Simulations reveal true odds of casino games

Some jurisdictions required a volatility analysis or a confidence interval report in order to have the game approved, if it is an electronic game. They make a certain amount of sense for games with no strategy to help prove that the game is functioning as it should.

I have a tougher time understanding the point for a game like Blackjack where the actual payback will bear no resemblance to any theoretical one. But, if this is what the regulators require, who am I to argue?

Essentially, I have to run some very large simulations that are broken down into sets of hands. So, for example, I might run 100,000 sets of 10,000 hands. Yes, that is 1 billion hands in total. You don’t actually need this many hands to accomplish the analysis, but the same set of hands is used on the larger sample sizes which do require a larger total number of hands.

The standard deviation is calculated on the payback of these 100,000 hands. I don’t expect you don’t know what this is and quite frankly, I let the spreadsheet do the calculation. This number is multiplied by 1.96 to get to the 95% confidence interval.

Now, this term you may have heard. Every time you hear about a poll (political or otherwise), you hear about the margin of error. What they don’t usually tell you is what the confidence interval is. Most political polls are done with a 90% confidence interval. What this means is that if they were to repeat the poll over and over again, that 90% of the polls would come back with results that or within the margin of error given. This column isn’t about polls, but the same principles apply to my simulations.

I recently ran one of these simulations for the player wager in Baccarat. The theoretical payback is 98.7649%. When I ran 1 billion hands, the average payback came in at 98.7623%. This simulation was broken down as mentioned previously. The standard deviation of the paybacks was 0.9475% and the 95% confidence interval was 1.8571%. This means that if I were to run a 10,000 hand simulation, that 95% of the time, I should find that the payback comes out to be between 96.9052% and 100.6193%. This is for a game with relatively low volatility and over 10,000 hands.

What I found my interesting in this was that over 10,000 hands of Baccarat, a player should expect to win some ‘significant’ number of times. The numbers presented so far do not really tell us how often. So I dug into the stats and found that 9.4% of the time, the player will win over 10,000 hands. That is a lot of hands for a player to still find himself on the winning side of the ledger for a game with a moderate house edge.

Of course, the banker wager is even better for the player and we find that he will still be winning at 12.73% of the time after 10,000 hands. The volatility of the wagers is remarkably similar. The biggest difference is that the banker wager has a house edge that is about 0.2% smaller. Imagine what numbers we might find if we did something similar for Blackjack with a house edge roughly half of that of the banker wager.

Unsurprisingly, the lower the payback goes, the less likely you’re going to see the player still winning after 10,000 hands. If I look at the data for a sidebet with a 90% payback, I will likely find zero or near zero times that the player is still winning. Also unsurprisingly, as the sample sizes increase, the number of sessions that the player will still be winning will also dwindle quickly.

As I move 100,000 hands, I will likely find that the player is unlikely to be winning at the end of any of them. But, it’s nice to know that over the not-so-short run, the little guy still stands a chance. 

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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