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The mathematics of gambling are a collection of probability applications encountered in games 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.


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Poker is any of a number of card games in which players wager over which hand is best Thus, while the outcome of any particular hand significantly involves chance, the The cards are dealt clockwise around the poker table, one at a time.


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Four Horsemen of the Apocalypse is the name given by gambling authors to the four U.S. Army engineers who first discovered in the s the best playing strategy in the casino game of Blackjack that can Doren, contained a pull-out strategy chart with sections on Draw or Stand, Doubling Down, and Splitting Pairsโ€‹.


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Four Horsemen of the Apocalypse is the name given by gambling authors to the four U.S. Army engineers who first discovered in the s the best playing strategy in the casino game of Blackjack that can Doren, contained a pull-out strategy chart with sections on Draw or Stand, Doubling Down, and Splitting Pairsโ€‹.


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The mathematics of gambling are a collection of probability applications encountered in games 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.


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Splitting aces and eights is part of blackjack basic strategy. Rules vary across gambling establishments regarding resplitting, doubling, multiple card draws, and.


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Four Horsemen of the Apocalypse is the name given by gambling authors to the four U.S. Army engineers who first discovered in the s the best playing strategy in the casino game of Blackjack that can Doren, contained a pull-out strategy chart with sections on Draw or Stand, Doubling Down, and Splitting Pairsโ€‹.


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The table limit is the minimum and maximum bet that a gambler can make at a gaming table. Although electronic versions of table games like blackjack have been available for years, The strategy worked for around 9 months until a high roller won $ million over a two-week period and wiped out their profit for a year.


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Card counting is a casino card game strategy used primarily in the blackjack family of casino Card counting, also referred to as card reading, often refers to obtaining a sufficient count on the number, distribution and high-card location of.


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Splitting aces and eights is part of blackjack basic strategy. Rules vary across gambling establishments regarding resplitting, doubling, multiple card draws, and.


