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Casino games mathematics

Casino Games Mathematics and Payout

What is a payout percentage?

Return to player or payout represents the percentage of the total bet amount that the casino has returned to the players, as wins. The value displayed on our website and inside our games, is the THEORETICAL AVERAGE PAYOUT. This is calculated using the formula of total win amount per total bet amount, after simulating all possible game results or a very large number of gameplays (over 10 million), using the current reel configuration of that game and the current paytable values.

The calculated PAYOUT is equal to the total amount of wins registered at your casino per total amount of bets registered at your casino.

The calculated PAYOUT within a game may not match the AVERAGE PAYOUT of that game, until the number of rounds played in that game at your casino will be very large. With the number of gameplays at your casino increasing inside a certain game, the calculated PAYOUT will tend to get more and more closer to the THEORETICAL AVERAGE PAYOUT of that game. For verification purposes, you can refer to the law of large numbers on WikiPedia.
We estimate that at least 10,000 gameplays are needed to be played in a game, in order for the calculated RTP to be within +-5% range of the theoretical RTP.

The PAYOUT may also go over 100% for some games, for a short period of time, depending on how the RNG worked and how it granted the prizes to the players. This is how any real money and licensed casino works and every casino owner must have a money reserve to cover these potential wins. Consider the scenario where a player wins the largest prize from a slot game... he hits 5 of a kind and wins bet x3000 for example. For the statistics to reflect a profitable RTP on that game, many gameplays will need to be played. With the number of gameplays increasing, the casino will gain back any amount that it lost and eventually make a considerable profit.
The risk can be minimized by lowering the bet values or by lowering the volatility of the game (this is done by requesting us to adjust the paytable values so that the highest paying symbols will pay less, but if we modify the paytable values, then the payout will change).

For non-regulated casinos (fun-currency, sweepstakes), the mathematics of the games may differ and the games may have a payout pool from which they pay money, as long as there are money in that pool.

What does RTP mean?

RTP means Return to player which is the same as the payout.

What does house edge mean?

The house edge represents the built-in advantage that the casino will gain, over time. A casino earns money by paying players at "house odds" (set by the game developers and mathematicians), an amount that is slightly less than the true odds.
For example, the true odds of hitting any given number (let's say number 21) in Single-Zero Roulette is 1:37 (you will hit your desired number once every 37 spins, in average if you play thousands of times - based on the law of large numbers). If you wager on all numbers, then you win no matter the extracted number, but you are paid back only 36$. So each time you lose 1$.
The House Edge is equal to 100% - PAYOUT. For example, if a game has a PAYOUT of 98.53%, then the house edge for the casino will be equal to 1.47%. For games that require a strategy to maximise the player's profits and payout (blackjack, caribbean poker, etc.), the house edge will usually get higher, because the theoretical RTP is based on the scenario where the player follows the perfect strategy, with no mistakes. But most ot the times, the player makes mistakes. If the player makes mistakes in following the perfect strategy, he will lose more money, therefore the house edge of the game will usually increase.

What is a hit frequency?

Hit/Win Frequency is a value that describes how often a win (with a minimum value=0.01 CREDITS) will be awarded to the player, in average. If a game has a hit frequency of 20%, this means that 20% of the times, or 1 out of 5 times, the player will win something (at least 0.01 CREDITS).

What is a bonus frequency?

The bonus frequency is a value that describes how often the bonus round of a current game will be triggered, in average. A value of 10% means that 10% of the times, a bonus round will occur, which means 1 out of 10 spins.

Can the RTP of a game be changed?

The RTP of each game is a number that is influenced by all possible game results and the game rules and paytable values (if the game uses a paytable - slots, videopoker, etc.).
In order to modify the RTP, the game rules and/or the paytable values need to be changed. For games that have a paytable (slots, video pokers), the RTP can be adjusted by us, one time, before your casino will go live. This can be done free of charge if your casino does not need GLI-19 certification. It is not possible to change the RTP of the other games (table card games, roulette) unless we make significant change to their rules.
Please note that it is not possible to change the RTP of any game from our backend administrator panel, because it is against GLI-19 certification requirements to be able to do this. GLI-19 certification is required to run legally a casino in any top jurisdiction. Companies who claim to offer the option of changing the RTP from backend, either do not have GLI-19 certified software/games, either modify the games to work with a manipulated RNG, which is unfair for the players.
If your casino does not require GLI-19 certification, then we can adjust the RTP of the games, in a fair manner by modifying the paytable values. Please contact us with your requirements.

What happens when the RTP(payout) of a game is modified?

