 Gaming mathematics

Gaming mathematics, also referred to as the mathematics of gambling, is a collection of probability applications encountered in games of chance and can be included in applied mathematics. From 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.
Contents
Experiments, events, probability spaces
The technical processes of a game stand for experiments that generate aleatory events. Here are 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, obtaining numbers with certain properties (less than a specific number, higher that 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 2size 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 2size 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 generate 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 6size 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 5card 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 2card 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 2card combinations from the 52, less the three cards you hold and less the two cards you discarded. This sample space counts the 2size combinations from 47.
The probability model
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 the set 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 longshot 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 nonindependent 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 or exclusive 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 nonindependent (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. These properties 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
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)(524)=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
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 predicted future 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, no longrun regular winning is possible for players of games of chance. Most gamblers accept this premise, but still work on strategies to make them win over the long run.
House advantage or edge
Casino games generally provide a predictable longterm advantage to the casino, or "house", while offering the player the possibility of a large shortterm 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 skilful play to minimize the house advantage, it is extremely rare that a player has sufficient skill to completely eliminate his inherent longterm disadvantage (the house edge or house vigorish) in a casino game. Such a skill set would involve years of training, an extraordinary memory and numeracy, and/or acute visual or even aural observation, as in the case of wheel clocking in Roulette.
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 nonzero 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 vary greatly with the game. Keno can have house edges up to 25%, 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 which 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 house edges below 0.5%.
Standard deviation
The luck factor in a casino game is quantified using standard deviation (SD). The standard deviation of a simple game like Roulette can be calculated using the binomial distribution. In the binomial distribution, SD = sqrt (npq ), where n = number of rounds played, p = probability of winning, and q = probability of losing. The binomial distribution assumes a result of 1 unit for a win, and 0 units for a loss, rather than 1 units for a loss, which doubles the range of possible outcomes. Furthermore, if we flat bet at 10 units per round instead of 1 unit, the range of possible outcomes increases 10 fold. Therefore,
SD (Roulette, evenmoney bet) = 2b sqrt(npq ), where b = flat bet per round, n = number of rounds, p = 18/38, and q = 20/38.
For example, after 10 rounds at $1 per round, the standard deviation will be 2 x 1 x sqrt(10 x 18/38 x 20/38) = $3.16. After 10 rounds, the expected loss will be 10 x $1 x 5.26% = $0.53. As you can see, standard deviation is many times the magnitude of the expected loss.
The 3 sigma range is six times the standard deviation: three above the mean, and three below. Therefore, after 10 rounds betting $1 per round, your result will be somewhere between $0.53  3 x $3.16 and $0.53 + 3 x $3.16, i.e., between $10.00 and $8.95. (There is still a 0.1% chance that your result will exceed a $8.95 profit, and a 0.1% chance that you will lose more than $10.00.) This demonstrates how luck can be quantified; we know that if we walk into a casino and bet $5 per round for a whole night, we are not likely to walk out with $500. However, any wise gambler (or "investor") should beware of the Black swan theory.
The standard deviation for the evenmoney Roulette bet is 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.
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 impossible for a gambler to win in the long term (if they don't have an edge). It is the high ratio of shortterm 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 evenmoney American Roulette bet is sqrt(18/38 x 20/38) = 0.499. The variance (v) is defined as the square of the VI. Therefore, the variance of the evenmoney American Roulette bet is 0.249, which is extremely low for a casino game. The variance for Blackjack is 1.2, which is still low compared to the variances of electronic gaming machines (EGMs).
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 inhouse expertise in this field, so outsource their requirements to experts in the gaming analysis field, such as Mike Shackleford, the "Wizard of Odds".
See also
 Gambling
 Game theory
 Mathematics of bookmaking
 Poker probability in general
 Statistical Soccer (Football) Predictions
Further reading
 The Mathematics of Gambling , by Edward Thorp, ISBN 0897460197 online version
 The Theory of Gambling and Statistical Logic, Revised Edition, by Richard Epstein, ISBN 012240761X
 The Mathematics of Games and Gambling, Second Edition, by Edward Packel, ISBN 0883856468
 Probability Guide to Gambling: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery and Sport Bets, by Catalin Barboianu, ISBN 9738752035 excerpts
 Luck, Logic, and White Lies: The Mathematics of Games, by Jörg Bewersdorff, ISBN 1568812108 introduction.
External links
Categories: Applied probability
 Games (probability)
 Gambling
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