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Another disadvantage is that some players may become irritated with players who enter in the middle of a game, and superstitiously believe that this interrupts the "flow" of the cards. The primary goal of a card counting system is to assign point values to each card that roughly correlate to the card's "effect of removal" or EOR that is, the effect a single card has on the house advantage once removed from play , thus enabling the player to gauge the house advantage based on the composition of cards still to be dealt. Taken to its ultimate conclusion, the Kelly criterion would demand that a player not bet anything at all when the deck does not offer a positive expectation; the "Wonging" strategy described above implements this. Stanford Wong first proposed the idea of back-counting, and the term "Wong" comes from his pen name. The Running count is the running total of each card's assigned value. When the count "cools off" or the shoe is shuffled resetting the count , the big player will "Wong out" and look for other counters who are signaling a high count. With Hi-Lo, the True count is essentially the Running count divided by the number of decks that have not yet been dealt; this can be calculated by division or approximated with an average card count per round times the number of rounds dealt. When the count is significantly high, the spotter will discreetly signal another player, known as a "big player," that the count is high the table is "hot". A range of card counting devices are available but are deemed to be illegal in most U. Early counter-strategies featured the dealers learning to count the cards themselves to recognize the patterns in the players. However, many variations of True count calculation exist. This allows big players to move from table to table, maintaining the high-count advantage without being out of action very long. The player may gauge the effect of removal for all cards dealt, and assess the current house advantage of a game based on the remaining cards. In such a team, some players called "spotters" will sit at a table and play the game at the table minimum, while keeping a count basically doing the back "counting". The following table illustrates a few ranking systems for card counting. Larger ratios between point values can better correlate to actual EOR, but add complexity to the system. Higher concentration of high cards benefit the player in the following ways:. Instead, card counters assign a point score to each card they see that estimates the value of that card, and then they track the sum of these values โ€” a process called keeping a "running count. Including aces in the count improves betting correlation since the ace is the most valuable card in the deck for betting purposes. These include: [31] [32] [33].{/INSERTKEYS}{/PARAGRAPH} Historically, blackjack played with a perfect basic strategy offered a house edge of less than 0. However, since the ace can either be counted as one or eleven, including an ace in the count decreases the accuracy of playing efficiency. The player may then raise their bets as their advantage increases, or lower their bets as their advantage goes down. Many others exist. Large variations in bet size are one way that casinos detect card counters, and this is eliminated with back-counting. While a single player can maintain their own advantage with back-counting, card counting is most often used by teams of players to maximize their advantage. In , Ken Uston , a Blackjack Hall of Fame inductee, filed a lawsuit against an Atlantic City casino, claiming that casinos did not have the right to ban skilled players. The reason for this is that the count is more stable in a shoe game, so a player will be less likely to sit down for one or two hands and then have to get up. The player will enter or "Wong in" to the game when the count reaches a point at which the player has an advantage. Since there is the potential to create an overtaxing demand on the human mind while using a card counting system another important design consideration is the ease of use. One is that the player frequently does not stay at the table long enough to earn comps from the casino. Monitoring player behavior to assist with detecting the card counters falls into the hands of the on-floor casino personnel " pit bosses " and casino-surveillance personnel, who may use video surveillance "the eye in the sky " as well as computer analysis, to try to spot playing behavior indicative of card counting. This was the system used by the MIT Blackjack Team , whose story was in turn the inspiration for the Canadian movie The Last Casino which was later re-made into the Hollywood version The main advantage of group play is that the team can count several tables while a single back-counting player can usually only track one table. For instance, the Hi-Lo system subtracts one for each dealt 10, Jack, Queen, King or Ace, and adds one for any value Values are assigned a value of zero and therefore do not affect the count. One way to deal with such tradeoffs is to ignore the ace to yield higher PE while keeping a side count which is used to detect addition change in EV which the player will use to detect additional betting opportunities which ordinarily would not be indicated by the primary card counting system. Instead it comes from the increased probability of blackjacks, increased gain and benefit from doubling, splitting and surrender, and the insurance side bet, which becomes profitable at high counts. The big player will then "Wong in" and wager vastly higher sums up to the table maximum while the count is high. Their resentment may not merely be superstition, though, as this practice will negatively impact the other players at the table, because with one fewer player at the table when the card composition becomes unfavorable, the other players will play through more hands under those conditions as they will use up fewer cards per hand, and similarly, they will play fewer hands in the rest of the card shoe if the advantage player slips in during the middle of the shoe when the cards become favorable because with one more player, more of those favorable cards will be used up per hand. The ideal system is a system that is usable by the player and offers the highest average dollar return per period of time when dealt at a fixed rate. A simple variation removes the loss of having spotters play; the spotters simply watch the table instead of playing and signal big players to Wong in and out as normal. Back-counting is different from traditional card-counting, in that the player does not play every hand they see. Some players will stay at the table until the game is shuffled, or they may "Wong out" or leave when the count reaches a level at which they no longer have an advantage. Card counting, also referred to as card reading , often refers to obtaining a sufficient count on the number, distribution and high-card location of cards in trick-taking games such as contract bridge or spades to optimize the winning of tricks. The goal of a card counting system is to assign point values that roughly correlate to a card's Effect of Removal EOR. Back-counting, also known as "Wonging," consists of standing behind a blackjack table that other players are playing on, and counting the cards as they are dealt. Since PE is more important in single- and double-deck games, and BC is more important in shoe games, counting the ace is more important in shoe games. Card counting is not illegal under British law, nor is it under federal, state, or local laws in the United States provided that no external card counting device or person assists