The probability depends on how many decks are played. Of course the essential ingredients to blackjack are an ace and a ten-value card (10, jack, queen, or king). In a single deck game there are 4 aces and 16 ten-value cards. You can get blackjack in one of. Blackjack is the name of the game. So what are the odds of getting dealt a blackjack? The probability depends on how many decks are played. Of course the essential ingredients to blackjack are an ace and a ten-value card (10, jack, queen, or king). In a single deck game there are 4 aces and 16 ten-value cards. You can get blackjack in one of. Blackjack Dealer Outcome Probabilities The dealer outcome probability charts on this page can be helpful in determining which of two tournament strategies is preferable. The values in these charts were calculated using a combinatorial analysis from a full deck after removing only the dealer’s upcard.
*Blackjack Probability Worksheet
May 24, 2017
Plan ahead before going on a casino tour: download our 6 Deck Blackjack Strategy Chart in PDF or image format and print it before taking a seat at any table!
Click to download PDF!
Sometimes it’s just more convenient to use a printed Blackjack Strategy Chart, instead of a digital one. For example, when you want to play online blackjack in full screen mode or on mobile. Therefore, we created a 6 Deck Blackjack Strategy Chart PDF and JPG to make your life easier. Just download the 6 Deck Blackjack Strategy Chart PDF to your computer, or open it in your browser and click on print. You can resize the documents based on your liking. If you plan to use it at a land-based casino, you probably want them to be smaller, but if you play from home it probably doesn’t matter.
Tags: 6 deck blackjack strategy, 6 deck blackjack strategy chart, 6 deck blackjack strategy chart download, 6 deck blackjack strategy chart print
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By Ion Saliu, Founder of Blackjack MathematicsI. Fundamental Probability Issue: True Odds at Blackjack, Software Calculator
II. Fundamental Myth of Blackjack Gambling: Counting Cards
III. Theory of Streaks: Foundation of Blackjack Gambling Strategy, Systems
IV. Blackjack Resources, Software, Systems, Basic Strategy Cards (Color Charts)1. The Fundamental Probability Issue: True Blackjack Odds - New Software CalculatorFirst captured by the WayBack Machine (web.archive.org) on November 21, 2002.
*They say roulette is the queen of casino games. Then, blackjack is the king of the casino. Many believe that Blackjack, or 21, or twenty-one is the most popular casino game in the world. Blackjack is also the most researched game ever. It is also the only casino game with fluctuating odds (or probability). The winning chance changes with the composition of the deck. This is more about blackjack mathematics than anything else.
* For gambling is such a mathematical phenomenon that the casinos would do .. you know what to you .. if you knew well about it! This casino game is so easy to win .. but it is NOT about card counting! In truth, card counting at blackjack lopsidedly serves the greedy interests of the casinos.
* The blackjack player is honestly served only by the Fundamental Formula of Gambling (FFG). And thus the casino henchmen will threaten you if you simply write down in a notebook what you lost and what you won. Hey, that’s a tax requirement in any jurisdiction!
Let me start by saying that the game of blackjack has caused me the most serious problems with casinos and gambling developers/authors/system vendors. Blackjack or Twenty-one (seen the movie 21?) is the most popular casino game and the most researched one. There are plenty of books dedicated to the so-called mathematics of blackjack.
There is worthiness in a few of such books or eBooks. For the most part, however, there isn’t much mathematics in all those blackjack studies. The heart of the matter is a worthless concept known as card counting.
I do have a strong interest in blackjack. It is well documented at my website. As a matter of fact, I consider myself the best blackjack player ever. As Muhammad Ali 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 — card-counting or not.
I wrote a book about the true mathematics of blackjack, insofar as precise probability calculations are concerned. You might be shocked to hear, but the mathematical truth is that your knowledge of blackjack probabilities or odds is dead wrong. Everything you had known was based on guesswork, albeit it educated guesswork.
