Abstract
This paper explores the use of mathematical concepts in gambling and playing games. The concepts employed when playing games of chance such as Blackjack are probability theory, combinations, and permutations. This paper tries to find ways of making mathematics appealing while shining light to possible research areas that may yield money as well as give new perspectives to certain areas. Interviews with tutors were carried out to get an understanding of what card counting entails. In addition, reviews of research done from as early as Thorp, to recent publications were looked into to provide direction. Card counting is considered illegal when devices are used meaning one can learn the skill and employ it without relying on a device for computation.
Mathematics has many beneficial applications in real life. The use of calculus in engineering and construction, employing statistics in establishing trends and understanding current relationships, and relying on probability to calculate chances for a wide range of activities, such as gambling. Gambling relies heavily on the chance occurrence of the preferred outcomes. However, gambling is not all chance. Games played with a deck of cards have demonstrated their use of with mathematical theory. This paper examines the application of mathematical concepts in the casino. That is, looking into card counting and the application of probability concepts in increasing game winnings at the casino table.
The field of mathematics is one considered hard to research on. Undergraduate students pursuing mathematical courses will research on various disciplines and only apply the mathematical principles in analyzing the research. However, card counting and probability theory have opened up avenues for research into purely mathematical phenomenon. Through the application of probability theory, one can simulate various games played in the casino using dies or a deck of cards. Kolber (2012) examines the legality of card counting using the brain as opposed to the agreed crime of using a device. The paper by Kolber is a good example of how mathematics is moving from research based on problems to real world activities affecting us.
Bărboianu in 2009 published a book that focused on gambling based on mathematical knowledge. The book helps simplify the concepts of combinations, permutations and the multiplication rule that students often struggle with (Nolan and Wroughton, 2012). In a journal of statistics article by Nolan and Wroughton (2012), the two explore how poker can help understand counting and probability. Their focus is to use a college level activity to explain mathematical concepts to high school students. Through such methods, mathematics can be embraced as much as people embrace the idea of going to Las Vegas to gamble and have fun. The reality is they do gamble but always loose to the house. Through such insight into the process of card counting, one can greatly increase their odds of winning.
Mathematicians such as Poisson and De Morgan saw the relationship between “chance” while playing poker and probability. The two renowned mathematicians each explored ways of demystifying gambling using probability. Poisson employed a 312-deck used in the casino game trente et quarante, where sampling took place without replacement, to reach useful conclusions on which the fundamental theory of card counting relies (Ethier and Levin, 2004, p 12). A research conducted by Thorp and Walden seeks to understand the fundamental theorem of card counting. In their research, they hypothesize that card counting is only beneficial to the game of Blackjack (1).
Card counting is essential when one is at the gambling table. It helps one to make decisions such as which combination of cards to keep or replace in a five draw poker, if to raise or not after a flop in Hold’em, or if you should ask for an additional card in Blackjack when you have good points (Bărboianu, 2009). However, it is not enough to know what one’s hand is, it is also important to calculate your opponent(s) hand while you play. This need to know is why one needs to be familiar with card counting before placing a bet at the Blackjack table.
The card counter relies on the game using the same deck of cards to play. Through this approach, one can employ the principles forwarded by Thorp (1966) and engage the house with significantly improved odds. Conrad and Smith (2002) simulated possible outcomes using a deck of cards in a game of Blackjack. In their simulation, they employed a High – Lo method of valuing cards where the card on the table was the pivot. Their simulation showed an increase in the chances of winning against the house as the game progressed. However, they noted that this increase was due to using the same deck of cards; a fresh deck used every time evened the odds against the house and even increased the chances of losing for card counters.
Card counting is usually a casino card game and strategy that is mostly used in black jack casino games (Simonson, 2011). The card counting strategy is mainly used to determine whether the hand that follows in the play is most likely to give an advantage probably to the player or to the dealer. The card counters are considered to form a class of players who are exceptional. They usually make attempts of decreasing the inherent casino house’s edge by tactically keeping a running tally of the low and high value cards that are seen by the players. Card counting gives the players the opportunity to place their bets more though at lower risks whenever the counts give a minimized likelihood of losses with increased levels of advantages whenever there is an unfavorable play count.
The card counting also provides players with an opportunity to change the decisions within the game that greatly depend on the composition of the cards remaining (Skar & Elizabeth, 2012). Card counting is also referred to as card reading. The card reading is also about the ability to obtain sufficient counts on the distributions, number and the highest card locations within any trick takings game so as to give an optimum point to the winning tricks. All these factors show that there is a relationship between card reading and mathematics. Card counting uses statistical evidences in its most but common variations of counts. It is especially visible in the play where higher cards do benefit the players more than they do benefit the dealers. The lower cards are also found to benefit the dealers more than the players.
As I study about mathematics and statistics, many of the teachers have connect their teaching with playing cards, and that interested me to study them. I talked to a couple of the teachers I had. One of them is my high school math teacher, Sam Calavitta, he has been teaching for about 25 years. He had taught all levels of mathematics of high school and University. He also received many math-teaching awards over the years; like the prestigious Siemen's Award for Advanced Placement. He said: “ Count cards are easier than you think, everyone with a master or higher degree in math should be able to win big in Vegas.” I believed it, went to talk to him after class; he told me, “Card counting uses mathematical concepts within its systems, as card counting is more of a calculated game of chance.” As I learn more and more of mathematics and statics. I went to interview my statics professor, Wenjian Liu, who is teaching psats 120A this quarter, and he used a lot of cards game examples in lectures. He told me “Gambling is all about probabilities, with more knowledge in math, you will have a better chance to know your situation, and more chance to win.”