the player in counting cards. This offers several advantages. Card counting with the mind is legal, although casinos in the US reserve the right to remove anyone they suspect of using the technique. Lastly, a player who hops in and out of games may attract unwanted attention from casino personnel, and may be detected as a card-counter. Advanced players might additionally maintain a side count separate count of specific cards, such as a side count Aces, to deal with situations where the best count for betting accuracy differs from the best count for playing accuracy. Detection of card counters will be confirmed after a player is first suspected of counting cards; when seeking card counters, casino employees, whatever their position, could be alerted by many things that are most common when related to card counting but not common for other players. When using Balanced count such as the Hi-Lo system , the Running count is converted into a "True count," which takes into consideration the number of decks used. {PARAGRAPH}{INSERTKEYS}Card counting is a casino card game strategy used primarily in the blackjack family of casino games to determine whether the next hand is likely to give a probable advantage to the player or to the dealer. The variance in blackjack is high, so generating a sizable profit can take hundreds of hours of play. The High-Low system is considered a level-one count, because the running count never increases or decreases by more than a single, predetermined value. This negatively impacts the other players, whether they are counting cards or not. Counting systems may be referred to as "level 1", "level 2", etc. Another reason is that many casinos exhibit more effort to thwart card counters on their pitch games than on their shoe games, as a counter has a smaller advantage on an average shoe game than in a pitch game. Back-counting is generally done on shoe games, of 4, 6, or 8 decks, although it can be done on pitch games of 1 or 2 decks. A bet ramp is a betting plan with a specific bet size tied to each true count value in such a way that the player is betting proportionally to the player advantage with aims to maximize overall bankroll growth. Many casino chains keep databases of players that they consider undesirable. The disadvantages include requiring multiple spotters who can keep an accurate count, splitting the "take" among all members of the team, requiring spotters to play a table regardless of the count using only basic strategy, these players will lose money long-term , and requiring signals, which can alert pit bosses. Under one set of circumstance, a player with a unit bet spread with only one-deck cut off of a six-deck game will enjoy an advantage of as much as 1. In addition, many casinos do not allow "mid-shoe entry" in single or double deck games which makes Wonging impossible. When a card of that value is dealt, the count is adjusted by that card's counting value. The disadvantage of higher-level counts is that keeping track of more information may detract from the ability to play quickly and accurately. Macau, the gambling capital of the world and the only legal location in China, [25] does not technically prohibit card counting but casinos reserve the right to expel or ban any customers, as is the case in the US and Britain. Card counting also provides the ability to alter playing decisions based on the composition of remaining cards. Still, casinos object to the practice, and try to prevent it, [21] banning players believed to be counters. Using this principle, a card counter may elect to vary their bet size in proportion to the advantage dictated by a count creating what is called a "Bet ramp" according to the principles of the Kelly criterion. For successful card counters, therefore, skill at "cover" behavior, to hide counting and avoid "drawing heat" and possibly being barred, may be just as important as playing skill. With this in mind, systems aim to achieve a balance of efficiency in three categories: [3]. In their pursuit to identify card counters, casinos sometimes misidentify and ban players suspected of counting cards even if they do not. Some card counters might earn more money by playing a simple count quicklyโ€”more hands per hour playedโ€”than by playing a complex count slowly. Contrary to the popular myth, card counters do not need unusual mental abilities to count cards, because they are not tracking and memorizing specific cards. Some strategies count the ace ace-reckoned strategies and some do not ace-neutral strategies. Card counters are a class of advantage players , who attempt to decrease the inherent casino house edge by keeping a running tally of all high and low valued cards seen by the player. The most commonly side counted card is the ace since it is the most important card in terms of achieving a balance of BC and PE. In practice, this means that the higher the count, the more a player should bet on each hand in order to take advantage of the player edge. Low cards increase the count as they increase the percentage of high cards in the remaining set of cards, while high cards decrease it for the opposite reason. For one, the player does not play hands at which they do not have a statistical advantage. The EOR is the estimated effect of removing a given card from play, and the resulting impact on the house advantage. This increases the total advantage of the player. The disadvantages of this variation are reduced ability of the spotter and big player to communicate, reduced comps as the spotters are not sitting down, and vastly increased suspicion, as blackjack is not generally considered a spectator sport in casinos except among those actually playing unlike craps , roulette and wheels of fortune which have larger displays and so tend to attract more spectators. As they are unable to ban counters even when identified, Atlantic City casinos have increased the use of countermeasures. Card counting allows players to bet more with less risk when the count gives an advantage as well as minimize losses during an unfavorable count. Advantages of up to 2. There are several disadvantages to back-counting. The New Jersey Supreme Court agreed, [23] ruling that "the state's control of Atlantic City's casinos is so complete that only the New Jersey Casino Control Commission has the power to make rules to exclude skillful players. As larger ratios between point values are used to create better correlation to actual EOR with the goal of increasing the efficiency of a system, such systems use more different numbers and are broken into classes depending on such as level 1, level 2, level 3, and so on, with regard to the ratio between the highest and lowest assigned point values. A mathematical principle called the Kelly criterion indicates that bet increases should be proportional to the player advantage. It also allows redundancy while the big player is seated as both the counter and big player can keep the count as in the movie 21 , the spotter can communicate the count to the big player discreetly as they sit down. Basic card counting assigns a positive, negative, or zero value to each card value available. Another advantage is that the player does not have to change their bet size as much, or at all if they choose. Some back-counters prefer to flat-bet, and only bet the same amount once they have entered the game. The most common variations of card counting in blackjack are based on statistical evidence that high cards especially aces and 10s, and to a lesser extent 9s benefit the player more than the dealer, while the low cards, 3s, 4s, 6s, and especially 5s, and to a lesser extent 2s and 7s help the dealer while hurting the player.