Law of Large Numbers in Poker

Whenever someone mentions the law of large numbers, usually they mean something along the lines of “if you try something enough, eventually it will work”. It’s a really dumbed down version of the law. Originally the law has been invented by Jakob Bernoulli, a Swiss mathematician, and we can translate that to mathematics:

Let’s take an experiment, where an A event’s probability is p. Let’s do the experiment n times and let us make zn the amount of times the event happened in the chain of experiments. At an e>0 and d>0 of your choosing, there is a n0 , wherein the case of n>n0 P (|zn/n-p|≥e) ≤d.

So, you’re feeling it that this isn’t what most people are talking about when they are talking about the law during a poker game, right? I’d wager that if someone’s aware of its practical application it won’t even put it in a poker context.

So the layman’s use of it and the mathematical understanding is hiding two separate things. There’s nothing wrong with that, there are other modified common phrases borrowed from sciences. On the other hand, there are multiple wrong assumptions based on the law, which are worth talking about.

The most common mistake

The law is misunderstood pretty often, sometimes even by poker players. The most important misunderstanding is about long-term results. It’s commonly believed that the random events will always somehow make up for themselves in the long term, so the probability will match the occurrence in the long run. Here’s an example.

It’s a rare, but great trait to have to not be bothered by a lost hand every now and then. Let us make this an 80-20% scenario, let’s say preflop all-in with aces against twos. One of my friends (unlike me) is one of these calm people, this is what he says about this:

  • “I know I’ll get it back, these situations work for me 80% of the time.”
  • “Yes, they will, but only from now. The devil took this one already.”
  • “I don’t think so, this one also counts in it!.”

My friend is a really solid poker player who has achieved much more success than I have, but this time he’s wrong.

Coinflip situations in the long-term

If you play many coinflip situations during your life, you think the losses and wins will even out eventually. Let us consider these situations 50-50 for now, it’s easier to count like this. Let’s say your career starts pretty, unfortunately, you lose the first 3 times. If the original assumption is correct, then you’ll have to win more than half of the coinflips coming later on, or else it won’t even out. You can feel that something’s not quite right here. Not the deck, nor the PokerStars RNG (random number generator) can remember how bad you ran earlier, so coinflips are going to be 50-50 even in the future.

So what’s the solution? Cards definitely don’t remember, but some kind of evening out will definitely be noticeable. Wins and losses won’t even out, but the ratio of the two numbers will come closer and closer to 1. Let us look at a larger sample of numbers, it’s going to be much more simple that way!

Let us say you won 45 out of the first 100 coinflip games and you lost 55. The losses are ahead by 10, and the ratio is 45-55%. If the bet has always been the same then you’re not in a pretty good spot economically either. Let’s say the bet is always 1 dollars, then -10 dollars is your current standing. This doesn’t ruin your mood, you continue the game. A couple months later you look at your statistics and can experience the following: You’re past 1000 coinflip games, where the number of wins is 490, the losses 510. This means that the loss-win ratio got a lot better, seeing as its 49-51%. Despite it getting better, you’re not better off, seeing as the losses are ahead by 20 at this point, so the final results are -20 dollars for us.

If you make the sample larger, it’s going to be true that the 49-51 will get closer to the 50-50, but the difference between losses and wins probably might not even out. Your -20 result could easily get worse, however, it might turn around and go into the positives.

There are no mathematical laws guaranteeing that your chips and dollars lost earlier will ever get back into your possessions. Your proper game knowledge, readiness, and good decisions might do that.

Let’s also take note of the fact that the 50-50 used in your example can very easily be changed to be 80-20, or anything else, the point won’t change.