Mathematical Formula For Blackjack

1. Mathematical Formula For Blackjack For Dummies
2. Mathematical Formula For Blackjack Games
3. Mathematical Formula For Blackjack Card Game
4. Mathematical Formula For Blackjack Rules

Probability of obtaining a natural blackjack is P = 8/663 = 1.20663% in the case of a 1-deck game and P = 16/1339 = 1.19492% in the case of a 2-deck game. Probability of obtaining a blackjack from the first two cards is P = 32/663 = 4.82654% in the case of a 1-deck game and P = 64/1339= 4.77968% in the case of a 2-deck game. Fundamental Formula of Gambling is the essence of gambling mathematics. The probability formula is a precise instrument in theory of games, gambling, randomness, inexistence of God. Failing to understand the basic mathematics of the games and their relationships to casino profitability. One casino owner would often test his pit bosses by asking how a casino could make money on blackjack if the outcome is determined simply by whether the player or the dealer came closest to 21. The answer, typically, was. Using Math in Baccarat. As mentioned above, there are a number of different formulas that you can try out in baccarat. One of these is a formula that Allan N. Wilson laid out in the book ‘The Casino Gambler's Guide'. The house advantages mentioned earlier in this article are derived by using this formula.

The mathematics of gambling are a collection of probability applications encountered in games of chance and can be included in game theory. From a mathematical point of view, the games of chance are experiments generating various types of aleatory events, the probability of which can be calculated by using the properties of probability on a finite space of events.

Experiments, events, probability spaces[edit]

The technical processes of a game stand for experiments that generate aleatory events. Here are a few examples:

• Throwing the dice in craps is an experiment that generates events such as occurrences of certain numbers on the dice, obtaining a certain sum of the shown numbers, and obtaining numbers with certain properties (less than a specific number, higher than a specific number, even, uneven, and so on). The sample space of such an experiment is {1, 2, 3, 4, 5, 6} for rolling one die or {(1, 1), (1, 2), .., (1, 6), (2, 1), (2, 2), .., (2, 6), .., (6, 1), (6, 2), .., (6, 6)} for rolling two dice. The latter is a set of ordered pairs and counts 6 x 6 = 36 elements. The events can be identified with sets, namely parts of the sample space. For example, the event occurrence of an even number is represented by the following set in the experiment of rolling one die: {2, 4, 6}.
• Spinning the roulette wheel is an experiment whose generated events could be the occurrence of a certain number, of a certain color or a certain property of the numbers (low, high, even, uneven, from a certain row or column, and so on). The sample space of the experiment involving spinning the roulette wheel is the set of numbers the roulette holds: {1, 2, 3, .., 36, 0, 00} for the American roulette, or {1, 2, 3, .., 36, 0} for the European. The event occurrence of a red number is represented by the set {1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36}. These are the numbers inscribed in red on the roulette wheel and table.
• Dealing cards in blackjack is an experiment that generates events such as the occurrence of a certain card or value as the first card dealt, obtaining a certain total of points from the first two cards dealt, exceeding 21 points from the first three cards dealt, and so on. In card games we encounter many types of experiments and categories of events. Each type of experiment has its own sample space. For example, the experiment of dealing the first card to the first player has as its sample space the set of all 52 cards (or 104, if played with two decks). The experiment of dealing the second card to the first player has as its sample space the set of all 52 cards (or 104), less the first card dealt. The experiment of dealing the first two cards to the first player has as its sample space a set of ordered pairs, namely all the 2-size arrangements of cards from the 52 (or 104). In a game with one player, the event the player is dealt a card of 10 points as the first dealt card is represented by the set of cards {10♠, 10♣, 10♥, 10♦, J♠, J♣, J♥, J♦, Q♠, Q♣, Q♥, Q♦, K♠, K♣, K♥, K♦}. The event the player is dealt a total of five points from the first two dealt cards is represented by the set of 2-size combinations of card values {(A, 4), (2, 3)}, which in fact counts 4 x 4 + 4 x 4 = 32 combinations of cards (as value and symbol).
• In 6/49 lottery, the experiment of drawing six numbers from the 49 generates events such as drawing six specific numbers, drawing five numbers from six specific numbers, drawing four numbers from six specific numbers, drawing at least one number from a certain group of numbers, etc. The sample space here is the set of all 6-size combinations of numbers from the 49.
• In draw poker, the experiment of dealing the initial five card hands generates events such as dealing at least one certain card to a specific player, dealing a pair to at least two players, dealing four identical symbols to at least one player, and so on. The sample space in this case is the set of all 5-card combinations from the 52 (or the deck used).
• Dealing two cards to a player who has discarded two cards is another experiment whose sample space is now the set of all 2-card combinations from the 52, less the cards seen by the observer who solves the probability problem. For example, if you are in play in the above situation and want to figure out some odds regarding your hand, the sample space you should consider is the set of all 2-card combinations from the 52, less the three cards you hold and less the two cards you discarded. This sample space counts the 2-size combinations from 47.

The probability model[edit]

A probability model starts from an experiment and a mathematical structure attached to that experiment, namely the space (field) of events. The event is the main unit probability theory works on. In gambling, there are many categories of events, all of which can be textually predefined. In the previous examples of gambling experiments we saw some of the events that experiments generate. They are a minute part of all possible events, which in fact is theset of all parts of the sample space.

For a specific game, the various types of events can be:

• Events related to your own play or to opponents' play;
• Events related to one person's play or to several persons' play;
• Immediate events or long-shot events.

Each category can be further divided into several other subcategories, depending on the game referred to. These events can be literally defined, but it must be done very carefully when framing a probability problem. From a mathematical point of view, the events are nothing more than subsets and the space of events is a Boolean algebra. Among these events, we find elementary and compound events, exclusive and nonexclusive events, and independent and non-independent events.

