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
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.
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.
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.
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.
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.
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.
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.
Michael Stevens has been researching and writing topics involving the gambling industry for well over a decade now and is considered an expert on all things casino and sports betting. Michael has been writing for GamblingSites.org since early 2016. ...
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