The Math Of Slot Machines

There was a question presented to you on 'mathematics in the slot machines'. Forget it, the slot machines are run by computers and can be tightened or loosened at the casino's whim. Best regards, Larry and Alice. Dear Larry and Alice, And Earth is flat and the center of the universe. Best of luck in and out of the casinos, John.

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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 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. Common belief is that 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. 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 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%.

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]

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 ${displaystyle {sqrt {npq}}}$ , where ${displaystyle n}$ is the number of rounds played, ${displaystyle p}$ is the probability of winning, and ${displaystyle q}$ 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 ${displaystyle 2b{sqrt {npq}}}$ , where ${displaystyle b}$ is the flat bet per round, ${displaystyle n}$ is the number of rounds, ${displaystyle p=18/38}$ , and ${displaystyle q=20/38}$ .

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 ${displaystyle 2cdot $1cdot {sqrt {100cdot 18/38cdot 20/38}}approx $9.99}$ . After 100 rounds, the expected loss will be ${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 ${displaystyle -$5.26-3cdot $9.99}$ and ${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 ${displaystyle {sqrt {18/38cdot 20/38}}approx 0.499}$ .

The variance ${displaystyle v}$ 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. 1.2, 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.

Slot Analysis Training

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.

References[edit]

^'Roulette'. britannica.
^'D'Alembert roulette system'.
^'Online slots Return to Player (RTP) explained - GamblersFever'.
^'Return to Player and Hit frequency - What do these mean? - GetGamblingFacts'.

External links[edit]

Retrieved from 'https://en.wikipedia.org/w/index.php?title=Gambling_mathematics&oldid=983192088'

You might ask yourself how much math you really need to know to play slot machine games and the answer is simple: none. You can sit there and push buttons without a care in the world. Many people do.

But despite their bad reputation among professional gamblers who prefer card games and table games, slot machine games have some things going for them. For millions of people they are fun to play and most professional gamblers have hit the slots at least once in their careers.

So if you’re going to play slots, why not get a little better insight into how the math works for them? You just may look at these games in a whole different light from now on.

The Truth about Random Number Generators

The conventional wisdom among gambling pundits for many years was that slot machine games are governed by random number generators and therefore they are not predictable. Reality began cashing in on this widespread misbelieve around 2014. A gang of Russian criminals acquired some old slot machine games and reversed engineered the random number generation algorithms on their chips. Using live camera feeds and a supercomputer the gang was able to win guaranteed jackpots on a number of games, catching the land based casino industry by surprise.

Until 2017 it was believed by mathematicians that a true random number generation algorithm would be impossible to design. No matter what we do to make our algorithms more random-like they are never truly random. Expensive, high-end random number generation hardware relies on external natural factors, such as the decay of certain heavy elements, to keep refreshing the algorithms with new unpredictable “seed” values. This is good enough for science but not economical enough for industry.

A new quantum random number generator device has now been designed that may revolutionize the gaming industry. Until large scale production of these chips begins, however, the gaming industry will have to rely on improved security practices and policies to protect their games. More recent RNG chips are also better at being hard to predict than the chips that were reverse-engineered, but the myth that slot machine games are truly randomized has been busted.

Let’s Begin with Some Simple Definitions

A Probability

This is computed from a distribution of values or scores. All of the probabilities in the distribution add up to 1. Probabilities never predict the outcomes of random events. We use probabilities to set expectations about the likelihood of a large number of future outcomes falling into a known or recognizable pattern.

You can compute probability distributions from the total number of distinct possible outcomes or from a historical record of actual outcomes. The historical distribution often differs from the distribution of distinct possibilities. Although neither type of distribution can predict future results the distribution of distinct possibilities is the one statisticians prefer.

The classic coin toss example illustrates the differences between these two types of distributions. A coin has two sides. If you toss it only one of the two sides can land face up. Hence, the probability of either side landing face up is calculated to be 1 divided by 2 or .5. But suppose you toss a coin 10,000 times. Because of many random factors that could affect your tosses, you may find the coin lands Heads up a total of 5,329 times. The laws of science and probability have not been violated. That is just the way random chance works.

Randomness is the Same as Unpredictability.

An event does not have to be truly random to be unpredictable. If you the player do not know all the factors that are used to calculate the outcome of a game, such as where the reels on a slot machine stop, then the event is considered to be random enough. We say “the more random a game is, the more unpredictable it is” to mean that the less we know about how the outcome of the game is produced the less likely we can predict the outcome if the game is repeated over and over.

The Return to Player is the Inverse of the House Edge.

In other words, you take all wagers played in a game and add them up. This is the “pot”. The House Edge is that percentage of the pot that the casino expects to keep, either as a fee for brokering the game (such as in sports betting or poker), or as a percentage of wagers that is deducted because of the odds paid on the game when the casino bets against the player.

The Odds Paid on the Game.

These are calculated to be less than the probability of distinct possibilities. In European Roulette there are 36 slots on the wheel, but the highest odds you can win are 35-to-1. That difference of 1 represents the House Edge on a single number bet. The more green slots there are on the wheel the worse the Return to Player becomes for any bet. In slot machine games the odds paid on the game are based on the number of possible permutations of the reels.