In the experiment of rolling a die:

• Event {3, 5} (whose literal definition is occurrence of 3 or 5) is compound because {3, 5}= {3} U {5};
• Events {1}, {2}, {3}, {4}, {5}, {6} are elementary;
• Events {3, 5} and {4} are incompatible orexclusive because their intersection is empty; that is, they cannot occur simultaneously;
• Events {1, 2, 5} and {2, 5} are nonexclusive, because their intersection is not empty;
• In the experiment of rolling two dice one after another, the events obtaining 3 on the first die and obtaining 5 on the second die are independent because the occurrence of the second event is not influenced by the occurrence of the first, and vice versa.

In the experiment of dealing the pocket cards in Texas Hold'em Poker:

• The event of dealing (3♣, 3♦) to a player is an elementary event;
• The event of dealing two 3's to a player is compound because is the union of events (3♣, 3♠), (3♣, 3♥), (3♣, 3♦), (3♠, 3♥), (3♠, 3♦) and (3♥, 3♦);
• The events player 1 is dealt a pair of kings and player 2 is dealt a pair of kings are nonexclusive (they can both occur);
• The events player 1 is dealt two connectors of hearts higher than J and player 2 is dealt two connectors of hearts higher than J are exclusive (only one can occur);
• The events player 1 is dealt (7, K) and player 2 is dealt (4, Q) are non-independent (the occurrence of the second depends on the occurrence of the first, while the same deck is in use).

These are a few examples of gambling events, whose properties of compoundness, exclusiveness and independency are easily observable. Theseproperties are very important in practical probability calculus.

The complete mathematical model is given by the probability field attached to the experiment, which is the triple sample space—field of events—probability function. For any game of chance, the probability model is of the simplest type—the sample space is finite, the space of events is the set of parts of the sample space, implicitly finite, too, and the probability function is given by the definition of probability on a finite space of events:

Combinations[edit]

Games of chance are also good examples of combinations, permutations and arrangements, which are met at every step: combinations of cards in a player's hand, on the table or expected in any card game; combinations of numbers when rolling several dice once; combinations of numbers in lottery and bingo; combinations of symbols in slots; permutations and arrangements in a race to be bet on, and the like. Combinatorial calculus is an important part of gambling probability applications. In games of chance, most of the gambling probability calculus in which we use the classical definition of probability reverts to counting combinations. The gaming events can be identified with sets, which often are sets of combinations. Thus, we can identify an event with a combination.

For example, in a five draw poker game, the event at least one player holds a four of a kind formation can be identified with the set of all combinations of (xxxxy) type, where x and y are distinct values of cards. This set has 13C(4,4)(52-4)=624 combinations. Possible combinations are (3♠ 3♣ 3♥ 3♦ J♣) or (7♠ 7♣ 7♥ 7♦ 2♣). These can be identified with elementary events that the event to be measured consists of.

Expectation and strategy[edit]

Games of chance are not merely pure applications of probability calculus and gaming situations are not just isolated events whose numerical probability is well established through mathematical methods; they are also games whose progress is influenced by human action. In gambling, the human element has a striking character. The player is not only interested in the mathematical probability of the various gaming events, but he or she has expectations from the games while a major interaction exists. To obtain favorable results from this interaction, gamblers take into account all possible information, including statistics, to build gaming strategies. The oldest and most common betting system is the martingale, or doubling-up, system on even-money bets, in which bets are doubled progressively after each loss until a win occurs. This system probably dates back to the invention of the roulette wheel. Two other well-known systems, also based on even-money bets, are the d'Alembert system (based on theorems of the French mathematician Jean Le Rond d'Alembert), in which the player increases his bets by one unit after each loss but decreases it by one unit after each win, and the Labouchere system (devised by the British politician Henry Du Pré Labouchere, although the basis for it was invented by the 18th-century French philosopher Marie-Jean-Antoine-Nicolas de Caritat, marquis de Condorcet), in which the player increases or decreases his bets according to a certain combination of numbers chosen in advance.^[1][2] The predicted average gain or loss is called expectation or expected value and is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds are repeated many times. A game or situation in which the expected value for the player is zero (no net gain nor loss) is called a fair game. The attribute fair refers not to the technical process of the game, but to the chance balance house (bank)–player.

Even though the randomness inherent in games of chance would seem to ensure their fairness (at least with respect to the players around a table—shuffling a deck or spinning a wheel do not favor any player except if they are fraudulent), gamblers always search and wait for irregularities in this randomness that will allow them to win. It has been mathematically proved that, in ideal conditions of randomness, and with negative expectation, no long-run regular winning is possible for players of games of chance. Most gamblers accept this premise, but still work on strategies to make them win either in the short term or over the long run.

House advantage or edge[edit]

Casino games provide a predictable long-term advantage to the casino, or 'house' while offering the player the possibility of a large short-term payout. Some casino games have a skill element, where the player makes decisions; such games are called 'random with a tactical element.' While it is possible through skillful play to minimize the house advantage, it is extremely rare that a player has sufficient skill to completely eliminate his inherent long-term disadvantage (the house edge or house vigorish) in a casino game. The common belief is that such a skill set would involve years of training, extraordinary memory, and numeracy, and/or acute visual or even aural observation, as in the case of wheel clocking in Roulette. For more examples see Advantage gambling.

The player's disadvantage is a result of the casino not paying winning wagers according to the game's 'true odds', which are the payouts that would be expected considering the odds of a wager either winning or losing. For example, if a game is played by wagering on the number that would result from the roll of one die, true odds would be 5 times the amount wagered since there is a 1/6 probability of any single number appearing. However, the casino may only pay 4 times the amount wagered for a winning wager.

The house edge (HE) or vigorish is defined as the casino profit expressed as a percentage of the player's original bet. In games such as Blackjack or Spanish 21, the final bet may be several times the original bet, if the player doubles or splits.

Example: In American Roulette, there are two zeroes and 36 non-zero numbers (18 red and 18 black). If a player bets $1 on red, his chance of winning $1 is therefore 18/38 and his chance of losing $1 (or winning -$1) is 20/38.

The player's expected value, EV = (18/38 x 1) + (20/38 x -1) = 18/38 - 20/38 = -2/38 = -5.26%. Therefore, the house edge is 5.26%. After 10 rounds, play $1 per round, the average house profit will be 10 x $1 x 5.26% = $0.53.Of course, it is not possible for the casino to win exactly 53 cents; this figure is the average casino profit from each player if it had millions of players each betting 10 rounds at $1 per round.

