Reddit Blackjack
2021年3月13日Register here: http://gg.gg/omz8w
*Reddit Blackjack Tips
*Reddit Blackjack App
*Reddit Pokerrr2
*Reddit Poker
*Blackjack Tips
*Basic Online Blackjack Rules and Strategy. Anyone new to the game of Blackjack will appreciate a brief run-down of the rules and strategy. Blackjack is a card game that pits you against the Dealer in a battle of the better hand. In order to win, you must have a higher score than the Dealer, up to a maximum of 21 points.
*The airspace above the nation’s capital and the area surrounding Washington, D.C. Is the most restricted in the nation. Coast Guard is responsible for intercepting any potential threats.
*Everything you wrote here is a lie. Number one: blackjack is not a type of poker. Blackjack is blackjack and poker is poker. They are both great games IMO but they are not the same. Number two: ’You can make a lot of money with these games’ and ’you can earn higher profits easily.’ In online gambling, you are far more likely to lose than.
Blackjack should need no introduction. It is the most popular table game in the United States, and is easily found in casinos throughout the world. The object of the game of Blackjack is simply to get more points than the dealer without going over 21. Rules; Hand Signals; Wizard’s Simple Strategy; Basic Strategy; Blackjack FAQ; Bad.
Poker, blackjack, dominoes and five finger fillet in RDR2 are different minigames you can play in the game. All of you that like to play gambling games will have many opportunities to play blackjack, dominoes, five finger fillet, and poker in Red Dead Redemption 2. The only real catch here is where to find RDR2 gambling games and when they unlock for you. With all of that in mind, our RDR2 Where to Play Poker, Blackjack, Dominoes, Five Finger Fillet guide will answer those questions for you.Where to Play Poker in Red Dead Redemption 2?
To play poker in Red Dead Redemption 2, there are several places you can visit. A grand total of five poker locations, to be exact. Depending on how far in the game you’ve come, some of those locations might not be open for you just yet. Before we get into the locations, one important side note. Playing poker in RDR2 opens a few missions into Chapter 2. Specifically, starting the mission “Who is Not without Sin” will open poker for you. In fact, that’s when you’ll play your first game of poker.
Anyway, poker locations. The first one is in Flatneck Station. That’s where the aforementioned main-story-mission poker game takes place. Another location is in the town of Valentine, aka what is basically the starting town. The third RDR2 poker location is the center of Saint Denis, in the southeast of the map. You can also play poker in the center of the town of Blackwater, in the east of Great Planes in West Elizabeth. Lastly, later in the game, you’ll be able to play poker in the small settlement of Tumbleweed, way in the west.
EDIT: It seems that you can also play poker at the first camp, the one near Valentine, if you visit your crew at the correct time of day. Thank you to our reader billyjoel for pointing it out.RDR2 Blackjack Locations – Where to Find?
To find blackjack locations in Red Dead Redemption 2, there are three locations you can go to and play. Blackjack should be available for you starting Chapter 2, just like basically everything else. First off, you can go to the south of the town of Rhodes in the south of Lemoyne. Another location is in Van Horn Trading Post (where you can also find a fence vendor). Van Horn Trading Post is in the east of the map, south of the mining city of Annesburg. The third and final location is Blackwater, in the Great Planes of West Elizabeth. Turns out that Blackwater is something of a gambling capital. As we’ve already mentioned, you can also play poker there, as well as dominoes. Speaking of…Dominoes Locations Red Dead Redemption 2 – Where to Play?
To play dominoes in Red Dead Redemption 2, you have three locations at your disposal. The one I assume you’re going to run into first is in Saint Denis. You should be able to find it just east of the N in the word Saint. Your second location is at Emerald Station / Emerald Ranch. Last, but most certainly not the least is Blackwater; where, as we’ve said above, you can also play poker and blackjack. The only difference is that you’ll have to go a bit to the southwest to find the dominoes game. Incidentally, much like blackjack, and, as we’ll see, five finger fillet, dominoes will unlock for you from the start of Chapter 2.Five Finger Fillet – Where to find in RDR2?
To find a game of Five Finger Fillet in Red Dead Redemption 2, the first place you can look is Valentine, the first town you’re going to come across. Since five finger fillet unlocks in Chapter 2, this is likely the first place you’ll find a game of stabby-fingers. The second location is the Van Horn Trading Post, which you can find in the east, if you head south from the city of Annesburg. Lastly, there’s the city of Strawberry. You can find it in central West Elizabeth, on the banks of Hawks Eye Creek. Now, why anybody would want to play this is beyond me, but hey, different strokes for different folks. Maybe Arthur does indeed have too many fingers.
If you end up getting stuck at another point, you might want to check out some of the other guides we have. We’ve written about mysteries like the missing person Gavin, the Rhodes Gunsmith prisoner, the locked door at Valentine doctor. We also have instructional guides that show how to rob stores without getting bounty or where to sell jewelry and gold bars. If it’s collectibles or key items you’re after, we’ve found all the Penny Dreadful comic books, Chick’s treasure map location, or that famous pipe for Dutch. Finally, if you don’t care to explore the map yourself, you could take a look at our Watson’s Cabin and Catfish Jackson’s Homestead locations guides. In the latest Red Dead Online Moonshiners Update you’ll have a chance to find Navy Revolver and Best Shack Locations.By Ion Saliu, Founder of Blackjack MathematicsI. Fundamental Probability Issue: True Odds at Blackjack, Software CalculatorII. Fundamental Myth of Blackjack Gambling: Counting CardsIII. Theory of Streaks: Foundation of Blackjack Gambling Strategy, SystemsIV. 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
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*Reddit Blackjack Tips
*Reddit Blackjack App
*Reddit Pokerrr2
*Reddit Poker
*Blackjack Tips
*Basic Online Blackjack Rules and Strategy. Anyone new to the game of Blackjack will appreciate a brief run-down of the rules and strategy. Blackjack is a card game that pits you against the Dealer in a battle of the better hand. In order to win, you must have a higher score than the Dealer, up to a maximum of 21 points.
