The Statistics of casino blackjack
The Statistics of casino blackjack The Statistics of Casino Blackjack is an artifact composed of a large spreadsheet and mathematical formula derivation with commentary. This document includes a factual introduction to basic strategy, the theory and purpose of the statistical processes, step by step derivation of the decision equation and, a comprehensive evaluation of the spreadsheet showing a worked example. There are additional chapters on: the historical background of Blackjack, card counting theory (including an example of a simple count), strategy comparison, discussion of legality issues and risk followed by a summarisation. All the Blackjack strategies mentioned in this report are genuine methods used by professional gamblers to win at Blackjack. center127000The Statistics of Casino BlackjackContents:Why Blackjack? Page 3An introduction to my topic and why I chose to explore itHistory of the GamePage 4Information about the origins of BlackjackBasic StrategyPage 5Introduction to basic strategyTheoretical explanation of the conceptDeriving the decision formulaComposing a spreadsheet for a worked exampleCard CountingPage 20Card counting theory – why it worksRed Seven count introductionApplications of card countingKelly CriterionPage 24ExplanationIs the player really unable to lose?Combining strategiesWhat’s the risk?Page 25Risk of ruin formulaConsequences of card countingDanger of game variationsWould I gamble?Page 27A summary of my investigationBibliographyPage 28Why Blackjack?My obsession with cards started at a young age. Magic tricks proved fascinating and presented me with the challenge of understanding the logic behind the apparently incomprehensible. My interest developed when I began to learn the tricks for myself and I gained a new appreciation for the skill behind the illusions. What appealed to me was how the minds of others could be manipulated, not with witchcraft, but with wit. Many card tricks use techniques that distract and divert from the reality of the situation. These often require many hours of practice to achieve something convincing and although ‘slight of hand’ illusionists are visually pleasing, I felt that this genre of ‘magic’ was less interesting because it didn’t require a lot of thought. As a result of studying Mathematics at school however, I became able to understand many more of the principles behind logic based tricks and, although perhaps not as aesthetically pleasing as the flourishes and false shuffles in ‘slight of hand’, I believe that the use of mathematics is far more impressive in carrying out a display. A couple of years ago, I managed to talk myself into a holiday job abroad. I knew some of the people I would be working with and immediately leaped at the opportunity. Over the six weeks we spent working, eating and living together, we became a close knit community. We would spend our evenings watching movie-marathons and playing card games (which would become increasingly competitive as the weeks went on). This experience sparked a development of my interest in all things card-based and I began indulging myself in books about the temptingly dangerous world of casinos and the individuals that beat them. My favourite casino games are poker and Twenty-one (Blackjack). The combination of chance with an element of strategy is an exhilarating one - there are so many elements to consider. It’s a rush to try and conceal your own reactions to the cards you are dealt as you try to study the reactions of others. You hope for a hint of a facial expression; willing them to give something away. Are they bluffing? Do they seem nervous? Is it all just a ruse? Meanwhile, you frantically calculate rough probabilities, all in the hope that it will contribute to the right decision. Poker is very popular in casinos and I can understand the addictive qualities that it possesses. The rush of the game may be appealing but I believe that the contribution of chance is far greater than that of strategy and it is therefore too difficult for anyone to play for profit (with the exception of memory masters or cheats). Blackjack is vastly different to poker. Depending on the game variation, players are able to seriously increase their advantage over the dealer using a combination of tactics. Theoretically, it is very possible to play Blackjack for profit, especially when working as part of a team. In practice, beating the game is relatively easy; however, the reality of the situation is often quite different. When playing Blackjack in real life, there is a lot more that can go wrong. In this project, I plan to investigate the mathematical principles that may be applied to a game of Twenty-one. I will also outline the adaptations that a player must make in order to accommodate the constantly changing casino environment. I will cover a range of Blackjack strategies including basic strategy and card counting, subsequently presenting a structured evaluation of my findings.The History of the GameThe first reference to Blackjack appeared in the literary works of Miguel de Cervantes (author of ‘Don Quixote’) with a game called ‘ventiuna’ (short for ‘veinte y uno’ meaning twenty-one in Spanish). This primitive version of the game has the same objectives as traditional Blackjack (although it uses a slightly different deck). Cervantes himself was a keen gambler and pays homage to his hobby in the tale ‘Rinconete y Cortadillo’- one of a dozen short stories in the series Novelas Ejemplares. ‘Rinconete y Cortadillo’ tells the story of a couple of teenage con-artists working in Sevilla. The boys are accomplished at cheating ‘ventiuna’ and use this talent as a form of income. This particular story was written between 1601 and 1602 implying that the origins of Blackjack precede this date; the game probably came about before the 17th Century. There are many other written examples of Twenty-one in both France and Spain throughout the 1700s.Cervante’s novella suggests the game was popular throughout Spain in the early 1600s. At this period in history, the Spanish currency consisted of Escudos, Reales and Maravadies. These coins are mentioned frequently in ‘Rinconete y Cortadillo’. This pre-decimal monetary system became the first form of international currency in around 1500 and remained as such for many centuries. (Silver Eight real coins were also known as ‘pieces of eight’ a phrase famously echoed by Long John Silver’s parrot, Captain Flint, in Robert Louis Stevenson’s novel, Treasure Island). The combination of international trade and the transfer of coins from mints throughout the Spanish colonies meant that Blackjack probably made its way to America through the pastimes of merchants and sailors. Before Twenty-one was played in casinos, the most popular game in the US was ‘craps’, a game of chance using multiple dice. Craps has no element of strategy or skill - qualities that, in my opinion, make a game more much more appealing. In Blackjack, the ability to apply some sort of mental process gives the player a greater sense of control. This element of reassurance presents individuals with the illusion that the game is less of a gamble. I believe that this would have been the main reason for Blackjack’s great success immediately after it was introduced. Initially, bonus payouts were offered for a ‘Black Jack’ (the combination of the ace of spades and either the jack of spades or the jack of clubs). Although this is no longer the case, the game has come to be universally known as Blackjack. Over the years, other payouts have been offered for different card combinations e.g. a trio of sevens, but these bonuses have since been removed by the casinos. Today, there are around 100 documented versions of Blackjack, many of which have an invariant house edge – unable to be