The concentration of the cards has a statistical determination of the winner in the game. When there is a higher concentration of 10s, the deck increases the probability of the player a natural blackjack (Simonson, 2011). The chance is also expresses in the form of a ratio in which the chance of the higher numbers leading to the player being favored is said to pay a 3:2, which is a ratio. The absolute values on the cards also count thus the card game essentially relies on a mathematical concept of counting. The systems in card counting also assign mathematical values such as positive values, zero value and negative values. When cards are dealt, their counts are usually adjusted depending on the value counts of the cards. Thus, the system of adjustment depends on the mathematical rules of addition and subtraction to deal with the values that are assigned to the cards.
The other mathematical aspect that card counting use is correlation. The primary goal of systems found in card counting aims at assigning various points and values that do have a correlation to the cards (Simonson, 2011). It is referred to as the ‘effect of removal’ of a card. It enables players make a guess of the house advantages based on the cards consumed. The correlation adds some level of complexity to the whole card counting system. Either, the betting correlation uses permutations especially on the card that have not been dealt with to offer expectations to the players that are positive. It offers optimum strategy to players in the game. There is also other mathematical aspect like the balanced and running counts that the card counting strategy uses. The balanced counts use more of a mathematical system called the Hi-Low system that helps in the conversion of the running counts to true counts all of which have mathematical meanings.
Therefore, it can conclusively be said that be stated that there is a strong relationship between card counting in casino games and mathematics (Skar & Elizabeth, 2012). The card counting systems uses mathematical concepts in allocating card values and even uses the concepts further when adjusting the card during any given count. Either, in the counting design and betting, the card counting system totally relies on permutation that is a mathematical concept. It, therefore, means that without these mathematical concepts card-counting game cannot proceed. The mathematical concepts thus form an important part of the card counting game. It helps various stages of the card counting game within casinos to proceed since they rely on them in totality. Therefore, it is clear that there is a direct relationship between the card counting game and mathematics.
The one per person (OPP) studies the card counting system and its advantages especially in gambling (Skar & Elizabeth, 2012). The OPP research studies reveal that the card counting system is one of the simplest mathematical works that put in practice that is most put used to count low cards between 2 and 6 indicating them as +1 which is then subtracted from the hands available. It includes those of the dealers and has greatly helped in betting and gambling. The research study relied heavily on simulations with the results agreeing to the fact that counting is more profitable either, it further shows that the earnings from card counting do form a significant fraction of returns that link with the conventional way of counting. The study analyzed the mathematics behind OPP through card counting explaining how it works. It further shows the reasons as to why certain modifications on the concept have never worked while there are those that have been successful.
The study further employed the usage of shuffling and sequencing that gave further information on the card combinations (Simonson, 2011). Most systems were found to count higher combinations with lower cards being subtracted from them. Correlation counting was also studied on the blackjacks and the busted hands. The authors further recommended that counting should be done on the ashtrays that had most of the dubious theories. This counting can be similar to the card counting. Through card counting observations, it was revealed that the OPP system could be approximated in counting of the cards. This study gave an explanation on the OPP system comparing with the Jake Smallwood’s KWIK Counts. The explanation was based on the removal effects, which concluded that the OPP system was more effective that Jake Smallwood’s systems explaining further, why the OPP system is mainly used in the hand held games.
There is a rhetorical strategy applied on this strategy study of the card count. An example is when counting sevens as 1 in a bid to improve the OPP counts (Simonson, 2011). The research shows that this has a fixed effect on the overall negative effects of the seven on the OPP count. Either, this procedure has a negative aspect of it in the sense that it creates an imbalance within the system later. The counting of sevens as 1 has the effect of unbalancing the system by a positive four. The volatility of the OPP counting system is not that prominent. The research shows that it is almost half-volatile as other counting strategies. It is rhetorical to say that the OPP is the most effective of all the counting systems yet when the sevens are counted as 1 the net effect has the tendencies of making the OPP even more volatile compared to any other counting method (Sklar &Sherr Sklar, 2012). The selective counting has also contributed to some level of rhetoric within the card counting systems. When determining whether a player will have some advantage over the dealer the numbers of the individual card are usual cards.
Based on the illumination provided by Conrad and Smith (2012) about changing of odds depending on the deck of cards used, further research should be conducted to ascertain the effectiveness of this strategy used by casinos in protecting their house winnings. Further research will not only enhance the field of probability and mathematics as a whole, but also provide an insight to how much one can determine their winnings in a game of Blackjack, or any other casino card game. Published research about beating the house using mathematical concepts like probability, combination and permutations will entice the youth to delve into mathematics and stop shunning it for its assumed inapplicability to their lives. Since most grown-ups gamble, either in sports bets or casino games, students can be encourage to increase their winning chances for future bets they may place through the comprehension of probability theory.
References
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