In this article, I will introduce how to build the mathematical model of slots with scatter feature
Brief Introduction
As a very common symbol in slots, the scatter symbols normally used to trigger free games. In most cases, It does not have combinations and payouts. Once there are enough scatters landing on any position of the windows, the free games can be triggered
When you build the mathematical model of slots with scatter symbols, not only do you need to calculate the base game RTP, you also have to calculate the Free Game RTP and the expectation of Free Game Times in each spin
Example Information
In order to clarify the modeling steps, this article using a 5-3 slots as example
This slot only have 4 symbols, which are J, Q, K and Scatter
When there are no less than 3 scatter symbols landing on the windows, free game will be triggered. More scatters can trigger more free games, and you can trigger extra free game times during the free game too
Mathematical Modeling - Classic Slot
The first few steps are similar to slots without free game feature, you just need to calculate the amounts of landing on the windows events and total events. Then, you can get the expected RTP of standard symbols
Next, you need to calculate the hit frequency of free game
In this example, there is only one scatter symbol on each reels, which means you don't need to consider the possibility that there are more than one scatter symbols on one reel and calculate the scatter amounts easily by calculate the chance that scatter symbol lands on that reel. Don't forget times the row numbers when you calculate the events of scatters, because scatter symbols can trigger free game in all positions.
Times the corresponding free game times, you can get the expectation of triggered free game times in each base game spin
After the mathematical part of base game, you can start calculating the counterparts of free game
Like base game, the RTP of free game also only includes payouts of standard symbols
Also, you can find in upper pictures that reels of base game and free game are different. This is a common design in slots, in order to provide different experience to balance the payouts and draw players attention to the features
In this example, the difference are randomly selected. You need spend much more time in actual work to make the difference works for game experience improvement
Next, owing to more free games can be triggered during the free game, you need put extra free game times into consideration
In this step, we need to use the method called Infinite series summation. It is unnecessary to dig into the complicate definition of Infinite series. We just need to know that owing to a stable triggering rate and corresponding free game times, we can simply use the summation formula
Total Free Game Times in Each Spin = Triggering Times in Base Game / (1 - Triggering Times in Free Game)
As you can see, if Triggering Times in Free Game is more than 1, the total times would be a negative number, which is not right. This is because when Triggering Times in Free Game is more than 1, the free game would be triggered repeatedly indefinitely. So, you can use whether Total Free Game Times in Each Spin is a positive number to figure out whether the triggering setting are proper
At last, using the RTP of standard symbols in free game times the total free game times, you can get the free game RTP. Adding the base game RTP, the total RTP of this slot can be calculated
In conclusion, the most important part of mathematical modeling process of slot with free game feature is the expected times of free games. In this article, the basic idea is introduced. In following articles, I will introduce some different example, such as video slots with free game features and how to model when there could be more than one scatters landing on the same reel
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