lowered by any mental methods. American casinos now make a net average of about 30 million dollars per annum, with Blackjack being one of the most popular games among the gambling community. Gambling houses currently have many countermeasures put in place to deter card counters including sophisticated surveillance systems and specially trained dealers. Beating the casinos has never been more of a challenge.Basic StrategyOptimum strategy or ‘Basic strategy’ is the statistical approach to Blackjack. By applying the laws of probability to specific situations experienced during play, it is possible to calculate the decision that is most likely to result in a positive outcome. It is, mathematically, the best way to play the game without counting cards. Basic strategy is based on the principle of a permanent dealer. The actions of the dealer are determined by the house rules and these rules are fixed regardless of the game situation. The permanent nature of the dealer’s strategy and the fact that this is known to the player are the factors that enable a thorough and detailed mathematical analysis of the game. Basic strategy tables dictate whether or not the player should ‘stand’ or ‘hit’ (whether or not to be dealt an additional card). This decision is dependent on the current total of the player’s cards in relation to the dealers ‘up card’ (information regarding the total value of the dealer’s cards) and the strategy can change due to game variations such as the number of decks currently in use. Nowadays we have a variety of Basic Strategy tables at the player’s disposal that can state the appropriate decisions to accommodate these variables. These tables can also contain additional information on options such as doubling down, splitting pairs or taking insurance etc. An individual may choose to use a table that is as simple or complex as their ability and playing style would warrant. The first version of Basic strategy can be found in ‘The Optimum Strategy in Blackjack’. This paper was a result of collaborative research by Rodger Baldwin and his associates Wilbert Cantey, Herbert Maisel, and James McDermott. It was published in 1956 in the Journal of the American Statistical Association and contains the first accurate strategy tables for Casino Blackjack. The following year, these mathematicians went on to privately publish limited numbers of the guidebook ‘Playing Blackjack to Win’. It was the first publication on the subject of basic strategy and even includes a chapter on ‘partial casing’ that should be recognised as the first valid counting system (although that merit is often credited to Edward Thorpe). The first editions of ‘Playing Blackjack to Win’ have since become a collector’s item and its authors are referred to, in the gambling world, as ‘The Four Horsemen of Blackjack’. Surprisingly, these men were not mathematicians by trade, but GIs. They spent the majority of their time off-duty working on the paper in the Aberdeen research lab. All their processing equipment was provided by the military. The U.S. Army were aware of their research and had no objections to the use of government resources for analysing Casino Blackjack. The four horsemen went on to complete thousands of hand computations resulting in their Basic Strategy. It was with these approximate calculations that Edward Thorpe, a professor at M.I.T, would use as a blueprint for his work on the game. He was able to utilise the University’s IBM 704 computer to calculate Basic Strategy to higher degree of accuracy and for many subsets of cards. Thorpe would go on to develop his research into the world of Blackjack. Today, he is more famously associated with his work on card counting. 'The Optimum Strategy in Blackjack’ covers the main decision elements of the game: when to stand or hit, when to double down and when to split pairs. These values are based on a set of rules that are ‘common but not universal’. The rules are also the conditions for the derived general equation to remain true. They are outlined at the beginning of the paper:‘1. The Number of Players. A dealer and from one to six players.2. The Pack. An ordinary 52 card deck.3. Betting. The players make their bets before any cards are dealt. The house establishes a minimum and maximum bet.4. The Deal. The players and dealer each receive two cards. Each player gets both cards face down. The dealer receives one card face up and one card face down. Cards received face down in the deal or draw are commonly known as ‘hole cards’. 5. The Numerical Value of Cards. The numerical value of an ace is 1 or 11 as the player chooses; the numerical value of a face card is 10; and the numerical value of all other cards is simply their face value. The numerical value or total of a hand is simply the sum of the numerical values of the cards in the hand.6. Object of the Player. To obtain a total which is greater than the dealer’s but does not exceed 21.7. Naturals. An ace and a face card or ten dealt on the first two cards to either player or dealer constitutes a ‘natural’ or ‘blackjack’. If a player has a natural and the dealer does not, the player receives 1.5 times his original bet from the dealer. If a player does not have a natural and the dealer does, the player loses his original bet. If both player and dealer have naturals, no money changes hands.8. The Draw. A player is not required to increase the number of cards in his hand and may look at his hole cards and elect to ‘stand’. Otherwise he may require that the dealer give him additional cards, face up, one at a time. If the player goes over 21 (‘busts’), he immediately turns up his hole cards and pays his bet to the dealer. After each player has drawn his cards, starting with the player at the dealer’s left and proceeding in a clockwise fashion, the dealer turns up his hole card. If his total is 16 or less, he must draw a card and continue to draw cards until his total is 17 or more, at which point he must stand. If the dealer has an ace, and counting it as 11 would bring his total to 17 or more without exceeding 21, he must count the ace as 11 and stand.9. The Settlement. If the player does not go over 21 (‘bust’) and the dealer does, the player wins an amount equal to his original bet. If neither the player nor the dealer busts, the person with the higher total wins an amount equal to the player’s original bet. If neither player nor dealer busts and both have the same total, no money changes hands.10. Splitting Pairs. In the following rules a pair is defined as two cards which are identical except for suit, such as two jacks, two aces or two tens. If the player’s hole cards form a pair, he may choose to turn them face up and treat them as the initial cards in two separate ‘twin’ hands. This strategy is known as ‘splitting pairs’. The original bet goes on one of the split cards, and an equal amount is bet on the other card. The player automatically receives a second card face down on each of the split cards and may continue cards face up to both twin cards as he desired. An exception to this rule is made in the case of split aces where the player may draw only one more card to each ace. Furthermore, if a face card or ten falls on one of the split aces, the hand is not counted as a natural but as ordinary 21 (Similarly the player splitting a pair of face cards or tens who draws an ace holds an ordinary 21). Finally, if a player splits a pair and receives a third card of the same type, he is not permitted to undertake further splitting.11. Doubling Down. After looking at his hole cards a player may elect to double his bet and draw only one more card. This strategy is known as ‘doubling down’. A player who elects to double down turns up his hole cards and receives his third card face down. A player splitting any pair except aces, after