Where a Knowledge of Math Helps the Average Slot Gamer

Although you will never be able to map the random sequences your favorite slot games use to determine results, there are other ways you can use math. We categorize these applications of math in four ways:

Cost per Game Analysis

This is the simplest use of math in slot games. There are two levels of slot game costs: the initial cost per spin and the cost per any additional game. The initial cost per spin is deducted from your stake or balance. The cost per additional game is deducted from your winnings. An example of an additional game that costs money would be a “gamble” feature that becomes available only after your initial spin wins a prize.

Money Management

You begin with your stake or bank balance. You decide whether to bet conservatively (minimum bets per game), aggressively (maximum bets per game), or somewhere in-between. In the worst case scenario you lose every bet. A conservative strategy allows you to play more games but you’ll win smaller prizes. An aggressive strategy allows you to win larger prizes but you may play fewer games. In practice it often does not matter but that is not guaranteed.

Progressive games may require that you play maximum bets to be eligible for the additional progressive jackpots. Games where you can change how many pay lines are active adjust your total bet amount.

Probability Assessment

If you cannot see the game’s inner workings then how can you estimate any probabilities? You have to make assumptions and try to err on the low side. It helps to have a basic familiarity with how mechanical slot games were designed. Most of today’s virtual slot games still follow their basic principles.

For example, look at the highest non-progressive prize a traditional 3 reel slot game pays. Say that is $1000. Now look at how many reels the game uses. If there is only one pay line the total number of possible combinations the reels can form must be somewhere above 1000. In other words, especially on older games, the maximum prize is usually paid only on 1 out of X combinations. If you have three reels then they each probably have no fewer than 11 slots (allowing for 1331 possible combinations).

Here is another example. If you play a slot game where you can deactivate pay lines, what is the optimum selection of active pay lines? The answer is all of them. The more pay lines you deactivate the fewer winning combinations the game will award you. This is because the game is designed to return the most money to players who play all the pay lines.

If a slot game uses multiple pay lines you can still calculate a minimum number of slots per reel based on the maximum possible prize. Just treat all the pay lines as a single active pay line. The more possible combinations your estimate gives you, the harder it will be to win that maximum prize.

Risk Assessment

We can look at risk in a simple way and a more complicated way. The simple definition of risk in a gambling game is your wager: how much are you prepared to lose on the outcome of the game? The more complicated solution combines the wager amount with your probability estimates. A game that pays a $50,000 non-progressive jackpot is more likely to take your wager than a game that pays a $1000 jackpot. Why? The jackpot has to be paid for out of player wagers.

Yes, smaller prizes must also be covered by player wagers. A game could be designed to pay more small prizes than large prizes. We use “volatility” to describe the ratio of prize sizes and frequencies. But the bottom line is that the more risk the casino accepts in terms of the jackpot the more likely the game pays for that jackpot by paying fewer or smaller “small” prizes.

Volatility Assessment

Gambling pundits are divided on how to define and measure volatility. They may use “variance” interchangeably with “volatility” as well. Confusing the issue further you’ll find that volatility has many different uses in science, math, and statistics. Variance is most often used in finance. A variance is computed for a set of scores by squaring their standard deviation. In gambling the more unpredictable the variance is the more volatile the game is said to be.

Another way to look at volatility and variance is to say that a game is more volatile if it pays prizes less often than other games. You can also measure a game’s volatility by how much the prizes differ. A game of roulette where all the players only make outside bets (Black, Red, Even, or Odd) illustrates low variance or volatility. The players should win frequently but they only win small prizes.

You can estimate the volatility of a slot game in several ways, depending on what information the casino shares with the players. Some slot games may bear a label indicating the ratio of expected prizes to spins. Such a label might say, “This game has a prize ratio of 4:1”. State-run lottery games throughout the US display similar ratios on their tickets. A high volatility game has a higher ratio.

Some slot games disclose their theoretical return to player in the rules or help screens. These returns are given as percentages. A game with a lower return to player and a larger jackpot most likely has high volatility.

If you spend a few minutes observing several people playing the same slot game on different machines, you can guess how volatile the basic game is by how often they win prizes. Player betting practices may influence how fast their balances change. Casinos may also program one machine to pay better than others.

Intuitive Math in Bonus Game Scenarios

Although slot machine gaming is largely automated and doesn’t require much skill, there are also some bonus games that allow players to develop skills. These are random shooter style bonus games that require “skill stop” timing. If the game directs you to press a button to fire a weapon, that is such a game. Your intuitive ability to time movements will help you. Other shooter style bonus games merely simulate the experience and require no player interaction.

Conclusion

Slot machine games can be fun to play but they rarely demand that players think the way good card games do. The need for applied math skills in slot machine gaming is minimal. However, savvy slots gamers learn to judge the quality of the games by the measures described above, and to manage their money and their risk accordingly.

Practice Slot Machines

Some players enjoy high risk, high volatility games. Sometimes it’s as simple as looking for the high roller versions of your favorite slots. But even among the general collection of games on the main floor of the casino there are ways to differentiate between the slot games. If you ever wondered why players congregate around certain machines, you can apply a little bit of math to analyze what makes the games so appealing.

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