The Law of Large Numbers (LLN) is a theory which states that over time, a large number of trials or experiments will result closer to the expected value or result. It was first theorised by a 17th century mathematician by the name of Jacob Bernoulli.
LLN is commonly analysed by casino managers who understand that while the occasional short-term loss will happen, in the long run the house wins.
Flipping a coin is a great example of LLN. When you flip a coin, there is a 50% expectancy it will land on heads, and a 50% expectancy it will land on tails. If you flip the coin once and it lands on heads you are at each end of the probability spectrum. Heads has occurred 100% of the time, and tails 0%.
The chance of this result occurring is 50%.
If you flip a coin 10 times the chances of turning up heads 10 times is 0.097%. Even though each flip has a 50/50 chance of turning up heads, to do it continuously increases the odds.
If you have already flipped 9 heads, what are the chances of flipping 10 heads in a row? The answer is 50%. This is because 9 heads have already appeared, and you simply need 1 more to get to 10.
Due to the extremely small chances of one result continuously occurring, the LLN comes into play. After thousands of coin flips you will likely see a very close to even split between heads and tails (percentage wise).
This is evident in the graph (pictured above), where you can see sporadic percentages at the start of the experiment, but as more and more coins are flipped, the overall percentage becomes closer and closer.
While it is technically possible for you to flip heads even 100 times in a row, you would have to run millions upon millions of simulations to find one that does.
How does the Law of Large Numbers relate to poker
Poker is a game of patience and variance. The players who understand poker variance and are able to apply patience to their game, will be winning players.
All players go through rough patches, even professionals. But what makes professionals so successful is that it doesn’t bother them. They know that soon enough, they will have a great patch which will outdo the rough patch. Take a look at any professional players results graphs, and you will see periods of what seems to be complete horror, but there is an overall trend of ‘up and to the right’.
If you have an edge in a poker game, you will eventually beat it.