receiving an additional card on each of the split cards, may elect to double down on one or both of his twin hands.’In my project, I am going to cover in detail the derivation of the general equation. I will also outline the processes used to analyse when to hit, stand, split pairs and double down. The Decision Equation:The purpose of the decision equation is to find ‘minimum standing values’ for specific situations. A minimum standing value is the smallest player’s total for which it is better to stand than to hit. These are based under the assumption that if it is better to stand rather than to hit for a specific integer value, then the same is true for all larger values. The dealer’s up-card holds specific information regarding the total of his hand. Our minimum standing values will change in relation to the dealers total and therefore up-card. For example, if the dealer’s up-card is high, then he is more likely to hold a high hand and winning may be more likely by hitting rather than standing. By developing this concept of the up-card and considering its value, we can derive minimum standing values that correlate to both the player’s total and the dealer’s up-card.Before deriving the equation, Baldwin’s paper states that it is essential to differentiate between ‘soft hands’ and ‘hard hands’. The value of hands that contain one or more ace can vary in accordance to whether the player chooses aces to be taken as 1 or 11. A hand in which the ace may be counted as 11 without busting is known as a ‘soft hand’. If when the ace is taken at a value of 1, the total of the hand does not exceed 21 but the player busts if value is counted at 11, this is known as a ‘hard hand’. If a hand does not include an ace, it is known as ‘unique’. We can treat hard hands in the same way as unique hands since the ace must be counted as 1 and there can only be one value that the players total can take. Since different strategies are required with soft and hard hands it is important that they are defined as such. If ‘D’ is the value of the dealer’s up-card, then we can say that M(D) is the minimum standing value for a specific integer D. To distinguish soft and hard hands we can define M(D) as the standing value for hard and unique hands, and M*(D) as the standing value for soft hands. The decision equation can help to obtain our minimum standing values. By considering specific values of the player’s total and the dealer’s up card, we can use probabilities to calculate whether the player is more likely to win by standing or hitting in this specific situation. The lowest value of the player’s total for which he is more likely to win by standing rather than hitting is our minimum standing value. The decision equation derived by the four horsemen is as follows: Ed,x – Es,x = -2P(T< x) - P(T= x) - 2P(T> 21)P(J> 21) + 2P(T< J< 21) + P(T= J< 21)This equation uses ‘expectations’ where the greater the value of the expectation, the greater the probability of winning. Ed,x is defined as the expectation of the player winning the game with a total of ‘x’ when he draws another card. Es,x is the expectation of a player winning the game when he stands on a total of x. This means that when (Ed,x – Es,x)<0 then we know that Es,x is greater than Ed,x and therefore the probability of the player winning the game by standing is greater than the probability of winning the game when he hits his total of x. In this particular situation, it is better for the player to stand rather than to hit. To determine our expectations, we resolve (probability of winning – probability of losing) for both standing and hitting situations. To form Es,x (standing), it is necessary to state any assumptions we make beforehand. For a player that does not draw another card, the following is true: If a player decides whether to stand or to hit, the player’s total must be stiff (larger than 11) as if (x< 11) then it is impossible for the player to bust (even if he receives an ace, the value of the card automatically falls from 11 to 1) so for values of x between 2 and 11, hitting is assumed. Since the player is able to stand, his total x must be less than or equal to 21 (x< 21). It is easier to work out the probabilities now that x is in the range of values (12< x< 21) as there is a significant reduction in the number of card combinations. The dealer’s rules force the dealer to hit if his total is 16 or less. This means that his total is always greater than or equal to 17. If the dealer hits on his maximum hitting value of 16, the largest total he can obtain is 26. Denoting the dealer’s total as ‘T’, we can say that T is in the range of values (17< T< 26). It is necessary to state that P(x< 21) =1 and P(x> 21) =0. Our next step is to look at the outcomes for winning and losing:The player wins when: The dealer is bust and the player does not. This outcome may be written as P(T>21). P(x< 21) however, since P(x< 21) =1 this reduces down just to P(T>21).The player’s total is greater than the dealers and they are both less than 21. We write this as P(x>T).P(T< 21).P(x< 21). Since P(x< 21) =1 and (x>T) then P(T< 21) is definite and also equal to 1. This means that the expression reduces down to just P(x>T).The total probability for winning when standing is P(T>21) + P(x>T).The player loses when:The player busts. We have already established that P(x> 21) =0 so we do not have to consider this outcome or the outcome where both the player and dealer bust as this will also be equal to 0.The dealer’s total is greater than the player’s and is less than or equal to 21. We can just state that the player loses when the dealers total is within the range: (x< T< 21).The total probability for losing when standing is P(x< T< 21).This means that the expectation for standing on a value of x is expressed as:P(T> 21) + P(X> T) – P(x< T< 21)Looking at P(x< T< 21), this can be written as P(x< T< 21) – P(x= T). We must negate the probability of a draw as P(x< T< 21) includes P(x= T) and P(x< T< 21) does not. We do this because P(x< T< 21) can be expressed using the product of other probabilities. This event occurs when both the player’s and dealer’s hands are less than or equal to 21 and the players total is less than or equal to the dealers. We can write this as P(x< 21).P(T< 21).P(x< T) but since we know that P(x< 21) = 1, this is just P(x< T).P(T< 21). The expectation is now:P(T> 21) + P(X> T) – [P(x< T).P(T< 21) – P(x= T)]This can be simplified:P(x< T)≡ 1 – P(x>T). i.e. the dealers total is bigger than the players total is the same as saying the player’s total is not greater than the dealer’s. Similarly, P(T< 21)≡ 1 – P(T>21).This means that P(x< T).P(T< 21) ≡( 1 – P(x>T)).