The Law of Large Numbers in Poker

One of the toughest things to get around in the game of poker, is actually losing a hand but knowing at the same time that you made the right play. Let's face it, losing a hand is never fun and hardly ever sparks a positive emotion. Your opponent hits his draw, somebody fills that inside straight, or a miracle river card turns his lower to pair into a full house. "That's poker." But that's no consolation - or is it?

 

The law of large numbers is a statistical reference to the imperfections of reality being smoothed out over a large number of repetitive scenarios. This too works in poker, because poker is a game based on probability. Empirical probability is the likelihood that an event will have a certain outcome in real life. Because the data is empirical that means that certain things are going to happen based on past real life data.

 

Currently there is poker software on the market, like Poker Academy or Wilson Turbo Poker that allows you to play a scenario out thousands and thousands of times, emulating real life results. Now when you play a hand 10,000 times, and one of these programs proves that you are correct in your assumption that it is a profitable play, you are simply confirming that you have a long term strategic edge. But a long-term strategic edge doesn't mean you win every hand. If it did you wouldn't need statistics and probability to analyze any poker situation.

Let's say you rolled one die 10 times and recorded the outcomes. We already know there are only six possible outcomes from each event. The outcomes are 1, 2, 3, 4, 5, and 6. These were your results:

1 = rolled 0 times.

2 = rolled 2 times.

3 = rolled 1 times.

4 = rolled 0 times.

5 = rolled 4 times.

6 = rolled 3 times.


On average, each of the numbers would roll between one and two times, but as you can see the results over such a small test are heavily skewed. Let's say you're at a 6 seated poker table, and you happen to be the number 4 just like in the test above. You have won nothing, while number 5 over there has won four times! With just 10 rolls of the die, number 5 has been extremely lucky hitting 40% of the roles! Have you ever played with a player like this in online poker? Absolutely, we all have but that certainly doesn't mean anything to the skill of player.

Let's do another test, only this time, we will roll the die 100 times. Here are the results:

1 = rolled 11 times.

2 = rolled 15 times.

3 = rolled 9 times.

4 = rolled 18 times.

5 = rolled 27 times.

6 = rolled 20 times.

While number 5 is still ahead, numbers 6 is closing the gap, and number 1 and 4 are now on the board. Number 5 has still gotten a larger share of the roles compared to all the other numbers. Now that's over 100 hands. If you extrapolate a situation like this over a session at a poker table, you can still see that, over several hours of play, one player could still have some exceptional luck. This is empirical statistical data that is much more accurate than our first test of only 10 throws, however, 100 throws still is not enough to say which number is the best number. It has only determined that number 5 is the luckiest number of this sampling.

Most poker players, who pay attention to win rates, will start taking credence in long-term numbers after 250,000 played hands. So when you factor in luck and skill to those players that really study their game, they will not give a pittance of thought to data collected over 100 hands, citing it as far too small a sample. In that spirit let's do one more test, and this time, we will run a sampling of a thousand roles. Here are the results:

1 = rolled 145 times.

2 = rolled 188 times.

3 = rolled 129 times.

4 = rolled 181 times.

5 = rolled 166 times.

6 = rolled 191 times.

Remember each of these roles are considered events, and due to our test size, the disparity between the results of each event is narrowing simply because of the most fundamental of loss in probability - the higher the number of events in each experiment will result tabulations approaching the theoretical mean.

So now you can see in our experiment, things are changing. Maybe number 5 was just a lucky player to start after all, we still don't know for sure, but now all the players [die] have had their wins, and the disparity between all of the players has diminished. When you play poker, and you're analyzing hands, situations and pots that come up, there are numerous factors to take into account for you to be able to determine if you actually played optimally.

Over time, thousands and thousands of hands, you are going to get a very accurate picture of how often you make correct decisions at the table, by utilizing the law of large numbers. Now all you have to do is ignore short-term results and concentrate on the big picture.