Adjusting the RTP percentage means to make modifications to the game rules and/or to the game paytable (if applicable). After that, the new game rules must be simulated in order to determine the new RTP percentage. Only our company has access to these simulators and they are not distributed with the purchase of the games.
For example, in the slot games, if we lower the paytable values, the RTP will become lower, but we do not know by how much unless we resimulate all possible combinations of the reels using the new paytable values. In roulette games, the only way to lower the RTP is to lower the multiplier values of the winning bets or by introducing a fee on every win. In the table card games, the RTP can be lowered by introducing more decks of cards (in some games) or by introducing a fee on every win or by lowering the multiplier values of the winning bets.

Can I adjust the RTP percentage myself?

You can adjust the paytable values and then, indirectly affect the RTP of the game, but you will not know by how much your game RTP increased or decreased, unless you run a simulation on a large pool of data. We do not provide any simulators with the purchase of our games. We provide no assistance for such modifications and we cannot tell you if the new payout percentage is correct.
If you want a different RTP for any game, we must do these changes for you. Please contact us for more details.

Can I adjust the HIT frequency percentage?

No. This statistic is mainly dependant on the composition of the reels in the slot games. In order to adjust it, you need to redesign the reels composition and run a simulator on a large pool of data, in order to determine the new value. Making changes to the reels composition are also not permitted unless you buy the games with source codes. We offer this adjustment as a service, the price being listed on our website.

What is Standard Deviation (SD)?

In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set (Theoretical RTP), while a high standard deviation indicates that the values are spread out over a wider range.
In a casino game, the luck factor is quantified using standard deviation (SD). SD is used to calculate the Volatility Index of a game.

What is VI (Volatility Index)?

"VI" stands for "volatility index," and is an indication of how much the game's payback percentage will deviate from its calculated value, for a given number of games played. When it refers to slots, the term is used to let players know how often they can expect the slot to pay them, and how big they can expect that win to be.
With more games, the slot will pay closer and closer to its calculated (and theoretical) payout percentage. With fewer games, the payout is going to vary even more.
Basically, the VI is one way to compare slot games based on how many spins it will take for the calculated payout to approach the theoretical payout. The higher the VI, the more spins it will take.
From the perspective of net win, most casinos will attempt to minimize volatility so that the win from the casino will remain predictable. However, by doing so, the game will be less attractive for the players. In order to attract "whale players", some casinos prefer to offer games with high volatility. It is a risk that not many can take. A whale can make the casino rich, or he can make the casino poor.
Casinos can also reduce volatility by limiting the size of the wagers that players can make. This is one of the reasons for bet limits.

Low volatility

A game with a low volatility will give the player often but small wins. This means that on the long term, the player will feel like he didn't win anything, but he will enjoy lots of win animations, bonuses, freespins.

High volatility

A game with a high volatility will give the player rare but large wins. This means that on the long term, the player will barely win anything, but when he does it will be like hitting the jackpot. A game with high volatility can lead to losing streaks of 20-30 spins without any win and then a very large win can hit the player. High Volatility leads to high emotions, which is why many players will chase after this kind of games.

Medium volatility

A game with a mix of both the above. This means that the player will experience a win every few spins and while most of these will be average wins, sometimes these can be larger wins that will cause the player to gain a solid a profit from the game.
Most of our games have a medium volatility.

How much money reserve should I have to pay the players?

Every casino operator needs to understand that because of RNG, a game cannot be profitable for the casino 24/7. At some moments, the game will tend to get further from the Theoretical RTP. This is usually displayed using statistics such as Standard Deviation (Volatility). The real data which is used to calculate the payout of your game (total wins per total bets) will be far from the Theoretical RTP (the RTP displayed on our website) in the first few hundred gameplays, but it will get closer to that value, when more rounds are played.
For example, a game that has a Standard Deviation value of 5% with 93% Theoretical RTP can fairly give you a calculated RTP anywhere between ~88% and ~98% after 50,000 spins and the Gambling Commission would still consider the game to be performing as expected. Therefore, giving it time, it will get closer and closer to the 93%.
The volatility of a game plays an important role in preparing your money reserve.
There are games that can offer a steady payout and then suddenly reward a player with wins equal to 2000$-3000$ on just a 1$ bet. Those are games with high risk factor for the casino and high volatility.
There are also games where the player can bet 10$ and win 1-2$ most of the times and then the player can rarely win 500-600$. Those are games with medium volatility.
The low volatility games will reward the player with small and often wins, so the risk for the casino is very low with these games.
Usually, the amount of money that you should have in order to cover the payouts of the players should be equal to MAX BET x MAX WIN. Let's assume that you have 5 casino games and the MAX WIN across all of them is bet per line x 5000. If the max bet per line is 0.10$ then you need to have a money reserve of 0.10$ x 5000 = 500$.

Other useful links

List of all games displaying the RTP(Payout), Hit Frequency, Max win and other statistics

List of games to configure your own package

List of prices



BY CASINOWEBSCRIPTS PUBLISHED 21 June 2020

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