( 1 – P(T>21)) and we can expand this to form 1 – P(x>T) – P(T>21) + P(x>T).P(T>21)But, if we look at the last part of this expansion in blue italics, P(x>T).P(T>21). This is saying ‘plus the probability of the dealer going bust (his total is above 21) and the player’s total being greater than that of the dealer’s’. If this is true then the player must also be bust which we know not to be the case. Since P(x>21) = 0 then P(x>T).P(T>21) will be equal to 0 and we can cancel it.Our new expression can now be balanced: P(T> 21) + P(X> T) – [1 – P(x>T) – P(T>21) – P(x= T)]Es,x = 2P(T> 21) – 1 + 2P(x> T) + P(x= T)The assumptions for Ed,x are different. When the player draws another card he is able to bust. When drawing a card we use a different notation ‘J’ for the player’s total as the range of values the total can take is no longer the same. The dealer’s rules are constant so T remains in the range of values (17< T< 26). If J can take two separate values less than 21 then we consider the larger value of J (e.g. if the player has an Ace and a 3 and receives a 6 his cards can value 10 or 20 so we take J as 20). For this derivation we also assume that J and T are independent of each other. This assumption carries a small error as we are negating the possibility that both the player and the dealer draw exactly the same card. It is necessary to state that P(T> 17) = 1 and P(T<17) = 0.To determine Ed,x, we look at the range of values that J can take. If we consider the outcomes of the game for specific values of J, we can see that by splitting the range of values, we can derive a set of probability rules that will remain true for all the integer values of J in that specific section. When (J<17):Since (T< 17), the only way that the player can win in this situation is if the dealer busts. So this is P(T>21).The player loses if the dealers total is less than or equal to 21 and greater than 17. This is P(T< 21).P(T> 17) but since P(T> 17) = 1 this reduces down to P(T< 21).For when (J<17) the expectation is: P(T>21) – P(T< 21)This can be simplified as P(T< 21)≡ 1– P(T>21).P(T>21) – [1– P(T>21)]2P(T>21) – 1When (17< J< 21):The player wins if the dealer busts, this is P(T>21).The player also wins if the his total is larger than the dealers P(J>T).The player loses if the dealers total is greater than the players and less than or equal to 21 so we say that T is in the range of (J< T< 21).For when (17< J< 21) the expectation is:P(T>21) + P(J>T) – P(J< T< 21)When (J>21):The player always loses in this situation regardless of whether the dealers also busts. The probability of losing is 1.For when (J>21) the expectation is:-1This means that the expectation for hitting to obtain a value of J is expressed as:P(J<17).[ 2P(T>21) – 1] + P(17< J< 21).[ P(T>21) + P(J>T) – P(J< T< 21)] + P(J>21).-1Since P(J< 17) + P(17< J< 21) + P(J> 21) encompasses all values of J then it is equal to 1 meaning that P(17< J< 21) ≡ 1 – P(J< 17) – P(J> 21). We can substitute this into our equation: P(J<17).[ 2P(T>21) – 1] + [1 – P(J< 17) – P(J> 21)].[ P(T>21) + P(J>T) – P(J< T< 21)] + P(J>21).-1 It is useful to note that P(J< T< 21) may be be written as P(J< T< 21) – P(J=T< 21). This is because P(J< T< 21) includes P(J=T< 21) but (J< T< 21) does not. Now we have obtained the inequality P(J< T< 21), we can express this as the product of the probabilities P(J< T) and P(T< 21). To reduce the number of coefficients in our expression we can say that since P(T< 21)≡ 1 – P(T> 21) and P(J< T)≡ 1 – P(T<J) this means that P(J< T< 21) = [1 – P(T> 21)].[ 1 – P(T<J)]. We can expand this to give 1 – P(T> 21) – P(T<J) + P(T>21)P(T<J). We now know that P(J< T< 21) = 1 – P(T> 21) – P(T<J) + P(T>21)P(T<J) – P(J=T< 21).Substituting this into our equation gives:P(J<17).[ 2P(T>21) – 1] + [1 – P(J< 17) – P(J> 21)].[ P(T>21) + P(J>T) –(1 – P(T> 21) – P(T<J) + P(T>21)P(T<J) – P(J=T< 21))] + P(J>21).-1P(J<17).[ 2P(T>21) – 1] + [1 – P(J< 17) – P(J> 21)].[ 2P(T>21) + 2P(J>T) –1 – P(T>21)P(T<J) + P(J=T< 21)] + P(J>21).-1Now we can multiply out our expression:P(J<17).[ 2P(T>21) – 1] =2P(J<17) P(T>21) – P(J<17)[1 – P(J< 17) – P(J> 21)].[ P(T>21) + P(J>T) –1 – P(T> 21) – P(T<J) + P(T>21)P(T<J) – P(J=T< 21)] can be multiplied using a grid:x-12P(T> 21)2P(J> T)P(J> T)P(T> 21).-1P(J= T< 21)1-12P(T> 21)2P(J> T)P(J> T)P(T> 21).-1P(J= T< 21)P(J< 17).-1P(J< 17)2P(T> 21)P(J< 17).-12P(J> T)P(J< 17).-1P(J> T)P(T> 21)P(J< 17)P(J= T< 21)P(J< 17).-1P(J> 21).-1P(J> 21)2P(T> 21)P(J> 21).-12P(J> T)P(J> 21).-1P(J> T)P(T> 21)P(J> 21)P(J= T< 21)P(J> 21).-1In the table, we can see that the coefficients in blue italics are contradictions and can be cancelled out. The reasons for this are as follows:If the dealer’s total is less than the player’s, it cannot be less than 17 as P(T< 17) = 0. So 2P(J> T)P(J< 17).-1 = 0.If the dealer’s total is greater than 21, the players total cannot be both greater than the dealer’s and less than 17. P(J> T)P(T> 21)P(J< 17) = 0.If the dealer’s total is greater than 21 and the player’s total is greater than the dealer’s then it must also be greater than 21. This means that in this situation, P(J> 21) = 1 and P(J> T)P(T> 21)P(J> 21) = P(J> T)P(T> 21).If the player’s total is less than 17 it cannot be equal to the dealer’s as P(T< 17) = 0. Therefore P(J= T< 21)P(J< 17).-1 = 0.Similarly, the player’s total cannot be greater than 21 and equal to the dealer’s total if it is less than 21. P(J= T< 21)P(J> 21).-1 = 0.After applying these changes we are left with:-1 + P(J<17) + P(J> 21) + 2P(T> 21) – 2P(T> 21)P(J< 17) – 2P(T> 21)P(J> 21) + 2P(J> T) – 2P(J> T)P(J> 21) – P(J> T)P(T> 21) + P(J> T)P(T> 21) + P(J= T< 21)-1 + P(J<17) + P(J> 21) + 2P(T> 21) – 2P(T> 21)P(J< 17) – 2P(T> 21)P(J> 21) + 2P(J> T) – 2P(J> T)P(J> 21) + P(J= T< 21)P(J>21).-1 just multiplies to give:-P(J>21)By combining the expressions above we are left with:2P(J<17)P(T>21) – P(J<17) – 1 + P(J<17) + P(J> 21) + 2P(T> 21) – 2P(T> 21)P(J< 17) – 2P(T> 21)P(J> 21) + 2P(J> T) – 2P(J> T)P(J> 21) + P(J= T< 21) – P(J>21)-1 + 2P(T> 21) – 2P(T> 21)P(J> 21) + 2P(J> T) – 2P(J> T)P(J> 21) + P(J= T< 21) We are able to factorise P(J> T) to get 2P(J> T).[1 – P(J>21)]. We know that 1 – P(J>21) ≡ P(J<21) so we can say that 2P(J> T).[1 – P(J>21)] = 2P(J> T)P(J<21). This is the probability that J is greater than T and less than or equal to 21 which can be expressed using the inequality 2P(T< J< 21). Our final equation for Ed,x is:Ed,x = -1 + 2P(T> 21) – 2P(T> 21)P(J> 21) + 2P(T< J< 21) + P(J= T< 21)Now we have Ed,x, we can subtract Es,x for the decision equation: 1 + 2P(T> 21) – 2P(T> 21)P(J> 21) + 2P(T< J< 21) + P(J= T< 21) – [2P(T> 21) – 1 + 2P(x> T) + P(x= T)]Ed,x – Es,x = -2P(T< x) - P(T= x) - 2P(T> 21)P(J> 21) + 2P(T< J< 21) + P(T= J< 21)For hard and unique hands, finding M(D) is a straightforward process. It is simply the smallest integer value of x for which Ed,x – Es,x <0. With M*(D), x can take two different values so, although we can continue to use the formula, there are a larger number of probability combinations. This makes calculating the expectation far more complex. It makes sense to calculate Es,x using the largest possible value of x. Ed,x uses the larger values of J for all probabilities where J< 21. When J exceeds 21, we then reduce the value of J by 10 so the player is no longer bust and continue to calculate Ed,x as a stiff total. When the player has a soft hand, it is less likely for him to bust. This is why the minimum standing values for soft hands are larger than for unique hands.When to split pairs or double down is more complex and requires us to make comparisons. The concept of doubling down and splitting pairs is the same. We compare |Ed,x – Es,x| of the cards before we split them with the sum of |Ed,x – Es,x| for both cards individually after it has been split. If the sum of |Ed,x – Es,x| for the individual cards is greater than |Ed,x – Es,x| of the cards when they are together then it is better to separate them. i.e. If |Ed,x – Es,x|i+ii - (|Ed,x – Es,x|i + |Ed,x – Es,x|ii )< 0 then you should double down.When splitting pairs as the value of the cards ‘i’ and ‘ii’ are the same, if i=ii we can write the expression for splitting pairs as:If |Ed,x – Es,x|2i - 2|Ed,x – Es,x|i <0 then you should split pairs.Using the decision equation:I wanted to attempt a decision calculation using the formula we derived. I assumed that it would be easier to calculate expectations for larger values of D as there would fewer combinations for the dealer’s total. Although Aces can be worth more than 10 valued cards, its value can change. This means that when the dealer would appear to have bust, the card value falls to 1, enabling an increasing number of card combinations. When D=10, there is only one situation when the Ace’s value can be 11, otherwise the Ace is always equal to 1. I decided to work out the expectations for when D=10 based on this theory. The value for x does not affect the calculation except for when it is a soft total. When the total is soft there are more possibilities for J as the ace can change its value. Assuming is easier to calculate my expectation for a hard or unique values of x, I decided to run my calculation for when x=19. 