Binomial Experiment Casino Slot Machine

When you think about the relatively modest amount of money players risk on a spin-to-spin basis, it might come as a surprise that slot machines are, by far, the biggest money-maker for the house.

Binomial Experiment Casino Slot Machines

When playing any online casino game for the first time, it is best to start simple and then progress to more complex versions. For instance, classic Vegas slots offer newcomers the chance to understand how a slot machine works, what each symbol represents, and the probability odds of different combinations. A binomial experiment is repeated until a specified number of successes have occurred. The random variable X is the number of trials necessary. There are two parameters for this distribution - the probability p of success on each trial, and the awaited number of successes k.

In fact, with the exception of Las Vegas Strip casinos (which often draw extremely high rollers to table games), slot machines are responsible for between 65% and 80% of your local casino’s gambling revenue. How have they managed to turn such a simple game into such a moneymaker?

In this article, I’ll explain how basic psychology is responsible for the massive profitability of real money slot machines.

Ease of Gambling

If you were dropped into a casino from outer space and sat in front of a slot machine, chances are it wouldn’t take long before you began to understand how slots work. When it comes to having a low barrier to entry, it’s hard to beat a slot machine.

Never ones to miss out on an opportunity to make a few more dollars, slot machine designers recognized that when it comes to a game that relies on a high volume of trials to make money, the easier it is, the better.

That’s why the “spin” button was added – it removed the need for the laborious task of pulling the lever (don’t worry, the lever is still there, mostly for nostalgic reasons).

The Casino RTP Illusion

When you look at the odds for a given slot machine, you might come across the acronym “RTP.” These initials, meaning “return-to-player,” is shown as a percentage which indicates how much money you’ll get back from your initial bet on average.

For example, you might find a machine with a 90% RTP. The idea of getting back 90% of the money you risk might sound comforting…but like everything else in the casino, it’s not quite as good as it seems.

The average number of spins in an hour for someone playing the slots is typically around 600. This might sound like a huge number, but it takes just seconds to complete the cycle of spin-stop, spin-stop.

For the sake of the example, I’ll assume that each spin is a modest, but realistic, $2 bet. That means over the course of an hour where you spin 500 times, you’ve wagered $1,000. If your RTP is exactly as advertised, you’d wind up losing about $150 over the course of your session.

Of course there’s a chance that you could walk away with a jackpot win, or even just a few extra dollars in your pocket, but it doesn’t happen often. If people won regularly, there wouldn’t be a need for the flashing lights and bells. The only reason for these often-annoying features is to indicate to everyone in the neighborhood that you can, in fact, win a jackpot.

Near Misses

Perhaps no aspect of slot machine psychology is more impactful than the “near miss.”

If you’re unfamiliar with the concept, a near miss is best-described, as you might expect, as a turn that ends in a player falling just short of getting the necessary images to show up on the slot machine. The question is: Why is this so effective in keeping people at the machine hoping for a jackpot?

The answer requires a look back to how the human brain was wired by evolution.

Back in the era of hunting and gathering, the idea of a near miss indicated that, at the very least, you were getting close to scoring a big meal for you and the rest of your tribe. If your arrow was just a bit off, or the animal got away at the last second, your senses were heightened at the prospect of accomplishing your goal.

Because your brain acknowledged that you were making progress, it was motivating and often led to eventual success. Unfortunately when it comes to gambling, the once-useful psychological trait has been hacked.

Whether you get four out of five or two out of five on the slot machine, the result is the same. You are no closer to hitting the jackpot, even though it might feel like it. The house relies on your innate sentiment of, “I’m this close, so I can’t stop now. Maybe the next spin will be the winner.”

Notably, a scientific study in 2009 by the journal ‘Neuron’ concluded that the reward center in our brains is triggered by a near miss in the same way that it is by an actual win while gambling. Furthermore, it was explained that the impact of the near miss triggering the reward center was most noticeable when there was less time between placing a bet and playing the actual game.

When you combine these two factors, the near miss and the enhanced reward effect when the pace of play is quicker, it’s easy to see why the slot machine could almost be considered irresistible.

Instant Gratification

Whereas the near miss effect seems to exploit an ancient feature of our brain, this one might be hitting on something. A growing feature in the future of human psychology: the concept of immediacy.

It’s likely that you’ve heard our desire for “instant gratification” has grown immensely over the past couple of decades. Things that used to require patience, such as travel, communication, and entertainment, have been transformed in modern times.

In order to observe this phenomenon, just look at the success of Amazon. Obviously there are other factors at play, but the main reason for their unprecedented growth is the fact that they adopted 2-day shipping. Online shopping was never the same again.

When it comes to getting your result (good or bad) right away, it’s impossible to beat a slot machine. Even the best blackjack dealers can’t keep up at a rate that can be up to 600 spins per hour.

For this reason, new gamblers like slots because they are looking to score a quick, easy, and relatively mindless win. “Just a few spins,” is probably the most common lie heard on a daily basis.

Slot Machines Prey on the Vulnerable

Excuse the dramatic header, but if you objectively look at what goals a slot machine is trying to accomplish (get as many people to spin the wheels as many times as possible), it’s not hard to see why it impacts some people differently.

A study that was conducted by the Centre for Gambling Research at UBC examined why slot machines were both the most popular way to gamble globally, and also the game that was most likely to be associated with addiction.

The experiment took two groups of people – first-time and experienced slot machine players – and examined which was more likely to end up in the “slot machine zone state.” This state, as defined by the researchers, is categorized by losing track of time, not noticing surroundings, and interestingly enough, not picking up on details of the slot machine that exist outside the screen.

The results of the study conclusively showed that those who had prior experience with slot machines fell into a slot machine trance state at a much faster rate than those who had not played before.

The main takeaway from this particular experiment, and others like it, is that slot machines both create and then continue to encourage compulsive gambling behavior in some people. The process of playing is in many ways mindless enough that it allows players to keep playing for hours while also letting their mind wander – perhaps this is responsible for the trance-like state so often observed at slot machines.

Conclusion

Slot machines aren’t going anywhere. For decades they’ve maintained their spot as the biggest earner the casino has on the floor. The human psychological component that was undoubtedly taken into account when building these games is largely to thank for the success…success for the house, not the players, that is.

At the end of the day, as long as players recognize that it’s easy to fall into the slot machine zone state and actively work to avoid letting it happen, slots can be a fun component of any day at the casino. Just remember to get up and walk around every few minutes.

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.

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.

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