*The airspace above the nation’s capital and the area surrounding Washington, D.C. Is the most restricted in the nation. Coast Guard is responsible for intercepting any potential threats.
*Everything you wrote here is a lie. Number one: blackjack is not a type of poker. Blackjack is blackjack and poker is poker. They are both great games IMO but they are not the same. Number two: ’You can make a lot of money with these games’ and ’you can earn higher profits easily.’ In online gambling, you are far more likely to lose than.
Blackjack should need no introduction. It is the most popular table game in the United States, and is easily found in casinos throughout the world. The object of the game of Blackjack is simply to get more points than the dealer without going over 21. Rules; Hand Signals; Wizard’s Simple Strategy; Basic Strategy; Blackjack FAQ; Bad.
Poker, blackjack, dominoes and five finger fillet in RDR2 are different minigames you can play in the game. All of you that like to play gambling games will have many opportunities to play blackjack, dominoes, five finger fillet, and poker in Red Dead Redemption 2. The only real catch here is where to find RDR2 gambling games and when they unlock for you. With all of that in mind, our RDR2 Where to Play Poker, Blackjack, Dominoes, Five Finger Fillet guide will answer those questions for you.Where to Play Poker in Red Dead Redemption 2?
To play poker in Red Dead Redemption 2, there are several places you can visit. A grand total of five poker locations, to be exact. Depending on how far in the game you’ve come, some of those locations might not be open for you just yet. Before we get into the locations, one important side note. Playing poker in RDR2 opens a few missions into Chapter 2. Specifically, starting the mission “Who is Not without Sin” will open poker for you. In fact, that’s when you’ll play your first game of poker.
Anyway, poker locations. The first one is in Flatneck Station. That’s where the aforementioned main-story-mission poker game takes place. Another location is in the town of Valentine, aka what is basically the starting town. The third RDR2 poker location is the center of Saint Denis, in the southeast of the map. You can also play poker in the center of the town of Blackwater, in the east of Great Planes in West Elizabeth. Lastly, later in the game, you’ll be able to play poker in the small settlement of Tumbleweed, way in the west.
EDIT: It seems that you can also play poker at the first camp, the one near Valentine, if you visit your crew at the correct time of day. Thank you to our reader billyjoel for pointing it out.RDR2 Blackjack Locations – Where to Find?
To find blackjack locations in Red Dead Redemption 2, there are three locations you can go to and play. Blackjack should be available for you starting Chapter 2, just like basically everything else. First off, you can go to the south of the town of Rhodes in the south of Lemoyne. Another location is in Van Horn Trading Post (where you can also find a fence vendor). Van Horn Trading Post is in the east of the map, south of the mining city of Annesburg. The third and final location is Blackwater, in the Great Planes of West Elizabeth. Turns out that Blackwater is something of a gambling capital. As we’ve already mentioned, you can also play poker there, as well as dominoes. Speaking of…Dominoes Locations Red Dead Redemption 2 – Where to Play?
To play dominoes in Red Dead Redemption 2, you have three locations at your disposal. The one I assume you’re going to run into first is in Saint Denis. You should be able to find it just east of the N in the word Saint. Your second location is at Emerald Station / Emerald Ranch. Last, but most certainly not the least is Blackwater; where, as we’ve said above, you can also play poker and blackjack. The only difference is that you’ll have to go a bit to the southwest to find the dominoes game. Incidentally, much like blackjack, and, as we’ll see, five finger fillet, dominoes will unlock for you from the start of Chapter 2.Five Finger Fillet – Where to find in RDR2?
To find a game of Five Finger Fillet in Red Dead Redemption 2, the first place you can look is Valentine, the first town you’re going to come across. Since five finger fillet unlocks in Chapter 2, this is likely the first place you’ll find a game of stabby-fingers. The second location is the Van Horn Trading Post, which you can find in the east, if you head south from the city of Annesburg. Lastly, there’s the city of Strawberry. You can find it in central West Elizabeth, on the banks of Hawks Eye Creek. Now, why anybody would want to play this is beyond me, but hey, different strokes for different folks. Maybe Arthur does indeed have too many fingers.
If you end up getting stuck at another point, you might want to check out some of the other guides we have. We’ve written about mysteries like the missing person Gavin, the Rhodes Gunsmith prisoner, the locked door at Valentine doctor. We also have instructional guides that show how to rob stores without getting bounty or where to sell jewelry and gold bars. If it’s collectibles or key items you’re after, we’ve found all the Penny Dreadful comic books, Chick’s treasure map location, or that famous pipe for Dutch. Finally, if you don’t care to explore the map yourself, you could take a look at our Watson’s Cabin and Catfish Jackson’s Homestead locations guides. In the latest Red Dead Online Moonshiners Update you’ll have a chance to find Navy Revolver and Best Shack Locations.By Ion Saliu, Founder of Blackjack MathematicsI. Fundamental Probability Issue: True Odds at Blackjack, Software CalculatorII. Fundamental Myth of Blackjack Gambling: Counting CardsIII. Theory of Streaks: Foundation of Blackjack Gambling Strategy, SystemsIV. 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
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