269176562865-514353415665To start, I used Excel to record all the outcomes for the dealer’s hand when his up-card is 10. I was able to use the dealer’s rule for hitting below 17 to work methodically. I would descend the value of the second card drawn until the value fell below 17. I would then add a third card, again working from highest to lowest values. I would add another card when required, always entered in descending value. This way I could be sure that I would not miss a single outcome. I used a column on the right hand side to record the total of the dealer’s hand, this made it easier to make comparisons between the dealer’s total and the players total x. I colour-coded my table to easily identify the regions where: the dealer is bust, the dealer wins, the player wins and there is draw. When I produced the first version of my table, I assumed that the probability of drawing each card was 1/52, regardless of the composition of the deck. This was, of course, a simplification and enabled me to quickly produce some rough probabilities in the column to the right of my totals. In a single deck game, there are 16 cards with a value of 10 and 4 cards of each of the other numerical values. For this specific situation, the dealer’s first card is always 10. I did not have to consider the probability of this card since it is fixed and equal to 1. I would then look at the following cards in the row. For a row (not considering D) where each of the card values were different, I would multiply the number of cards with a specific value in the deck by the probability of 1/52. For example, the probability of getting a card with a value of 2 is 4 multiplied by 1/52 or 4/52. To calculate the probability of a hand, I would combine probabilities using the ‘and’ rule. i.e. If the dealer’s hand was composed of ‘D’ as well as card valued 2 and 10, this would be 1*4/52*16/52 or 4*16*1/52*1/52. For hands containing cards of the same value, our combined probabilities are different. If we have previously drawn a card of the same value, the number of the cards of that value remaining in the pack has been reduced. This means if we have already drawn a card with a value of 3, the probability that we draw another is no longer 4/52 but 3/52. E.g. the probability of a hand of ‘D, 3, 3, 2’ is 1*4/52*3/52*4/52 or 4*3*4*1/52*1/52*1/52. The same is true for our probabilities if we consider permutations. Looking at the rows on my table, I would visualise the cards in a line:1094740-6350D233D is fixed so we do not need to consider it. A pack contains 4 3-valued cards and there are several ways in which these four cards can be arranged within the 2 spaces available. This is ‘4P2’ - equal to 12. There is only one position for a card of 2 and one way in which each of the four cards of that value can be arranged. This is 4P1 or just 4. The probability of this arrangement is 12*4*1/52*1/52*1/52*1/52 which gives us the same result as above. I found that this mental image correlated well to the format of my table and it was easier to visualise these permutations than to consider which cards had been removed from the pack. -4711701086485After I had completed calculating the probabilities of all of the outcomes on my table, I used my colour coding to group my table into sections and form new columns. I added columns that contained probabilities for all the outcomes that would result in: a draw, the player winning, the player losing, the dealer busting, and the dealer obtaining totals of 20 and 21. -534035321310I then used the ‘SUM’ function to add all the probabilities in each column:The dealer’s rules and the Aces’ ability to change value are just a couple of examples of game complications that contribute to errors. I converted each column total to real decimals by dividing by the summation of the total outcome probabilities. 4664075599440I then composed a separate table. I recorded a table for hitting on a total of 19 that listed all the values that J could take. Before computing my probabilities for J, I returned to the decision equation.Ed,x – Es,x = -2P(T< x) - P(T= x) - 2P(T> 21)P(J> 21) + 2P(T< J< 21) + P(T= J< 21)By looking at the equation and my table, I could see that already I was able to calculate P(T< x), P(T= x) and P(T> 21). I needed to calculate:P(J> 21), this was the sum of the probabilities that J takes a value from 22-29 inclusive. P(T< J< 21). There are only two values of J that are less than or equal to 21 and these are 21 and 20. If J is 21, T can be 17, 18, 19 or 20. If J is 20, T can be 17, 18 or 19. I can use the tables to find all the probabilities for T so I am only required to find P(J=20) and P(J=21).P(T= J< 21) which is when T and J are both equal to either 20 or 21. Again, I need to find P(J=20) and P(J=21). I then added columns for the probabilities of J and specifically for when J was bust, equal to 20 or equal to 21. I did all the calculations using the probability of each card being 1/52 and summed each probability column in the row below. Since the probability that x=19 in this specific situation is 1, I only had to consider the probability of drawing one card of each particular value.Ed,x – Es,x = -2P(T< x) - P(T= x) - 2P(T> 21)P(J> 21) + 2P(T< J< 21) + P(T= J< 21)I could then calculate the values of each of my probability coefficients:P(T<x) is when T is 17 or 18. I can calculate this by taking the probability of the dealers busting from the probability that the player wins. P(T=x) is the same as the probability of a draw.P(T>21)P(J>21) is the probability the dealer busts multiplied by the probability J busts.P(T< J< 21) is when T is 17,18 or 19 and J is 20 or T is 17, 18 or 19 and J is 21. P(T= J< 21) is the probability that J and T are equal to each other and less than or equal to 21.center4059555After I had found each of my probability coefficients, I was able to substitute them into my decision equation. My result was -0.67355 to five significant figures. This value is below zero which would suggest that it is more advantageous for the player to stand in this specific situation.I had completed the prototype for my table and now that I had some results, I could compare them to the values given in ‘The Optimum Strategy in Blackjack’. When I consulted the paper, I found that it contained no raw data, only column totals. They had used different column titles to me, but I could work out the equivalent by combining columns. This is a spreadsheet of my comparison: 14554206985My results from this spreadsheet were similar to those in the paper but I wanted to improve the accuracy of my results. Although ‘The Optimum Strategy in Blackjack’ had no raw data available in the PDF file or online, it did give a vague description of the methods they used to gather results. They mentioned that they first found ‘three card probabilities’. I understood this to imply that the first three cards are dealt to the dealer meaning we could use this information to find more accurate results. They then continued to describe that they used these ‘three card probabilities’ to find the results for hands of more than three cards. By assuming the probability of receiving any card after first three was 1/52 regardless of the number of cards drawn. In other words, this secondary step uses the principles of ‘equiprobability’ and ‘sampling with replacement’. I decided to make a separate spreadsheet for my own ‘three card probabilities’. I then combined these into my original spreadsheet and made the necessary alterations to the probabilities for hands with more than three by modifying the ‘three card probabilities’ I had already found. I started by changing my multiples of 1/52 to x/52*y/51*z/50. Since I had entered the formula as x*y*z*1/52*1/52*1/52, I only had to alter the second half which took less time. -4191001397635I knew that my result for the probability of the dealer obtaining a 10 valued card and an Ace was correct because the Optimum Strategy in Blackjack had given the value for a ‘natural’ as one of their column totals. We both obtained the same result of 0.78431 to five significant figures. I was unable to confirm that my other results were correct, but this gave me some assurance that I was on the right track with the method that I had chosen. After I had combined my three card probabilities with my spreadsheet and completed the necessary changes my column totals had altered quite considerably. When I made the same comparison as I made before, with my previous spreadsheet, I found that I had obtained the same results as Baldwin and his associates to two significant figures. 