The house edge of casino games varies greatly with the game. Keno can have house edges up to 25% and slot machines can have up to 15%, while most Australian Pontoon games have house edges between 0.3% and 0.4%.

The calculation of the Roulette house edge was a trivial exercise; for other games, this is not usually the case. Combinatorial analysis and/or computer simulation is necessary to complete the task.

In games that have a skill element, such as Blackjack or Spanish 21, the house edge is defined as the house advantage from optimal play (without the use of advanced techniques such as card counting or shuffle tracking), on the first hand of the shoe (the container that holds the cards). The set of the optimal plays for all possible hands is known as 'basic strategy' and is highly dependent on the specific rules, and even the number of decks used. Good Blackjack and Spanish 21 games have to house edges below 0.5%.

Online slot games often have a published Return to Player (RTP) percentage that determines the theoretical house edge. Some software developers choose to publish the RTP of their slot games while others do not.^[3] Despite the set-theoretical RTP, almost any outcome is possible in the short term.^[4] RTP is the percentage of the jackpot or prize money that can be regurgitated from the slot machine. Say the dealer does a 20% set, then there is a potential for jackpot 1: 5 to come out, but the fact is of course not this big considering the patterns or symbols that exist in online slot games are so diverse. ^[5]

Standard deviation[edit]

The luck factor in a casino game is quantified using standard deviation (SD). The standard deviation of a simple game like Roulette can be simply calculated because of the binomial distribution of successes (assuming a result of 1 unit for a win, and 0 units for a loss). For the binomial distribution, SD is equal to npq{displaystyle {sqrt {npq}}}, where n{displaystyle n} is the number of rounds played, p{displaystyle p} is the probability of winning, and q{displaystyle q} is the probability of losing. Furthermore, if we flat bet at 10 units per round instead of 1 unit, the range of possible outcomes increases 10 fold. Therefore, SD for Roulette even-money bet is equal to 2bnpq{displaystyle 2b{sqrt {npq}}}, where b{displaystyle b} is the flat bet per round, n{displaystyle n} is the number of rounds, p=18/38{displaystyle p=18/38}, and q=20/38{displaystyle q=20/38}.

After enough large number of rounds the theoretical distribution of the total win converges to the normal distribution, giving a good possibility to forecast the possible win or loss. For example, after 100 rounds at $1 per round, the standard deviation of the win (equally of the loss) will be 2⋅$1⋅100⋅18/38⋅20/38≈$9.99{displaystyle 2cdot $1cdot {sqrt {100cdot 18/38cdot 20/38}}approx $9.99}. After 100 rounds, the expected loss will be 100⋅$1⋅2/38≈$5.26{displaystyle 100cdot $1cdot 2/38approx $5.26}.

The 3 sigma range is six times the standard deviation: three above the mean, and three below. Therefore, after 100 rounds betting $1 per round, the result will very probably be somewhere between −$5.26−3⋅$9.99{displaystyle -$5.26-3cdot $9.99} and −$5.26+3⋅$9.99{displaystyle -$5.26+3cdot $9.99}, i.e., between -$34 and $24. There is still a ca. 1 to 400 chance that the result will be not in this range, i.e. either the win will exceed $24, or the loss will exceed $34.

The standard deviation for the even-money Roulette bet is one of the lowest out of all casinos games. Most games, particularly slots, have extremely high standard deviations. As the size of the potential payouts increase, so does the standard deviation.

Unfortunately, the above considerations for small numbers of rounds are incorrect, because the distribution is far from normal. Moreover, the results of more volatile games usually converge to the normal distribution much more slowly, therefore much more huge number of rounds are required for that.

As the number of rounds increases, eventually, the expected loss will exceed the standard deviation, many times over. From the formula, we can see the standard deviation is proportional to the square root of the number of rounds played, while the expected loss is proportional to the number of rounds played. As the number of rounds increases, the expected loss increases at a much faster rate. This is why it is practically impossible for a gambler to win in the long term (if they don't have an edge). It is the high ratio of short-term standard deviation to expected loss that fools gamblers into thinking that they can win.

The volatility index (VI) is defined as the standard deviation for one round, betting one unit. Therefore, the VI for the even-money American Roulette bet is 18/38⋅20/38≈0.499{displaystyle {sqrt {18/38cdot 20/38}}approx 0.499}.

The variance v{displaystyle v} is defined as the square of the VI. Therefore, the variance of the even-money American Roulette bet is ca. 0.249, which is extremely low for a casino game. The variance for Blackjack is ca. 1. Grand casino gulfport buffet. 2, which is still low compared to the variances of electronic gaming machines (EGMs).

Additionally, the term of the volatility index based on some confidence intervals are used. Usually, it is based on the 90% confidence interval. The volatility index for the 90% confidence interval is ca. 1.645 times as the 'usual' volatility index that relates to the ca. 68.27% confidence interval.

It is important for a casino to know both the house edge and volatility index for all of their games. The house edge tells them what kind of profit they will make as percentage of turnover, and the volatility index tells them how much they need in the way of cash reserves. The mathematicians and computer programmers that do this kind of work are called gaming mathematicians and gaming analysts. Casinos do not have in-house expertise in this field, so they outsource their requirements to experts in the gaming analysis field.

Mathematical Formula For Blackjack For Dummies

See also[edit]

References[edit]

1. ^'Roulette'. britannica.
2. ^'D'Alembert roulette system'.
3. ^'Online slots Return to Player (RTP) explained - GamblersFever'.
4. ^'Return to Player and Hit frequency - What do these mean? - GetGamblingFacts'.
5. ^https://www.okslot88.com/hal-yang-perlu-di-ketahui-tentang-mesin-slot-online/

Further reading[edit]

• The Mathematics of Gambling, by Edward Thorp, ISBN0-89746-019-7
• The Theory of Gambling and Statistical Logic, Revised Edition, by Richard Epstein, ISBN0-12-240761-X
• The Mathematics of Games and Gambling, Second Edition, by Edward Packel, ISBN0-88385-646-8
• Probability Guide to Gambling: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery and Sport Bets, by Catalin Barboianu, ISBN973-87520-3-5excerpts
• Luck, Logic, and White Lies: The Mathematics of Games, by Jörg Bewersdorff, ISBN1-56881-210-8introduction.

