How Long is the Long Run?

9 March 2006

By Elliot Frome

Much of the criticism of Expert Strategy is based on the idea that the ‘long run’ in video poker would take either years of habitual play or several lifetimes for the more casual player. To be honest, I’ve never really seen a complete analysis of this concept. Each version of video poker will also have its own ‘long run’ based on the volatility of the game. In relatively simple terms, a game will have higher volatility if a larger percentage of its payback comes from rarely occurring hands. Thus, double double bonus poker, which has what amounts to 2 jackpots, is far more volatile than the original jacks or better. The higher the volatility, the more hands it will take to get to the long run.

At the same time, it’s not as if everything that occurs BEFORE you get to the long run is meaningless. Basic laws of probability tell us that as you play more hands the more likely you will approach the theoretical payback. This doesn’t mean you should play more and more hands. I don’t advocate playing for hour upon hours. For almost everyone reading this, video poker is a hobby, and no hobby should be all consuming. Rather, it means that no matter how often you play or how much you play, each hand that you play brings you that much closer to the long run, and that much closer to approaching the theoretical payback if you start counting from the moment you first started playing.

To illustrate this, think of a simple coin toss. After 1 toss, you’ll have either 1 head or 1 tail. If you were to wager even money on the outcome, we know that the payback of this game would be 100%, yet after 1 toss, you’ll either be at 0% or 200%. If you toss it again, you’ll have a 25% chance of being at 0%, a 25% chance of being at 200% and a 50% chance of being at 100%. If we continue the math, we’ll find that as you flip the coin more and more, the probability of being at each extreme will diminish, while the probability of being at the theoretical payback will increase.

While the math associated for video poker is far more complex, the basic concepts are still the same. If you play one hand of Video Poker, you might get nothing and have a payback of 0% or you might hit a Royal and have an 80,000% payback. Or, you will get one of the hands in between and have that particular payback. In fact, after a single hand it’s not even possible to have the exact theoretical payback. After 2 hands, the odds of having a 0% payback will diminish. Most versions of video poker have a losing hand occurring about 55% of the time. So after one hand, you have a 55% chance of having a 0% payback. After 2 hands, this decreases to about 30%. As you play more and more hands, you’ll find your payback begins to approach the theoretical payback. Unlike our coin toss, which will very quickly approach the theoretical payback, video poker will take a bit longer. There are far more possible combinations in video poker, and with the Royal, which occurs roughly every 40,000 hands, accounting for 2% of the overall payback, we’re certainly not going to reach the long run in just a couple thousand hands.

To get a sense of just how long the long run is, I started with the most common hand in video poker, the Low Pair. These account for just a bit below 30% of all hands. I ran a simulation of 1 million Low Pairs, and I tracked the results both of each 1000 hand ‘session’ and the total after each 1000 hand session. I found the results to be rather interesting.

First of all, the theoretical payback (i.e. expected value) of a Low Pair is 0.82368. After 1 million hands, the overall payback of the simulation was 0.82326. After 1000 hands, the simulation showed a payback of 0.857. This was close, but still clearly ‘different’. After 5000 hands, the simulation showed a total payback of 0.839. Getting closer. After 10,000 hands, the simulation showed a total payback of 0.8124 as the machine went ‘cold’. At 100,000 hands, the simulation showed a payback of 0.82031, and from that point never went below 0.82 or as high as .827. I took this to mean 100,000 hands was a good indication of where the long run for this particular type of hand.

The other statistic I gathered was the payback of each 1000 hand session within the 1 million hands. This showed that 66% of all sessions had a payback between 0.77 and 0.87. Also, 93% of the sessions had a payback of between 0.72 and 0.92.

Does this mean that after 100,000 hands we can expect a video poker machine to be very close to the theoretical payback? Absolutely NOT. The hand I chose was the most common one, but also one of the less volatile ones. Hands that can turn into a Royal will have much higher volatility and will take more hands to get to the long run. At the same time, these hands make up a much smaller percentage of the total possible hands. After a Low Pair, the most common hands are a High Pair, 1 High Card and 2 High Card hands. A High Pair will behave fairly similarly to a Low Pair (actually it is LESS volatile because every hand is at least a Pair), while the High Card hands will have quite a bit higher volatility because they can wind up as so many possible other hands.

So, what can we take from this? A few things. First, the long run is NOT a few thousand hands. It is also NOT several lifetimes of play. The average expert player can play 600-800 hands per hour. So, even playing 100,000 hands will take ‘only’ about 165 hours of play. If you’re a professional, you might do this every month. If you’re a casual player who comes to Las Vegas a few times a year and plays mostly video poker, it might take 3-4 years. It’s not a week, but it’s not a lifetime either.

More importantly is the lesson that you don’t have to EVER reach the long run to approach the theoretical payback. As we saw from the example, even after 5,000 or 10,000 hands, we begin to get very close to the number. By using expert strategy, you make sure that the number you are approaching is maximized. If you choose to deviate from the strategy, the payback you will approach must by its very definition be something less than expert strategy.