136334534290My mean difference score had also reduced by almost thirty times its original value. Using my new column totals with the J values I had derived before, I could obtain a new result for Ed,x – Es,x. With my latest results, it remains that it is better to stand than to hit when D is 10 and x is 19. Looking at any basic strategy table, this is always the case for hard or unique totals of 19, so I know that my value should be negative. Because I had no raw data to compare, I cannot be sure whether my J values or other probabilities are correct. I can only estimate that the source of the difference in results is down to variation in method or assumptions; I have more than likely misinterpreted the description of the method used in the manuscript. There is a slight possibility that my own data is more accurate than Baldwin’s; the technology they had at their disposal in 1956 was almost certainly less advanced than Microsoft Excel.The minimum standing values found by the four horsemen of blackjack are as follows...For hard and unique hands:12 when D is 4, 5, 6.13 when D is 2, 3.17 when D is 7, 8, 9, 10, A.For soft hands:18 when D is 2, 3, 4, 5, 6, 7, 8, A.19 is D is 9, 10.This can be clearly expressed in a table: 1424940393702769870714375When combined with their findings for splitting pairs and doubling down, they were able to publish several basic strategy tables in their book ‘Playing Blackjack to win’. Here is an example of a strategy table used when playing blackjack:All the hands listed with an ‘A’ are soft hands. S – StandH – HitD – Double downSP – Split PairsSoon after the manuscript had been published, it attracted attention from no other than Edward Thorpe, professor at Massachusetts Institute of Technology. Baldwin’s team had initially analysed the house edge over the player to be approximately 0.6% but upon investigation by Thorpe, he identified the actual house edge to be -0.09% (meaning the player is more likely to win). This advantage is slight but nevertheless it is still an advantage especially when we take into account the 3:2 pay out for a natural. When combined with an effective betting strategy, these tables alone could be successfully used to win at Blackjack. I decided to test the use of basic strategy tables for myself using the tables on page 45 - 46 of ‘Blackbelt in Blackjack’ by Arnold Snyder. I had an idea to use random number simulations to test the tables, an example of this can be found in the appendix. I found that due to the nature of the probabilities changing each time a card was removed from the pack, this method was very slow and it was much faster to complete these simulations by hand. I played a total of 100 single-deck games (I treated both splitting pairs and doubling down as two separate games). This only took around forty minutes to play by hand and record a tally of games won, lost and drawn. My results were as follows: I won 59 games, drew 11 and lost 40. My simulations correlated to the advantage predicted by Thorpe but surpassed the expected advantage by a large margin (19.19% is far greater than 0.09%). Due to this comparison, I think that the successful results from my simulations would be difficult to repeat.Card CountingCard counting is a method that monitors the probability of the player receiving preferable cards (cards that give the player an advantage). This is based on the principle of the ratio of high to low-valued cards. This ratio is important because in blackjack, low cards favour the dealer and high cards favour the player. In blackjack, the dealer is bound by the game rules to hit his ‘stiff hands’ regardless of the composition of the deck. Hands are called ‘stiff’ if they have a total of below 17 and are able to bust; 12, 13,14,15,16. A hand of 11 is not stiff because if the player receives an ace, the value of the card automatically falls from 11 to 1 so the player cannot bust.If the deck has a greater proportion of high valued cards, this gives a player advantage because the dealer is more likely to bust when he hits his stiffs. High cards also make gaining a natural more likely. Although the probabilities of the player and the dealer getting a natural are the same, this is still more advantageous for the player due to the bonus payout for a blackjack. When the player obtains a blackjack he is paid an amount equal to one and a half times his original bet whereas the dealer only receives the initial value. Low cards favour the dealer because he is more likely to obtain winning totals on his hands when he is stiff. When counting cards, we assign each card a positive or negative value (usually +1 or -1) depending on whether the card is high or low. If when all the cards in a deck have been counted and the total is equal to zero, this count is said to be balanced. In the same way, if the sum of all cards in a deck is not equal to zero, this is known as an unbalanced count. During the game, an individual keeps a running count of all the cards that have been played. In this way, he knows whether there are a larger proportion of high or low cards in the deck and can use this information to tell whether his advantage had changed and whether he should alter his bet. Counting systems not only tell us what we should bet, but also dictate some of the decisions we make while playing. We can also use the information from our count to make slight alterations to our generic basic strategy. Over the years many counting systems have been developed with a range of complexities. As with basic strategy, an individual may choose a system that suits their own unique playing style. One of the simplest and most effective counting systems is the ‘Red Seven Count’. This was devised by Blackjack Hall of Famer Arnold Snyder. Unlike the other aficionados that I have mentioned in this project, Snyder had no mathematical background whatsoever and initially had a job as a postman. Today, he is a world renowned Blackjack player and consultant, and is recognised for several books about the game as well as his contribution to various playing strategies. The Red Seven CountThe red seven count assigns point values to specific cards as they are removed from the deck. These values are as follows:Aces and 10s (-1)9s, 8s and Black 7s (0)Red 7s, 6s, 5s, 4s, 3s and 2s (+1)You can see that if you add up all the values of the cards in the pack, we are left with a total of +2. This means that the count is unbalanced and we can use +2 as a pivot. Of course, if there is more than one deck, the pivot is multiplied by the number of decks in play (e.g. if there are four decks in play, the pivot is 2x4 which is 8. To start counting cards take a normal 52 card deck and begin, one by one, turning cards face-up. After all the cards have been discarded, the running count should be +2. (If your remaining total is not +2, you have made a