Gaming mathematics

# Gaming mathematics

Discussion
 Ask a question about 'Gaming mathematics' Start a new discussion about 'Gaming mathematics' Answer questions from other users Full Discussion Forum

Encyclopedia
Gaming mathematics, also referred to as the mathematics of gambling, is a collection of probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

applications encountered in games of chance and can be included in applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

. From mathematical point of view, the games of chance are experiments generating various types of aleatory
Aleatory
Aleatoricism is the incorporation of chance into the process of creation, especially the creation of art or media. The word derives from the Latin word alea, the rolling of dice...

events, the probability of which can be calculated by using the properties of probability on a finite space of events.

## Experiments, events, probability spaces

The technical processes of a game stand for experiments that generate aleatory
Aleatory
Aleatoricism is the incorporation of chance into the process of creation, especially the creation of art or media. The word derives from the Latin word alea, the rolling of dice...

events. Here
are few examples:
• Throwing the dice in craps
Craps
Craps is a dice game in which players place wagers on the outcome of the roll, or a series of rolls, of a pair of dice. Players may wager money against each other or a bank...

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
Roulette
Roulette is a casino game named after a French diminutive for little wheel. In the game, players may choose to place bets on either a single number or a range of numbers, the colors red or black, or whether the number is odd or even....

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
Blackjack
Blackjack, also known as Twenty-one or Vingt-et-un , is the most widely played casino banking game in the world...

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
Lottery
A lottery is a form of gambling which involves the drawing of lots for a prize.Lottery is outlawed by some governments, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of regulation of lottery by governments...

, 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 6-size combinations of numbers from the 49.

• In draw poker
Draw poker
Draw poker is any poker variant in which each player is dealt a complete hand before the first betting round, and then develops the hand for later rounds by replacing, or "drawing", cards....

, 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

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 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 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 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. These
properties are very important in practical probability calculus.

The complete mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...

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
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

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)(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

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

Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, to build gaming
strategies. The predicted future gain or loss is called expectation
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

or
expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

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 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 over the long run.

## House advantage or edge

Casino games generally 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 skilful 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. 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
Eudaemons
The Eudaemons were a small group headed by graduate physics students J. Doyne Farmer and Norman Packard at the University of California Santa Cruz in the late 1970s. The group's immediate objective was to find a way to beat roulette, but a loftier objective was to use the money made from roulette...

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
Blackjack
Blackjack, also known as Twenty-one or Vingt-et-un , is the most widely played casino banking game in the world...

or Spanish 21
Spanish 21
Spanish 21 is a blackjack variant owned by Masque Publishing Inc., a gaming publishing company based in Colorado. Unlicensed, but equivalent, versions may be called Spanish blackjack. In Australia and Malaysia, an unlicensed version of the game, with no dealer hole card and significant rule...

, the final bet may be several times the original bet, if the player doubles or splits.)

Example: In American Roulette
Roulette
Roulette is a casino game named after a French diminutive for little wheel. In the game, players may choose to place bets on either a single number or a range of numbers, the colors red or black, or whether the number is odd or even....

, 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 vary greatly with the game. Keno can have house edges up to 25%, slot machines can have up to 15%, while most Australian Pontoon
Pontoon (game)
Pontoon is an unlicensed variant of the American game Spanish 21 that is played in Australian, Malaysian and Singaporean casinos, in Treasury Casino, Brisbane, it is known as Treasury 21...

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
Blackjack
Blackjack, also known as Twenty-one or Vingt-et-un , is the most widely played casino banking game in the world...

or Spanish 21
Spanish 21
Spanish 21 is a blackjack variant owned by Masque Publishing Inc., a gaming publishing company based in Colorado. Unlicensed, but equivalent, versions may be called Spanish blackjack. In Australia and Malaysia, an unlicensed version of the game, with no dealer hole card and significant rule...

, the house edge is defined as the house advantage from optimal play (without the use of advanced techniques such as card counting
Card counting
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. Card counters, also known as advantage players, attempt to decrease the inherent casino house...

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
Blackjack
Blackjack, also known as Twenty-one or Vingt-et-un , is the most widely played casino banking game in the world...

and Spanish 21
Spanish 21
Spanish 21 is a blackjack variant owned by Masque Publishing Inc., a gaming publishing company based in Colorado. Unlicensed, but equivalent, versions may be called Spanish blackjack. In Australia and Malaysia, an unlicensed version of the game, with no dealer hole card and significant rule...

games have house edges below 0.5%.

## Standard deviation

The luck factor in a casino game is quantified using standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

(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, even-money 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
Black swan theory
The black swan theory or theory of black swan events is a metaphor that encapsulates the concept that The event is a surprise and has a major impact...

.

The standard deviation for the even-money 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 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 sqrt(18/38 x 20/38) = 0.499.
The variance (v) is defined as the square of the VI. Therefore, the variance of the even-money 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
Slot machine
A slot machine , informally fruit machine , the slots , poker machine or "pokies" or simply slot is a casino gambling machine with three or more reels which spin when a button is pushed...

(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 in-house expertise in this field, so outsource their requirements to experts in the gaming analysis field, such as Mike Shackleford
Michael Shackleford
Michael "The Wizard of Odds" Shackleford a trained actuary , now makes his living analyzing casino games...

, the "Wizard of Odds".

## See also

• Gambling
Gambling
Gambling is the wagering of money or something of material value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods...

• Game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

• Mathematics of bookmaking
Mathematics of bookmaking
In betting parlance, making a book is the practice of laying bets on the various possible outcomes of a single event. The term originates from the practice of recording such wagers in a hard-bound ledger and gives the English language the term bookmaker for the person laying the bets and thus...

• Poker probability
Poker probability
In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands.-Frequency of 5-card poker hands:...

in general
• Poker probability (Texas hold 'em)
Poker probability (Texas hold 'em)
In poker, the probability of many events can be determined by direct calculation. This article discusses computing probabilities for many commonly occurring events in the game of Texas hold 'em and provides some probabilities and odds for specific situations...

• Poker probability (Omaha)
Poker probability (Omaha)
In poker, the probability of many events can be determined by direct calculation. This article discusses how to compute the probabilities for many commonly occurring events in the game of Omaha hold 'em and provides some probabilities and odds for specific situations...

• Statistical Soccer (Football) Predictions
Statistical Soccer (Football) Predictions
Statistical Football prediction is a method used in sports betting, to predict the outcome of Association football matches by means of statistical tools. The goal of statistical match prediction is to outperform the predictions of bookmakers, who use them to set odds on the outcome of football...

## Further reading

• The Mathematics of Gambling , by Edward Thorp, ISBN 0-89746-019-7 online version
• The Theory of Gambling and Statistical Logic, Revised Edition, by Richard Epstein, ISBN 0-12-240761-X
• The Mathematics of Games and Gambling, Second Edition, by Edward Packel, ISBN 0-88385-646-8
• Probability Guide to Gambling: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery and Sport Bets, by Catalin Barboianu, ISBN 973-87520-3-5 excerpts

• Luck, Logic, and White Lies: The Mathematics of Games, by Jörg Bewersdorff, ISBN 1-56881-210-8 introduction.