External links[edit]

Ask any casual gambler about the best way to beat the house, and you'll inevitably hear tales involving intrepid card counters.

From the famed MIT Blackjack Team — which was immortalized in the film 21 (2008) starring Kevin Spacey as a math professor turned count team leader — to the instructional material published by the legendary gambling author Stanford Wong, card counting has been embedded in the public consciousness for decades.

And at first glance, that makes perfect sense…

All it takes is a quick back-of-the-napkin lesson to get the gist of counting cards at the blackjack table.

When you see plenty of high cards hit the felt as the dealer does their thing, you know the odds of hitting high totals like 18, 19, 20, and 21 decrease accordingly. Thus, you should lower your bets to the table minimum.

Conversely, when your scans show a ton of low cards getting dealt, it's time to ramp up the betting because your odds of hitting premium totals have now gone up.

After hearing this brief introduction, thousands of gamblers strike out to Sin City each and every year hoping to try their hand at the blackjack tables. Many even make a point to study card counting systems, of which literally hundreds have been devised over the last few decades.

It all began back in 1962 when math genius Edward O. Thorp invented the first effective method for counting cards and adjusting one's blackjack plays accordingly. Thorp's 'Ten Count' counting system established the template, and from there, dozens of blackjack sharps and math whizzes have taken their own shot at creating the best way to count cards.

Most Popular Card Counting Systems

• Ace 5
• Hi Lo
• Hi Opt I
• Hi Opt II
• Halves
• Knockout KO
• Omega II
• Red 7
• Speed Count
• Ten Count
• Zen

Each of these systems takes a particular path to the same destination — gaining insight into whether the deck is 'optimal' or 'suboptimal' going forward. When a proficient card counter can pin down which cards have been previously dealt, they can size their bets accordingly to take advantage — and even flip the house's inherent 1% edge in their own favor — the minute a deck's count creates a higher likelihood of landing the winning hand.

But which card counting strategy reigns supreme?

Well, that's all a matter of subjective interpretation, of course, because all card counters are created differently.

Many folks who are blessed with supercomputer minds and mathematical acumen prefer balanced systems like the Omega II because they don't mind running through complex calculations like long division and such on the fly.

Others out there who aren't ready for such taxing mental math might take a liking to an unbalanced system like the Red 7, as it cuts the process down to simple addition and subtraction.

Of course, the more complex the system, the better it will perform on a statistical basis. On the other hand, those slight gains in the player advantage don't really matter all that much if you're unable to keep things sorted in your head during a hectic game of blackjack in the casino setting.

Earlier, we covered one of the more widely used card counting methods out there — the Golden Touch Speed Count invented by Henry Tamburin and Frank Scoblete. The Speed Count lives up to its name by making the process streamlined and efficient, but doubts remain concerning its overall effectiveness.

Today, it's time to tackle another beloved member of the card counting family — the Knockout KO system.

Introduction to the Knockout KO Count System and Its Origins

The Knockout 'KO' card counting system was created in 1998 when Olaf Vancura published 'Knock-Out Blackjack: The Easiest Card-Counting System Ever Devised' (1998).

Having earned his Ph.D. in physics from the prestigious Johns Hopkins University in 1992, Vancura found himself studying the origins of the universe at the Harvard-Smithsonian Center for Astrophysics by day.

Mathematical Formula For Blackjack Games

By night, however, the mathematically inclined Vancura liked to visit the casino and gamble on card games — especially blackjack. Eventually, he combined his love for cards with his academic bona fides, creating and teaching a course at Tufts University on the math behind gambling games. Convened as part of the Tufts Experimental College, Vancura's course quickly wound up becoming one of the most popular on campus.

This acceptance from students and colleagues alike prompted Vancura to write a textbook entitled 'Smart Casino Gambling: How to Win More and Lose Less,' which was published in 1996.

As he later told the American Physical Society (APS) during a retrospective interview, Vancura was primarily interested in dispelling the common myths and misconceptions surrounding concepts like card counting:

'I couldn't find [a textbook] without junk.'

'I didn't want to expose my students to the nonsense that existed in current tomes about card counting.'

Shortly after 'Smart Casino Gambling' hit the university bookstore's shelves, Vancura extended an invitation to the director of casino operations at Foxwoods — a Connecticut casino resort which ranks as one of the largest in all of America — asking him to speak to students.

This connection led Vancura into the wider world of the gaming industry, and he soon decided to leave astrophysics behind for a new career as a consultant tasked with creating new casino games for Mikohn Gaming.

After relocating to Las Vegas in 1997, Vancura eventually met up with Ken Fuchs, who worked with Motorola as a senior electrical engineer specializing in cryptographic encryption. Fuchs also had a gambling-oriented hobby, one which involved the development of computer simulations designed to study optimal blackjack strategy.

And just like that, a marriage made in gambling heaven was born.

One year later, the pair's collaborations began focusing on perfecting the 'unbalanced' count, a concept originally conceived by card counting icon Arnold Snyder in 'Blackbelt in Blackjack' (1983). When using the 'Red 7' count introduced by Snyder, players relied on the following values to develop their running count.

Card Rank Values in the Red 7 Count

• Aces and 10s = -1
• 2s, 3s, 4s, 5s, and 6s = +1
• Red 7s = +1
• Black 7s, all 8s, and all 9s = 0

With 20 cards in the deck valued at -1 but 22 cards worth +1, Snyder's Red 7 count was deemed unbalanced as opposed to the 'balanced' systems of old which divided the deck's values into even proportions.

In the preface to 'Knock-Out Blackjack,' Vancura and Fuchs trace the lineage of their Knockout KO count directly to Snyder and his innovative Red 7 system:

'The Red 7 system was a breakthrough in simplicity, a pseudo-Level 1 system with all cards valued at +1, 0, or -1.'