mistake.) Your ability to count cards should improve quickly with practice –after a while, visual recall kicks in and you will begin to automatically associate cards with their count value. As you become faster at counting you will find that you are able to count more than one card at a time, looking for pairs that might cancel each other out. Before you begin to count in casinos, Arnold Snyder recommends that you are able to count down a pack of cards in under 40 seconds. (Most professional counters are able to do this in under 30). You can choose to make your pivot equal to zero, by starting your count at -2 multiplied by the number of decks in use. When you are using a balanced count and your running total is equal to zero, this indicates a normal house edge. When using an unbalanced count such as this one, zero indicates that the player advantage has risen by approximately 1% over the house edge that you started with. According to ‘Blackbelt in Blackjack’, about 80% of games have a starting advantage of 0.4 – 0.6%. This means that when the running count is zero, the player has an advantage of around 0.5%. Tables are available to help memorise more accurate advantages for specific games and numbers of decks.17786351101090We can analyse the game using a count system and use our running count to help us bet strategically. When our advantage over the house increases, so should our bet. As a general rule, betting using a system of units; if the running count is 0 or less, we bet our unit amount. If the running count is above zero, we bet a multiple of units. We can learn the multiple of units we should bet by memorising betting tables such as this one:A ‘shoe’ is a device used by casinos to hold multiple decks (normally six or eight).Count systems can also be used to modify our playing strategy and adapt to the constantly changing composition of the deck. For example, if our running count is negative then we know that there are a greater proportion of low valued cards in the deck. This means that we (and the dealer) are less likely to bust when we hit our stiffs. In this situation, our minimum standing values could have changed, meaning it is necessary to make alterations to our basic strategy. To know when to change our existing strategy, we use specific count strategy tables. The red seven count is considered one of the easiest to use because there are few index changes to remember (a table is not needed):In games with one or two decks, take insurance at a running count of 0 or more.In shoe games take insurance at a running count of +2.In any game, when the running count is 0 or more, stand when x=16 and D=10.In any game, when the running count is 0 or more, stand when x=12 and D=3.In any game, when the running count is +2 or more, stand when x=12 and D=2.In any game, when the running count is +2 or more, stand when x=15 and D=10.In any game, when the running count is +2 or more, double down when x=10+Y and D=10. These red seven index changes allow the player to take advantage of around 80% of possible gains from card counting. The player will play according to basic strategy except for these few specific situations when changes are required. Advanced versions of the red seven strategy tables are available, but the simplicity of the system is merit to its success. To calculate your increase in advantage more accurately at any point in the game, you must be able to accurately estimate the number of decks remaining to the nearest half a deck. The advantage (%) is defined as:running count2(number of decks remaining)This shouldn’t be difficult but if you feel that your mental maths is weak, you could memorise all the values for 1/2(number of decks remaining), then just multiply by your running count. If you find that you have a fraction such as 5/9, you just need to be able to tell whether that fraction is greater than or less than 1/2. You should be able to estimate your advantage to the nearest 1/2%. Remember that this value is the increase in advantage over the advantage at your pivot. If when your count is 0, you have an advantage of 1/2% and your increase in advantage is 1/2% then your total advantage over the house is 1%. It is important to be able to calculate your advantage for a number of betting systems and strategies. Other counting systems are available: Here is a chart showing a comparison between several counting systems. The table gives information about the following measurements:Card counting techniques include the following measurements:Playing Efficiency?–?‘PE’?indicates how well a counting system handles changes in playing strategy. Playing efficiency is particularly important in hand-held games (one or two decks.)Betting Correlation?–?‘BC’ is defined as the correlation between card point values and the effect of removal of cards. It is used to predict how well a counting system predicts good betting situations and can approach 1.00 (100% correlation.) BC is particularly important in shoe games (six or eight decks.)Insurance Correlation?–?‘IC’?is defined as the correlation between card point values and the value of cards in Insurance situation. A point value of -9 for tens and +4 for all other cards would be perfect for predicting if an Insurance bet should be placed.Level?–?The level of a strategy refers to the number of different values assigned to cards. Level 2 and 3 counting strategies are more efficient, but quite a bit more difficult for most people. Level 4 & 5 counting strategies also exist. But this is overkill.Side Counts?–?There are several methods of side counting used to increase strategy efficiency. Ease – ease is rated from 1(hard) to 10(easy).Type?–?The TYPE column key follows:B?–?Balanced count requiring the calculation of a True CountU?–?Unbalanced count with no True CountS?–?Suit Aware count requiring different counts for red and black cardsC?–?Compromise indexes used for greater ease 1, 2, 3, 4?–?Level. That is the highest tag value (index assigned to a card).StrategyA23456789TBCPEICEaseTypeCanfield Expert00111110-1-1.87.63.766B1Canfield Master01122210-1-2.92.67.854B2Hi-Lo-111111000-1.97.51.766B1Hi-Opt I001111000-1.88.61.856.5B1Hi-Opt II011221100-2.91.67.914B2KISS 200/11111000-1.90.62.877US1KISS 3-10/11111100-1.98.56.787US1K-O-111111100-1.98.55.787.5UC1Mentor-11222210-1-2.97.62.804B2Omega II01122210-1-2.92.67.854B2Red Seven-1111110/100-1.98.54.787USC1REKO-111111100-1.98.55.788UC1Revere Adv. Plus-Minus01111100-1-1.89.59.766B1Revere Point Count-212222100-2.99.55.784B2Revere RAPC-42334320-1-31.0.53.711B4Revere 14 Count02234210-2-3.92.65.821B4Silver Fox-11111110-1-1.96.53.696B1Unbalanced Zen 2-112222100-2.97.62.846.5U2Uston Adv. Plus-Minus-101111100-1.95.55.766.5B1Uston APC01223221-1-3.91.69.902.5B3Uston SS-22223210-1-2.99.54.734.5U3Wong Halves-1.5111.51.50-.5-1.99.56.722.5B3Zen Count-111222100-2.96.63.854B2Information regarding all of these counts can be found online or in subject specific books. True count conversions take into account the removal of cards in proportion to the number of cards being played. True count conversion requires the player to estimate the number of remaining decks and be able to form fractions without use of a calculator. There should be information on true count conversion in any card counting guidebooks and online.Kelly CriterionIn theory, Kelly betting is a system where you are unable to go bankrupt. By only betting a small proportion of your total funds (equal to the percentage advantage of the player), it is technically impossible for your entire bankroll to be depleted. This would seem like the ideal betting strategy but it isn’t. When the Kelly Criterion is applied as an algorithm, the player will always be able to achieve his desired profit however; the reality of this betting system is quite different. Here are some of the major issues