Math ultimately wins with casino games

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We’ve all the heard expression “On any given Sunday.” The implication is that on any given Sunday afternoon, any football team can beat any other football team.

There may be favorites and underdogs, but those represent long-term probabilities. In the case of sports, long-term probabilities don’t reflect how the teams will do over time, but rather if there were some way for the teams to play countless times at a specific moment, what we could expect to happen. Where sports are concerned, this is just a theoretical math model that is rather difficult to prove.

Where casino games are concerned, it is far more simple.

My father, Lenny Frome, used to recommend that people avoid betting on any event in which the participants had two legs. Humans are unpredictable. We are prone to emotion and to being affected by all sorts of outside events. 

Casino games do not have these issues. In the poker room, you are actually wagering on an event involving humans. On the main casino floor, you are wagering how inanimate cards (or similar objects) will behave.

Let’s take a look at an obvious situation. In the game of roulette, if you pick a single number you get paid 35-1. So, the payback of a single number is 36/37 for a single 0 wheel. This is 97.3 percent. Yet, on any given spin, there are only two possibilities. The number comes up, which has a payback of 3600 percent or it doesn’t, for a payback of zero.

But, unlike our sports comparison, the passage of time doesn’t change the specific results that we can expect. We can spin the wheel now, an hour from now, a day from now or a year from now and the expected results do not change. Thus, we can rely on the theoretical payback as a guide to what to expect.

Payback is no doubt a critical part of the equation in what to expect for the player. Some people have tried to criticize the idea of using the “long run” payback as anything meaningful for the player.

Just how long is the long run? That depends on the wager. I don’t think it takes a math genius to realize that a wager like a single number might have a different long term than say picking Red/Black. The one constant about long term is that the longer you play, the more likely you approach the theoretical payback.

So, at one spin, you’re at 0 or 3600 percent. At 100 spins, you’re going to be somewhere in between and likely a bit closer to the 97.3 percent. At 1000 spins, you’re likely to be even closer to that theoretical payback. If you go to a million spins, you’ll likely be right on it.

If we were to track the results for our Red/Black wager we’d probably find after 1000 spins that we’re very close to our theoretical payback. Even at 100 spins, we will be far closer to that number than we would be with our single number wager at the same number of spins. This is called volatility.

Volatility has a huge impact on the player’s experience. Imagine two different players who come up to our Roulette table with $100. Each wagers $5 per spin. One on Red, the other on the number 21. The payback of both wagers is the same — 97.3 percent.

What is the likelihood that each player goes bankrupt after 20 wagers — that is, walks away losing on every spin?  For the player playing 21, he has a better than 57 percent chance of losing all 20 spins. Our Red player has a 1 in 615,000+ chance of the same.

Conversely, our player playing 21 has a much better chance to double his payroll in that time. If he hits his number once, he will nearly double his money. He has a 32 percent chance of doing this. If he hits his number twice, he will more than double his money. He has an 8.5 percent chance of doing this.

Our Red player would need to hit his wager 14 out of 20 times in order to win just over $100. There is no way for him to win exactly $100. He has a 3 percent chance of winning exactly 14 of the 20.

Obviously, he could also win more than 14 which would also double his money. In total, the Red player has about a 5 percent chance to double his money vs. a 10+ percent chance for the single number player.

This was over 20 spins.  If they were to sit there for a few hours and play 100 spins, the numbers would begin to converge. Sit down for a week and keep doing it and slowly, the players will move closer and closer to the 97.3 percent payback.

Of course, the probability of our single number player going bankrupt first is also much higher and he might not make it to the end of the week. This is another way volatility impacts the player. His bankroll must be prepared for it.

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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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