'Due to the unbalanced nature of the system, no true count conversion, hence mental multiplication and division, is ever necessary.'

'The Knockout system is predicated on many of the important principles introduced by Snyder.'

As they alluded to in their tribute to Snyder, the purpose of an unbalanced count like the Red 7 or Knockout KO is to remove the need to convert your 'running' count into a 'true' count.

Games of chance are not merely pure applications of probability calculus and gaming situations are not just isolated events whose numerical probability is well established through mathematical methods; they are also games whose progress is influenced by human action. In gambling, the human element has a striking character. The player is not only interested in the mathematical probability of the various gaming events, but he or she has expectations from the games while a major interaction exists. To obtain favorable results from this interaction, gamblers take into account all possible information, including statistics, to build gaming strategies. The oldest and most common betting system is the martingale, or doubling-up, system on even-money bets, in which bets are doubled progressively after each loss until a win occurs. This system probably dates back to the invention of the roulette wheel. Two other well-known systems, also based on even-money bets, are the d'Alembert system (based on theorems of the French mathematician Jean Le Rond d'Alembert), in which the player increases his bets by one unit after each loss but decreases it by one unit after each win, and the Labouchere system (devised by the British politician Henry Du Pré Labouchere, although the basis for it was invented by the 18th-century French philosopher Marie-Jean-Antoine-Nicolas de Caritat, marquis de Condorcet), in which the player increases or decreases his bets according to a certain combination of numbers chosen in advance.^[1][2] The predicted average gain or loss is called expectation or expected value and is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds are repeated many times. A game or situation in which the expected value for the player is zero (no net gain nor loss) is called a fair game. The attribute fair refers not to the technical process of the game, but to the chance balance house (bank)–player.

Even though the randomness inherent in games of chance would seem to ensure their fairness (at least with respect to the players around a table—shuffling a deck or spinning a wheel do not favor any player except if they are fraudulent), gamblers always search and wait for irregularities in this randomness that will allow them to win. It has been mathematically proved that, in ideal conditions of randomness, and with negative expectation, no long-run regular winning is possible for players of games of chance. Most gamblers accept this premise, but still work on strategies to make them win either in the short term or over the long run.

House advantage or edge[edit]

Casino games provide a predictable long-term advantage to the casino, or 'house' while offering the player the possibility of a large short-term payout. Some casino games have a skill element, where the player makes decisions; such games are called 'random with a tactical element.' While it is possible through skillful play to minimize the house advantage, it is extremely rare that a player has sufficient skill to completely eliminate his inherent long-term disadvantage (the house edge or house vigorish) in a casino game. The common belief is that such a skill set would involve years of training, extraordinary memory, and numeracy, and/or acute visual or even aural observation, as in the case of wheel clocking in Roulette. For more examples see Advantage gambling.

The player's disadvantage is a result of the casino not paying winning wagers according to the game's 'true odds', which are the payouts that would be expected considering the odds of a wager either winning or losing. For example, if a game is played by wagering on the number that would result from the roll of one die, true odds would be 5 times the amount wagered since there is a 1/6 probability of any single number appearing. However, the casino may only pay 4 times the amount wagered for a winning wager.

The house edge (HE) or vigorish is defined as the casino profit expressed as a percentage of the player's original bet. In games such as Blackjack or Spanish 21, the final bet may be several times the original bet, if the player doubles or splits.

Example: In American Roulette, there are two zeroes and 36 non-zero numbers (18 red and 18 black). If a player bets $1 on red, his chance of winning $1 is therefore 18/38 and his chance of losing $1 (or winning -$1) is 20/38.

The player's expected value, EV = (18/38 x 1) + (20/38 x -1) = 18/38 - 20/38 = -2/38 = -5.26%. Therefore, the house edge is 5.26%. After 10 rounds, play $1 per round, the average house profit will be 10 x $1 x 5.26% = $0.53.Of course, it is not possible for the casino to win exactly 53 cents; this figure is the average casino profit from each player if it had millions of players each betting 10 rounds at $1 per round.

The house edge of casino games varies greatly with the game. Keno can have house edges up to 25% and slot machines can have up to 15%, while most Australian Pontoon games have house edges between 0.3% and 0.4%.

The calculation of the Roulette house edge was a trivial exercise; for other games, this is not usually the case. Combinatorial analysis and/or computer simulation is necessary to complete the task.

In games that have a skill element, such as Blackjack or Spanish 21, the house edge is defined as the house advantage from optimal play (without the use of advanced techniques such as card counting or shuffle tracking), on the first hand of the shoe (the container that holds the cards). The set of the optimal plays for all possible hands is known as 'basic strategy' and is highly dependent on the specific rules, and even the number of decks used. Good Blackjack and Spanish 21 games have to house edges below 0.5%.

Online slot games often have a published Return to Player (RTP) percentage that determines the theoretical house edge. Some software developers choose to publish the RTP of their slot games while others do not.^[3] Despite the set-theoretical RTP, almost any outcome is possible in the short term.^[4] RTP is the percentage of the jackpot or prize money that can be regurgitated from the slot machine. Say the dealer does a 20% set, then there is a potential for jackpot 1: 5 to come out, but the fact is of course not this big considering the patterns or symbols that exist in online slot games are so diverse. ^[5]

Standard deviation[edit]

The luck factor in a casino game is quantified using standard deviation (SD). The standard deviation of a simple game like Roulette can be simply calculated because of the binomial distribution of successes (assuming a result of 1 unit for a win, and 0 units for a loss). For the binomial distribution, SD is equal to npq{displaystyle {sqrt {npq}}}, where n{displaystyle n} is the number of rounds played, p{displaystyle p} is the probability of winning, and q{displaystyle q} is the probability of losing. Furthermore, if we flat bet at 10 units per round instead of 1 unit, the range of possible outcomes increases 10 fold. Therefore, SD for Roulette even-money bet is equal to 2bnpq{displaystyle 2b{sqrt {npq}}}, where b{displaystyle b} is the flat bet per round, n{displaystyle n} is the number of rounds, p=18/38{displaystyle p=18/38}, and q=20/38{displaystyle q=20/38}.