associated with Kelly Betting:As the goal profit increases toward infinity, so does the number of games needed to be played in order to achieve that goal and therefore the number of playing hours. When applied as an algorithm the player is unable to go bankrupt, however there will be large fluctuations in the size of the bankroll before achieving the goal profit. Although it is improbable, these fluctuations can be so large that the bankroll decreases to less than that of the minimum betting amount. If you are part of a group that is funded by a third party, they may pull the plug on the investment if the size of the bankroll falls below a critical value.If there is a house edge, then the percentage advantage is negative. The ideal bet in this situation is zero (or negative but taking insurance is risky and the conditions for taking insurance have been outlined in your count strategy). In casino Blackjack, you are unable to bet less than the minimum betting amount. The necessity to bet when the house holds the advantage increases the probability for you to bankrupt.Due to the complications above, Kelly betting is considered too risky by many professionals. In general, many experts choose to bet a proportion of the Kelly bet – usually half or a quarter of the defined amount. Full, half and quarter Kelly charts are available to accurately determine player decisions. These can easily be found online or in many books relating to Blackjack strategy.What’s the risk? Gambler’s ruinThe ‘Risk of Ruin’ formula is an equation that determines the probability of an individual either doubling his bankroll or going bankrupt during the process. It is often used in trading and other casino games such as poker. Gamblers ruin takes into account his average advantage during the game, the number of units that compose his bankroll, and the size of his bets and often the number of hands to play. Basing bets on, for example, a 10% element of ruin i.e. nine times out of ten you would double your bankroll, would present a gambler with a large profit over a long term investment period. However, say there are 500 players playing at a 10% RoR, this means that 50 players will go broke trying to do so! This may seem like a small proportion but of these 50 players, how many of them can afford to replicate another bankroll? This formula may be available online (though I can only find the version regarding trading) and is discussed in detail in ‘Million Dollar Blackjack’ by Ken Uston (according multiple sources). Tables regarding bet sizing based on the Risk of Ruin formula, number of units and hands to play can be found at The Wizard of Odds online (address found in bibliography). The risk of ruin is said to be greatest at the start of the game.Do the casinos let you count cards?Playing Blackjack as a profession is a risk in itself. It is not illegal to think in casinos, however, casino employees will try to prevent you from counting cards. Gambling houses are at liberty to ban anyone that they think may have an unfair advantage whilst playing and warn other casinos in the area. It is fairly common for those with a large wining streak to be ‘back roomed’ so it is important for players to know their rights (these may vary depending on your location and the country/state laws that are in place). Back rooming generally just involves questioning followed by verbal warning but on rare occasions, casinos have been known to resort to more physical methods of deterrence. Surveillance systems will not only enable the casinos to send images of players to neighbouring businesses, but will also contribute towards monitoring and catching card counters during play. Certain casino employees will be trained in game strategies so that they can identify individuals that make all the appropriate counting decisions. Professional Blackjack players lean how to play in a way that they go undetected by the casinos. This often involves combining and alternating certain strategies – they may even make mistakes deliberately in order to seem like more of a reckless gambler. One technique, ‘shuffle tracking’, when used effectively, is able to discourage casino surveillance whilst increasing your advantage. This technique is very difficult to master, but involves counting cards in order to identify regions of the deck that are highly saturated with tens and aces. When the cards are cut and/or shuffled, the player can suddenly increase his bet when these regions come into play. To any casino official, raising bets after shuffling or cutting the deck would seem like a thoughtless decision – certainly not one that any card counter would make. The concept would seem almost impossible, but it has been proved that this technique (when able to be carried out successfully) has a detrimental effect. Since developing this strategy, Arnold Snyder has written an entire report ‘The Blackjack Shuffle Tracker’s Cookbook’ dedicated to the subject. Blackbelt in Blackjack also features a fairly detailed section on this technique.Covering or concealing facial features from security cameras is a topic that causes conflict among professional players. Although disguising yourself will mean that you can return to a casino after being removed from the premises, the nature of your appearance may arouse suspicion if it seems as if you have something to hide. ‘Card Counter’s Guide to Casino Surveillance’ by D.V Cellini is advertised by Cadoza Publishing as a specialist guide on the inner workings of the Casinos. Game variationsSome game variations have different rules to those outlined in the basic strategy chapter. Some of these versions are unable to be beaten by mental methods however, most of these games just require a slightly different approach. Game variations to be avoided include:Double exposureSuperFun 21Spanish 21No-Bust BlackjackPlayer-Banked BlackjackAdvice and strategy changes for common game variations that offer side bets can be found online. Some of these games can have a lager opportunity for profit than traditional variations if played correctly.Would I Gamble?Before starting this project, I was under the impression that, with practice, I would be able to make money by playing Blackjack. Now that I have completed my investigation, I still believe that I could profit from playing the game. I would never gamble as a profession, but as a recreational activity, I would like to test the concepts that I have explored and analysed this year. I could never have predicted how much my knowledge would have expanded during the process of my EPQ. I enjoyed the challenging nature of all the maths involved in this report and believe that the problems I managed to overcome have contributed to the pride I feel upon completion.I will continue to explore my love of Blackjack by publishing this project as an online blog. I hope that this will present the opportunity for mutual exchange of information between me and any followers I might obtain. This should allow me to develop my understanding further and perhaps locate the source of error in my spreadsheet. I feel that there are no existing Blackjack websites that break down the derivation of basic strategy in a way that is both in-depth and easily understood. I believe that the mathematical approach to my research and personal nature of my report could allow my blog to gain success in the years to come. BibliographyBaldwin, R. et al. (2008). Playing Blackjack to Win. Las Vegas: Cardoza Publishing.Mezrich, B. (2004). Bringing Down the House. London: Arrow Books. Schneider, A. (2012). Blackbelt in Blackjack. Las Vegas: Cardoza Publishing.Thorpe, E. (1966). Beat the Dealer. New York: Vintage Books, Random House Inc. active (01/03/2015) active (01/03/2015) active (01/03/2015) ................
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