Standard deviation accurate tool for casino gaming

Over the years, I have used an expression that essentially says the longer you play the more likely you will approach the theoretical payback.

What does this really mean? Some naysayers have tried to claim that using something like Expert Strategy is useless because the long run is so far out there that it is meaningless for any human being.

I would dispute this notion for a variety of reasons for most wagers in the casino.

There is no single point that turns any wager from the short run to the long run. We cannot say that after 1,000 wagers or 10,000 wagers you are now in the long run.

As my opening sentence states, the more you play the more you approach the long run and the more likely you approach the theoretical payback.

I’m going to use my favorite subject — math — to try and show you how this works.  I’ll do my very best to explain it in the simplest of terms.

I created three simulations to prove my point. Each played 10 million hands of Blackjack, using fairly common rules. The only difference between the runs is at what point I collected the data.

In the first run, I collected after every 100 hands. In the second, 1,000 hands and in the last, 10,000 hands. Besides being round numbers, I picked these totals because 100 hands is roughly how many you might play in an evening.  One thousand hands is about what you might play if you came to Las Vegas for a week and the 10,000 might represent a couple of years’ worth of play.

Let’s start with some very basic numbers. The high and the low. The table below shows the lowest and highest payback for any single session within each of our simulations:

I have to admit, even I was surprised when I saw the numbers for 100 hands.  I could believe a 70 percent payback over a few hours. But I almost couldn’t believe a nearly 178 percent payback! Even in this simple table, you begin to see how the numbers begin to head towards the theoretical payback.

But, minimum and maximum are not very good indicators necessarily either.  Casino games tend to be skewed a bit. Generally speaking, players have a better chance to have big winners, but a lower chance of winning at all.

The numbers above bare this out. If we were to take a simple average of each set of two numbers, one might believe that the payback of Blackjack was over 100 percent. But, the overall paybacks of each of the sessions were in the 99.3-99.4 percent range.

A far more important tool to tell us what is happening is called the standard deviation. This is rather complex formula which thankfully a spreadsheet makes rather easy. The standard deviation gives us an idea of how the data spreads from the average. This in turn can tells us what percent of our samples we can expect to fall within a certain range.

This essentially is what is done when we hear about a margin of error or confidence intervals when we hear about polls being done. The table below shows the average and standard deviation for each of our samples:

So, as you can see, the overall average of each sample did not vary much.  Just a little over 0.1 percent between them. You’ll note that as the sample sizes went up, the standard deviation went way down.

What does this number mean? We can expect that just over 2/3 of our sessions will have a payback of +/- of that standard deviation.

So, for a 100-hand session, 2/3 of them will have a payback between about 88.7 and 110.2 percent. For our 1,000-hand sessions, this gap closes to about 96.1 — 102.6 percent. The 10,000-hand sessions have a range of only 98.1 — 100.5 percent.

To take this a bit further, the math teaches us that if we use roughly double the standard deviation, we would find that 95 percent of our sessions fall into this range. I’ll let you do the math for that one. But, what does all this really tell us?

Well, first of all, as I stated at the beginning, there is no point at which the short run becomes the long run. The larger the sample, the more you will head towards the theoretical payback.

All of these simulations were done assuming that you play using proper strategy. If you decide to never double on soft hands, the theoretical payback might drop from 99.4 percent down to 98 percent.

In a short session of 100 hands, you may not really notice the impact of this, with such a wide range of potential results. As you get to 1,000 hands, you might notice it. There is a chance that you’ll do better using this strategy, but it is a small one.

If we take it to 10,000 hands, the probability that you’ll do better using this alternate strategy decreases even further.  This is why it is imperative that you follow the proper strategy regardless of whether you are playing for the short run or the long run — because they really are the same thing.

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About the Author

Elliot Frome

Elliot Frome’s roots run deep into gaming theory and analysis. His father, Lenny, was a pioneer in developing video poker strategy in the 1980s and is credited with raising its popularity to dizzying heights. Elliot is a second generation gaming author and analyst with nearly 20 years of programming experience.

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