After enough large number of rounds the theoretical distribution of the total win converges to the normal distribution, giving a good possibility to forecast the possible win or loss. For example, after 100 rounds at $1 per round, the standard deviation of the win (equally of the loss) will be 2⋅$1⋅100⋅18/38⋅20/38≈$9.99{displaystyle 2cdot $1cdot {sqrt {100cdot 18/38cdot 20/38}}approx $9.99}. After 100 rounds, the expected loss will be 100⋅$1⋅2/38≈$5.26{displaystyle 100cdot $1cdot 2/38approx $5.26}.

The 3 sigma range is six times the standard deviation: three above the mean, and three below. Therefore, after 100 rounds betting $1 per round, the result will very probably be somewhere between −$5.26−3⋅$9.99{displaystyle -$5.26-3cdot $9.99} and −$5.26+3⋅$9.99{displaystyle -$5.26+3cdot $9.99}, i.e., between -$34 and $24. There is still a ca. 1 to 400 chance that the result will be not in this range, i.e. either the win will exceed $24, or the loss will exceed $34.

The standard deviation for the even-money Roulette bet is one of the lowest out of all casinos games. Most games, particularly slots, have extremely high standard deviations. As the size of the potential payouts increase, so does the standard deviation.

Unfortunately, the above considerations for small numbers of rounds are incorrect, because the distribution is far from normal. Moreover, the results of more volatile games usually converge to the normal distribution much more slowly, therefore much more huge number of rounds are required for that.

As the number of rounds increases, eventually, the expected loss will exceed the standard deviation, many times over. From the formula, we can see the standard deviation is proportional to the square root of the number of rounds played, while the expected loss is proportional to the number of rounds played. As the number of rounds increases, the expected loss increases at a much faster rate. This is why it is practically impossible for a gambler to win in the long term (if they don't have an edge). It is the high ratio of short-term standard deviation to expected loss that fools gamblers into thinking that they can win.

The volatility index (VI) is defined as the standard deviation for one round, betting one unit. Therefore, the VI for the even-money American Roulette bet is 18/38⋅20/38≈0.499{displaystyle {sqrt {18/38cdot 20/38}}approx 0.499}.

The variance v{displaystyle v} is defined as the square of the VI. Therefore, the variance of the even-money American Roulette bet is ca. 0.249, which is extremely low for a casino game. The variance for Blackjack is ca. 1. Grand casino gulfport buffet. 2, which is still low compared to the variances of electronic gaming machines (EGMs).

Additionally, the term of the volatility index based on some confidence intervals are used. Usually, it is based on the 90% confidence interval. The volatility index for the 90% confidence interval is ca. 1.645 times as the 'usual' volatility index that relates to the ca. 68.27% confidence interval.

It is important for a casino to know both the house edge and volatility index for all of their games. The house edge tells them what kind of profit they will make as percentage of turnover, and the volatility index tells them how much they need in the way of cash reserves. The mathematicians and computer programmers that do this kind of work are called gaming mathematicians and gaming analysts. Casinos do not have in-house expertise in this field, so they outsource their requirements to experts in the gaming analysis field.

Mathematical Formula For Blackjack For Dummies

See also[edit]

References[edit]

1. ^'Roulette'. britannica.
2. ^'D'Alembert roulette system'.
3. ^'Online slots Return to Player (RTP) explained - GamblersFever'.
4. ^'Return to Player and Hit frequency - What do these mean? - GetGamblingFacts'.
5. ^https://www.okslot88.com/hal-yang-perlu-di-ketahui-tentang-mesin-slot-online/

Further reading[edit]

• The Mathematics of Gambling, by Edward Thorp, ISBN0-89746-019-7
• The Theory of Gambling and Statistical Logic, Revised Edition, by Richard Epstein, ISBN0-12-240761-X
• The Mathematics of Games and Gambling, Second Edition, by Edward Packel, ISBN0-88385-646-8
• Probability Guide to Gambling: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery and Sport Bets, by Catalin Barboianu, ISBN973-87520-3-5excerpts
• Luck, Logic, and White Lies: The Mathematics of Games, by Jörg Bewersdorff, ISBN1-56881-210-8introduction.

External links[edit]

Ask any casual gambler about the best way to beat the house, and you'll inevitably hear tales involving intrepid card counters.

From the famed MIT Blackjack Team — which was immortalized in the film 21 (2008) starring Kevin Spacey as a math professor turned count team leader — to the instructional material published by the legendary gambling author Stanford Wong, card counting has been embedded in the public consciousness for decades.

And at first glance, that makes perfect sense…

All it takes is a quick back-of-the-napkin lesson to get the gist of counting cards at the blackjack table.

When you see plenty of high cards hit the felt as the dealer does their thing, you know the odds of hitting high totals like 18, 19, 20, and 21 decrease accordingly. Thus, you should lower your bets to the table minimum.

Conversely, when your scans show a ton of low cards getting dealt, it's time to ramp up the betting because your odds of hitting premium totals have now gone up.

After hearing this brief introduction, thousands of gamblers strike out to Sin City each and every year hoping to try their hand at the blackjack tables. Many even make a point to study card counting systems, of which literally hundreds have been devised over the last few decades.

It all began back in 1962 when math genius Edward O. Thorp invented the first effective method for counting cards and adjusting one's blackjack plays accordingly. Thorp's 'Ten Count' counting system established the template, and from there, dozens of blackjack sharps and math whizzes have taken their own shot at creating the best way to count cards.

Most Popular Card Counting Systems

• Ace 5
• Hi Lo
• Hi Opt I
• Hi Opt II
• Halves
• Knockout KO
• Omega II
• Red 7
• Speed Count
• Ten Count
• Zen

Each of these systems takes a particular path to the same destination — gaining insight into whether the deck is 'optimal' or 'suboptimal' going forward. When a proficient card counter can pin down which cards have been previously dealt, they can size their bets accordingly to take advantage — and even flip the house's inherent 1% edge in their own favor — the minute a deck's count creates a higher likelihood of landing the winning hand.