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) = 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!
In the case of the first 2-card hands, the combinations-generating method greatly simplifies the problem at no additional cost. 10-7 is equivalent to 7-10. The problem comes to life beginning with 3-card hands. A hand like 10-6-7 is not equivalent to 10-7-6: Dealer must stop at 10-7.
*’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 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 has never explained how he came up with that 169, kind of a new number of the beast! (There are 13 cards in each of the 4 suits in blackjack; 13 to the power of 2 equals 169.. what’s the relation?!)
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 found the project in the year of grace 2009 and also the code to generate sets from a list (last update: 2014). In this case, the list is a 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.
I generated blackjack hands as both combinations and 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 wrote a special Web page dedicated to the topic of calculating precisely mathematically the bust-odds at blackjack following the Dealer’s rules. There are lots of details, plus screenshots of the probability programs:
*Blackjack Dealer Bust: Software to Calculate Probability, Odds, House Edge, Advantage HA.
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 bj hands.
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!
I have seen lots of search strings in the statistics of my website related to the probability to get a blackjack (natural). This time the request was personal and directed to me:
*“In the game of blackjack determine the probability of dealing yourself a blackjack (ace face-card or ten) from a single deck. Show how you arrived at your answer. If you are not sure post an idea to get us started!”
* Oh, yes, I am very sure! As specified in this eBook, the blackjack hands can be viewed as combinations or arrangements (the order of the elements counts; like in horse racing trifectas).
1) Let’s take first the combinations. There are 52 cards in one deck. There are 4 Aces and 16 face-cards and 10s. The blackjack (or natural) can occur only in the first 2 cards. We calculate first all combinations of 52 elements taken 2 at a time: C(52, 2) = (52 * 51) / 2 = 1326.
We combine now each of the 4 Aces with each of the 16 ten-valued cards: 4 * 16 = 64.
The probability to get a blackjack (natural): 64 / 1326 = .0483 = 4.83%.
2) Let’s do now the calculations for arrangements. (The combinations are also considered boxed arrangements; i.e. the order of the elements does not count).
We calculate total arrangements for 52 cards taken 2 at a time: A(52, 2) = 52 * 51 = 2652.
In arrangements, the order of the cards is essential: King + Ace is distinct from Ace + King. Thus, total arrangements of 4 Aces and 16 ten-valued cards: 4 * 16 * 2 = 128.
The odds to get a blackjack (natural) as arrangements: 128 / 2652 = .0483 = 4.83%.
The generalized formula is:
Probability of a natural blackjack = (A * T) / C(R, 2)
* A = number of Aces remaining in the deck;
* T = number of 10-valued cards remaining in the deck;
* C = combination formula;
* R = total cards Remaining in the deck.
* Read a whole lot deeper analysis:
*Calculate Probability (Odds) for a Blackjack or Natural.2. The Fundamental Myth of Blackjack Gambling: Card CountingYou might have seen that movie 21. It had absolutely no success in theaters. A DVD was released in 2008 with much more success. The 21 DVD reopened the huge gambling appetite for the so-called sure-fire strategy of counting cards at blackjack. The movie also introduced a powerfully symbolic ghost: The MIT Blackjack Team.
If you watch all the features of the DVD, you will see the author of the original book that inspired the 21 movie. In his interview, the book author and the screenplay writer admitted that his book was the result of a rumor! How could people with the heads on their shoulders believe that an MIT blackjack team was even possible?! MIT (Massachusetts Institute of Technology), such a prestigious institution, would even accept the rumor of a gambling team on the premises? Let alone a real gambling team consisting of faculty and student body?! But adding MIT to a blackjack team did wonders!
The legend of card counting started with a well-written book: ’Beat the Dealer!’ The author, Edward O. Thorp, was a mathematician working for IBM. He also learned computer programming in order to prove his theory on blackjack card counting.
