Advantage Tactic: Scatters vs. Pay Lines

In my article about counting spins and symbols, I took a lazy shortcut in the analysis when I assumed the winning combination was a scatter pay as opposed to a pay line.  Fortunately, a reader called me out and I will add the relevant details. First, let me thank SB for asking this question and for reading my blog.

SB wrote:

Let me start by giving a quick answer to the question.

I personally would log each reel, not each position, and let math take care of the rest.  This is a shortcut, and it involves taking some risk.

-          There are 27 different ways to land 3 common symbols on a traditional 3x3 machine.

-          9 pay lines means, 9 of the 27 possible combinations are winners and 18 of 27 combinations are losers.

9 winning combinations / 27 possible combinations = 33.333…% chance of the target symbol landing on any given pay line.  If you take the 33.333…% chance of landing on a pay line and multiply by the probability of landing the target symbol anywhere on the reels, you get the probability of the event you are looking for.

Could this shortcut hurt me?  Maybe…  It’s all about how much work I’m willing to put in.  I also have to consider how many spins I’m willing to take to gather data.

Now for a longer answer (with more detail)…

I would not bother tracking symbols by position on a reel.  You won’t go wrong doing that, and I’m sure as soon as I post this someone will comment or tweet with an example of a machine that behaves in a way that will make me eat my words, but I am not aware of a machine that has a bias towards one payable position over another payable position on a given reel.  I do believe each reel is independent and can have a different frequency from an adjacent reel… I also believe that top paying symbols often appear just off of a payable position more often than you would expect…  i.e. just off the screen or just above or below a pay line to give the gambler a near miss experience.  But on a machine where 9 positions can make a pay line, I would expect the placement of those symbols to be random in those positions.

The truth of the matter is that I track top line symbols very rarely.  Usually when I’m doing this type of analysis, I’m looking for the frequency of a bonus game.  Bonus game triggers are typically scatters, which is why I took the shortcut in the first place.

Anyway, let’s start with the number of spins you are willing to take.  With the long frequencies of a top pay line symbol, you are not seeing a bunch of these things land in a payable postion anyway.  You need more spins to get meaningful data if you are tracking to 9 payable positions vs 3 reels.  Just to say it again, you will not go wrong with tracking each position; it’s just more work and more expense.  If I knew how to construct a game with code and simulate the spins, I’d do the 9 position analysis every time.  Because I use money and time to do my analysis, I take shortcuts.

Anyway, I want to get into the math because I think that will help people understand what we are talking about.  I’ll use my assumption of equal pay position distribution, but I’ll work though it in a way where if you choose to do a by position analysis, you can see how that works.

I mentioned that there are 27 ways to land 3 symbols on a 3x3 grid. 

Here are the 27 ways:

If you are dealing with more payable space, you can calculate the number of payable positions with a simple formula.

# of ways = # positions on reel 1 * # positions on the reel 2  * # of positions on reel 3….

Let’s say you come across Mr. Dean Martin.  How many ways are there to line up a symbol on this mess of a machine?

Spoiler Alert:  2 * 2 * 4 * 4 * 4 = 256 ways to line up 5 symbols.  Ain’t that a kick in the head?!?

For the sake of space, I’m going to invent a 3x3 machine that has 5 pay lines. My make believe machine will have a probability of 0.000025 of landing the top paying symbol across all three reels in a scatter pattern (about the same as a royal flush).  

Here are the probabilities of landing the top symbol by reel:

If we believe the symbol is equally likely to appear in each position on each reel, then the probabilities by position look like this.

The only thing that makes me nervous with the last assumption is that some spots on that reel are indeed more valuable than others based on the arrangement of the pay lines.  If you are tracking by position, you would just update the table above with your actual observations.

Pay Line View 1, Patterns:

Pay line View 2, Aggregated:

Notice the middle position has more payable combinations that all other positions.  Could a slot designer make a game where the jackpot symbol just doesn’t land in the middle spot as much as other positions?  Sure they could.  If you are really that worried about it, track all 9 spots…  There would be no other way.  I think it’s more likely the game designer would just make it so reel 2 lands less frequently than reels 1 and 3.

Anyway, I will do math in long form so either the by position or by reel approach works. Let’s define the payable positions as follows:

With these definitions, I can do math by line to get probabilities by line, then add to get the probabilities for all lines:

What’s great about math is you can test my easy approach to confirm you get the same number.

Oh look, the totals match.  Sooooo cute!

Anyway, I hope I understood your question properly and the answer made sense.