But which card counting strategy reigns supreme?

Well, that's all a matter of subjective interpretation, of course, because all card counters are created differently.

Many folks who are blessed with supercomputer minds and mathematical acumen prefer balanced systems like the Omega II because they don't mind running through complex calculations like long division and such on the fly.

Others out there who aren't ready for such taxing mental math might take a liking to an unbalanced system like the Red 7, as it cuts the process down to simple addition and subtraction.

Of course, the more complex the system, the better it will perform on a statistical basis. On the other hand, those slight gains in the player advantage don't really matter all that much if you're unable to keep things sorted in your head during a hectic game of blackjack in the casino setting.

Earlier, we covered one of the more widely used card counting methods out there — the Golden Touch Speed Count invented by Henry Tamburin and Frank Scoblete. The Speed Count lives up to its name by making the process streamlined and efficient, but doubts remain concerning its overall effectiveness.

Today, it's time to tackle another beloved member of the card counting family — the Knockout KO system.

Introduction to the Knockout KO Count System and Its Origins

The Knockout 'KO' card counting system was created in 1998 when Olaf Vancura published 'Knock-Out Blackjack: The Easiest Card-Counting System Ever Devised' (1998).

Having earned his Ph.D. in physics from the prestigious Johns Hopkins University in 1992, Vancura found himself studying the origins of the universe at the Harvard-Smithsonian Center for Astrophysics by day.

Mathematical Formula For Blackjack Games

By night, however, the mathematically inclined Vancura liked to visit the casino and gamble on card games — especially blackjack. Eventually, he combined his love for cards with his academic bona fides, creating and teaching a course at Tufts University on the math behind gambling games. Convened as part of the Tufts Experimental College, Vancura's course quickly wound up becoming one of the most popular on campus.

This acceptance from students and colleagues alike prompted Vancura to write a textbook entitled 'Smart Casino Gambling: How to Win More and Lose Less,' which was published in 1996.

As he later told the American Physical Society (APS) during a retrospective interview, Vancura was primarily interested in dispelling the common myths and misconceptions surrounding concepts like card counting:

'I couldn't find [a textbook] without junk.'

'I didn't want to expose my students to the nonsense that existed in current tomes about card counting.'

Shortly after 'Smart Casino Gambling' hit the university bookstore's shelves, Vancura extended an invitation to the director of casino operations at Foxwoods — a Connecticut casino resort which ranks as one of the largest in all of America — asking him to speak to students.

This connection led Vancura into the wider world of the gaming industry, and he soon decided to leave astrophysics behind for a new career as a consultant tasked with creating new casino games for Mikohn Gaming.

After relocating to Las Vegas in 1997, Vancura eventually met up with Ken Fuchs, who worked with Motorola as a senior electrical engineer specializing in cryptographic encryption. Fuchs also had a gambling-oriented hobby, one which involved the development of computer simulations designed to study optimal blackjack strategy.

And just like that, a marriage made in gambling heaven was born.

One year later, the pair's collaborations began focusing on perfecting the 'unbalanced' count, a concept originally conceived by card counting icon Arnold Snyder in 'Blackbelt in Blackjack' (1983). When using the 'Red 7' count introduced by Snyder, players relied on the following values to develop their running count.

Card Rank Values in the Red 7 Count

• Aces and 10s = -1
• 2s, 3s, 4s, 5s, and 6s = +1
• Red 7s = +1
• Black 7s, all 8s, and all 9s = 0

With 20 cards in the deck valued at -1 but 22 cards worth +1, Snyder's Red 7 count was deemed unbalanced as opposed to the 'balanced' systems of old which divided the deck's values into even proportions.

In the preface to 'Knock-Out Blackjack,' Vancura and Fuchs trace the lineage of their Knockout KO count directly to Snyder and his innovative Red 7 system:

'The Red 7 system was a breakthrough in simplicity, a pseudo-Level 1 system with all cards valued at +1, 0, or -1.'

'Due to the unbalanced nature of the system, no true count conversion, hence mental multiplication and division, is ever necessary.'

'The Knockout system is predicated on many of the important principles introduced by Snyder.'

As they alluded to in their tribute to Snyder, the purpose of an unbalanced count like the Red 7 or Knockout KO is to remove the need to convert your 'running' count into a 'true' count.

When utilizing a balanced count, players must keep track of two different mathematical processes at the same time.

First, you need to track exposed cards and tabulate their values to determine your running count. Concurrently, the counter needs to maintain a separate count of all cards dealt to determine how many decks are left in the shoe. Finally, knowing their running count number, a balanced count practitioner divides it by the number of decks in play to arrive at the true count.

Like Snyder before them, Vancura and Fuchs sought to eliminate that tedious mental math by limiting the player's focus to the running count only. This was accomplished by fusing the balanced count methodology with flexible starting count figures, which fluctuate depending on how many decks the shoe contains at the start.

Introductory Lesson on How the KO Count Works at the Table

Taking a look at the point values used in the Knockout KO system below, you should be able to spot a certain sense of familiarity.

Card Rank Values in the Knockout KO System

• Aces and 10s = -1
• 2s, 3s, 4s, 5s, 6s, and 7s = +1
• 8s and 9s = 0

That's right, other than the provision about valuing the red 7 of hearts and 7 of diamonds as +1 — and thus separate from their black 7 of spades and 7 of clubs cousins and their 0 value — the Knockout KO and Red 7 systems are essentially identical.

This similarity holds true when examining how the Knockout KO count manages to avoid those pesky true count conversions. Instead of assigning players with the task of counting the decks remaining and dividing their running count by that number, both the Red 7 and Knockout KO systems factor deck density into the starting count.

Here's how it works.

When you're using the Knockout KO approach at a single-deck table, you'll go ahead and start with a 0 count like you would in the classic balanced systems.

But up the deck density to a two-deck shoe, and your starting count now shifts to -4. Make it a six-deck shoe, and the starting count drops even lower to -20, while an eight-deck shoe requires the count to start at -28.