If the player keeps track of the cards that were dealt, there will be variable situations for the player. Thorp speculated that the situations were favorable to the player when ten-valued cards and Aces (high cards) were predominant in the remainder of the card deck. Reversely, the situation was unfavorable to the player when there were more small cards (2 to 6) compared to high cards. The 7, 8, and 9-valued cards were considered neutral.
The same John Scarne we talked about before puts jokingly the advantage of card counting. Suppose there is a one-deck blackjack game with 100% penetration (i.e. all cards are dealt). The player tracked the entire deck absolutely precisely. There are 5 cards remaining in the deck: 3 eights and 2 sevens. The player would bet the maximum immediately (actually, millions if it were possible!) There is NO way the player can lose! The player would always stay on two cards (it doesn’t matter if it is 7+7, or 7+8) or 8+8)! On the other hand, the dealer would always bust. It doesn’t matter: 7+7 (under 17); draws an 8 and busts. Or, 8+8 (still under 17); draws a 7 and busts. Or, 7+8 = 15 (under 17); either 7 or 8 as the third card would bust the dealer’s hand!
A situation like that would have occurred, but extremely rarely. To have 2 sevens and 3 eights at the bottom of a 52-card deck has a degree of certainty in the same category as the moon colliding with the earth! Keep in mind, total possible permutations of 52 cards is calculated by factorial of 52 (52! = 1 x 2 x 3 x 4 x 5 … x 50 x 51 x 52). Who can say that number?!
As a matter of fact, John Scarne challenged Edward O. Thorp to a real blackjack game in a casino. I quote from Scarne’s New Complete Guide to Gambling (pg. 361):“In 1964, in an effort to test Professor Thorp’s ’winning Black Jack’ statements I challenged him to a $100,000 contest to be staged in Las Vegas. Thorp’s reply was a big ’No’.”
This excerpt is from page 348:
*’… if he [Thorp] would like to team up with me and my partner to beat the Nevada Black Jack tables by making use of his unbeatable system. Thorp agreed and after the first three days of play in Reno, Nevada, we realized that Thorp knew nothing about the science of Black Jack play, and his countdown system never seemed to work… Thorp later admitted to us that he never really gambled…”
There is one more notable name to be mentioned here: Ken Uston. He became really famous when he appeared on CBS’s newsmagazine 60 Minutes. The curiosity of the TV news division was triggered by a successful lawsuit that Ken Uston had won. He charged that he was barred from playing blackjack because of his skills as a blackjack card counter. The court system decided that no player may be discriminated against based on the skills of the player.
The curious thing is that Ken Uston became a spokesperson for the casino he had filed lawsuit against: Resorts. He appeared in TV commercials aired in New York and other big cities of the eastern board of the U.S. Even more curiously, the 60 Minutes Ken Uston segment was shot inside the same casino! Moreover, Ken Uston was allowed to win big money, as if were a fictional movie! Meanwhile, if a regular casino patron tries to take a benign photo on the premises, he/she might as well be thrown in jail! Not to mention that doing card counting can result in harassment and eviction by force, sometimes!
Card counting, as devised by Edward E. Thorp is a footnote to gambling history now. It offered a slight advantage in one-deck games, and especially towards the end of the deck. Ideally, a player could destroy the blackjack game if knowing the composition of the deck and the sequence of the remaining cards in the deck. The latter part is the real problem: Nobody will ever be able to know the sequence of the remaining cards in the deck. The count may be +5, but often the sequence is Low card, High card, Low, Low, High, High, Low, Low, High, etc. The dealer has the same probability to get the high cards. It is even more complicated when one considers that there are several blackjack players at the table.
The cards, Low and High, will be distributed randomly among them. What makes a particular player believe he/she would be the one to get the High cards? In one-deck games, and only playing head-to-head against the dealer, there is a slightly higher chance for the player to get a blackjack. The dealer also has an equally higher chance to get a blackjack. The difference is th
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