Ask The Slot Expert

John Robison,

I like to work with some of the various math problems as they relate to slot machines. I have a question about some of the new machines and I wonder if you could help me with it.

I have calculated the probability for obtaining, say, 3 Jackpot symbols on a one-line 3-reel slot, assuming of course that we know the number of stops on the Virtual Reels and the number of Jackpot symbols per reel.

But, there are getting to be a lot of new machines where you can play multi-lines. A lot of these new machines have 5 reels and a window where you can get "Scatter Wins." How does one go about calculating the odds of getting a Scatter Symbol anywhere in the window? The window would consist of 5 reels and 3 symbols showing on each reel. So if a scatter symbol would show up in any of the 3 positions, AND on all 5 reels, then you would get the Scatter Win... or sometimes go to a Bonus Level.

Assume we know the number of Scatter symbols on each reel and the total number of stops on each of the virtual reels. I'm trying to understand how to calculate the probability of getting a Scatter Win.

Thanks,
Rosario

Dear Rosario,

Here is the general formula for figuring out the number of ways to hit a scatter pay (sp):

# of symbols visible * # of sp symbols on reel 1 * # of sp symbols on reel 2 * .... * # of sp symbols on reel n

To prove the formula, try a very simple test case. Take a 4-stop reel with 1 sp symbol on it and assume there are two such reels. There are 16 total results that can be displayed, 7 of which do not have a scatter pay and 9 of which do. You can prove this by writing out all the possible results.

If you don't want to write out the results, consider this: there is only one way a reel can stop such that a scatter pay symbol does not appear. If reel 1 stops there, then it doesn't matter how reel 2 stops. There will never be a scatter pay. That gives us 4 no scatter pay results. Now, there are three ways reel 1 can stop such that a scatter pay symbol is visible. There is only one way reel 2 can stop such that a scatter pay symbol is not visible. That gives us 3 more no scatter pay results for a total of 7. The remaining 9 results must be scatter pay results.

So, if you know the number of scatter pay symbols on each reel and the number of virtual stops on each reel, the probability of getting a scatter pay in which a scatter pay symbol appears on each reel is:

P = # ways to hit scatter pay / total outcomes

where "# ways to hit scatter pay" was given above, and total outcomes is the product of the number of virtual stops of each reel.

Best of luck in and out of the casinos,
John

John,

Thank you very much for getting back to me. Actually, the solution you have outlined is for determining how to get a Bonus Round Win... which is what happens if you get all 5 of the bonus symbols anywhere in the 3 x 5 window.

The problem I want to solve, which is what I would call the Scatter win, is when 3 or 4 of the scatter win symbols appear anywhere in the window area.

Perhaps there is a way to derive the solution from the one you came up with. I'll keep looking at it.

Thanks again,
Rosario

To figure out the number of ways to hit 4 scatter pay symbols on 5 reels, the formula is:

Let sp(n) be the number of ways to hit a scatter pay symbol on the nth reel. It is the number of symbols visible times the number of scatter pay symbols on the nth reel.

Let t(n) be the total number of symbols on the nth reel.

t(1) * sp(2) * sp(3) * sp(4) * sp(5) + sp(1) + t(2) * sp(3) * sp(4) * sp(5) + .... + sp(1) * sp(2) * sp(3) * sp(4) * t(5) - # of ways to hit 5 sp symbols

In words, you start off by saying that you don't care what is on the first reel, but you want a scatter pay symbol somewhere on the remaining reels. So you multiply the total number of symbols on the first reel by the number of scatter pay symbols on the second times the number of scatter pay symbols on the third, and so forth. The reel without a scatter pay symbol can be the first reel, the second reel, or the nth reel, so you repeat the multiplication for the remaining reels and add the products together.

Now, this sum includes some combinations that show a scatter pay symbol on each reel, so we have to subtract the number of ways to hit a scatter pay symbol on each reel to be left with the number of ways to hit a scatter pay symbol on four reels.

The same procedure is used to find the number of ways to hit a scatter pay symbol on three reels. You'll have many more addends because each reel has to be paired up with every other reel to be the two reels without scatter pay symbols. Once you've found that sum, in addition to subtracting the number of ways to hit a scatter pay symbol on each reel, you also have to subtract the number of ways to hit a scatter pay symbol on four reels.

John

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