Using these neutral and negative starting counts, players sift through the deck's exposed cards and adjust their running count accordingly, hoping to see the count eventually climb to the low negatives and positives. As the count increases, the Knockout KO system uses a set of 'Key Count' numbers to let you know exactly when to pivot and pump up the betting.

Using the table below, you can see the start counts — deemed initial running counts (IRCs) in advantage play vernacular — for various deck densities, alongside the key count needed to begin betting bigger.

Knockout KO Initial Running Counts (IRCs) and Key Counts

As you can see, more decks in the shoe means more positive progress needed in your running count to finally reach the key count.

A single-deck grinder only needs to see the count climb by +2 (from 0 to +2) in order to size their wagers higher. But if you're sitting in a double-deck game, you'll need to count +5 (from -4 to +1) before going for the gusto. Six-deck shoes have a whopping 312 cards to wade through, so players must count upward by +16 (from -20 to -4) before firing max bets. And finally, eight-deck tables can't be pounded until you see a +22 push (from -28 to -6).

We can use an example hand to see how the Knockout KO count really works in action. Owing to the rarity of single- and double-deck blackjack in modern casinos, we'll roll with the standard six-handed shoe found in most Sin City games. The table features yourself playing Hand 1, along with three other players trying to beat the dealer.

Because we're using a six-deck shoe, the initial running count (IRC) begins at -20.

Most counters try to separate cards into 'sides,' rather than doing a straight up and down count, so we'll start with the low-ranked cards (2, 3, 4, 5, 6, 7) valued at +1. You see six of these low cards on the table, good for +6, so your start count moves from -20 to -14.

Next, we'll do the high cards (A, K, Q, J, 10) worth -1, and with three of them out there for -3, the count drops from -14 back to -17.

The next step in the hand is players acting on the cards, followed by the dealer, so we'll run through one more count before this deal is done.

Mathematical Formula For Blackjack Card Game

You play the 11 perfectly with a well-timed double-down plus a face card for a 21, but while that's a nice treat, the name of the game here is counting cards.

Our count stands at -17, so we'll start with the low card side and see where we get. Five of the baby cards equals +5, so we're up to -12 for the moment. After subtracting -1 for the single high card to come out, the count slides back to -13.

Moving from the -20 start count to -13 represents significant progress, so if things keep up on this track, we'll hit the key count of -4 before long.

When you do cross over the key count, try to utilize a progressive wagering scale to take advantage of increasingly favorable deck conditions.

Mathematical Formula For Blackjack Rules

In other words, when you're right at the -4 key number in this six-deck scenario, feel free to double your initial minimum bet. Get the count to -2, and you can triple the table minimum. Find your way to a 0 count and make it 4x the minimum. And bet the maximum whenever the count reaches positive numbers.

Using the Knockout KO system correctly, even inexperienced card counters are capable of creating a player edge between 1-2%.

What Actual Players Are Saying About the KO Count

Researching and reading about card counting systems is an essential first step, but there's nothing quite like firsthand experience.

You'll be out there taking the Knockout KO count to your favorite casino sometime soon, but before you do, let's see what actual blackjack enthusiasts have to say about the system.

Robert Leeroy Parker — author of Magnum Blackjack and frequent contributor to the Blackjack Insider Newsletter — had this to say while reviewing the Knockout KO system:

'Knockout Blackjack was published in 1998, and it has quickly become the second most popular system, undoubtedly due to its simplicity.'

'KO was viciously attacked by many when it first appeared. People simply could not accept that anything that simple could work.'

'However, extensive computer simulations have shown that KO and Hi-lo are neck-and-neck, with KO out-performing Hi-lo under some game conditions.'

In a review of the original Knockout KO book titled 'Counting Cards Can Be Simple, This Book Shows the Way' — posted to Amazon by a reader named Richard from Florida — the average player's perspective is captured perfectly:

'If you have tried counting cards using other systems and have failed, then Knock-out Blackjack is the book for you. Not only is the system uncomplicated, but the authors layout a step by step progression of easily understandable techniques.'

'The data supporting the K-O system is included in various chapters, yet you do not need to know or memorize the math. There is no dividing and they show you how to make a simple adjustment so you there are no negative number to work with. And it is easily adjustable to any number of decks.'

'When they say it is the easiest card counting system every devised, it is true. I have tried other systems and failed. But with K-O system the average person will have the advantage over the house for a change.'

Meanwhile, fellow Amazon reviewer Ian Jeffreys wasn't as impressed, writing in 'Authors Have Overstated the Method' that the Knockout KO system is overrated:

'The KO system presented in this book depends on a set of simulation data I'm sure the authors believe is valid, but shows serious evidence of sample bias.'

'Recent work done on optimal betting theory, which allows a fair comparison of blackjack systems, shows that the KO method is extremely weak. The claim that KO can, for example, seriously challenge the dominance of Bryce Carlson's complex and powerful AOII system is just ridiculous…'

'Just look at some of the independent studies archived in the rec.gambling.blackjack archives to see how weak KO really is.'

Like I said, though, learning to appreciate any particular card counting method is all a matter of personal preference.

If you're new to the world of advantage play blackjack, so-called 'Level 1' unbalanced systems like the Knockout KO are the perfect entry point. You won't have to get bogged down in maintaining dual counts or converting from running to true, which lets you focus on the task at hand — beating the house with well-timed big bets.

Of course, other systems out there have been shown to be more effective, but as a result, these counts are far more difficult to use in the casino setting.

Conclusion

The Knockout KO card counting system offers the perfect compromise between complexity and usability. Rookie card counters who find themselves struggling with some of the old-school balanced counts — most of which were invented by legitimate math geniuses — will definitely appreciate how the Knockout KO boils things down to basic addition and subtraction. And as you grow more skilled with your advantage play abilities, the Knockout KO system provides an effective jumping-off point toward learning the more advanced concepts.

If you're interested in counting cards without dealing with intimidating math formulas, the Knockout KO system created by Vancura and Fuchs has everything you need to succeed.