If you do something that is subject to random chance a lot of times, the observed average outcome will converge on the theoretical average outcome.

Example: the theoretical average outcome of a six-sided die is 3.5 ((1 + 2 + 3 + 4 + 5 + 6) / 6). If you roll it 10,000 times, you’ll end up with an average that is very close to that.

There are 2 parts to this law.

The first is that if you do something many many times, it will tend to the average. For example if you flip a fair coin 10 times, you might get 70% heads and 30% tails. But if you flip it a million times, you might get 50.001% heads and 49.999% tails. Side note, if you flip a coin enough times and it does not tend towards 50%, you can calculate that the coin is unfair.

The second, known as Law of Truly Large Numbers in Wikipedia, is that if you do something enough times, even very unlikely events become likely. For example, if you flip a coin 10 times, it is very unlikely that you will get heads 10 times in a row. But if you flip a coin a million times, it is very likely that you will get heads 10 times in a row, and even 100 times in a row is still quite likely.

Imagine you have a fair sides coin. If you flip it once and it lands on head. So the average result is 1 (heads=1, tails=0). If you flip it again, you might get another head average is still 1. The third you might get a tails. Now the average is 0.66, much closer to what the true value.

The law of large numbers states, as you take more samples, the average of the samples will get closer to the true value.

This is because the chances of getting 2 heads and 1 tail after 3 flips is 0.375. However getting 20 heads and 10 tails after 30 flips is 0.028. And so on.

You can calculate how likely landing any number of heads is with the formula

0.5^(sample size) * (sample size) choose (no heads)

The choose function states how many different ways you can rearrange the heads in the sample.

So the formula is saying,

number of possibilities that the result happened times likelihood of each possibility.

Statistician here. Others have given good answers about *what* the law of large numbers is, I want to give perspective on *why it matters*.

If you’re trying to find some underlying truth about numbers, you need to gather a lot of data points to eliminate “randomness” and chance.

– consider a baseball team. The best baseball team might win 100 games and lose 62 games during the long regular season. We can confidently say they are a *good team* because we’ve seen them play enough that we are confident we’ve seen enough wins and losses to know they are truly good, *they didn’t just get lucky*. When the baseball team gets to the playoffs, they might lose 2 out of 3 games to not advance to the next round. *We cannot say this makes them a bad team*, because they may have just been unlucky or had a couple fluke games.

– or consider rolling a six sided die. It’s not unreasonable you roll a 2 then another 2. Does this mean the average roll of a six sided die is 2? No of course not! You need to roll the die a lot more to get the actual average; the more you roll, the closer you’ll get to the actual underlying value.

– or in political polling – if you ask 3 random people their political preference and they all say they’re going to vote for the same political party, you can’t say “oh this means that party is guaranteed to win the next election!” because you have randomness in that small sample. You’d need to ask lots more people before you start to get an accurate guess about who will actually win.

– or say you’re playing poker with a friend and he deals himself 4 aces (a very good hand). Should you accuse him of cheating? No, he probably just got lucky. But if he deals himself 4 aces every time he deals 20 times in a row should you accuse him of cheating? Probably, because you’ve seen enough deals to know this probably isn’t random, he’s stacking the deck somehow. Don’t play games for money with this friend, he’s a cheater.

This is why the law of large numbers matters. With large enough data, the actual underlying truth is revealed.

The law of large numbers is a way of saying that given enough attempts, results will tend towards expected averages.

Fire example: if you have a 100 sided dice and roll it once. You expect every number to have a 1-in-100 chance of occurring, but after 1 roll only one value will come up… say 61. While every other value doesn’t get rolled.

Roll the dice 1,000,000 times however (a large number) and now each value will get rolled about 10,000 times +/- a couple.

This can be really valuable in say insurance where a company wants to insure thousands and thousands of people/cars/houses so that what actually happens is close to what you would expect for long term averages. If they only insured a single house then claims would be all over the place and much harder to predict.

When working with large numbers, probabilities converge to their theoretical value.

If event A has a 1% change of happening when I do B, if I do B 10 times, it’s very unlikely A happens. If I do B 1 million times, now it’s very likely that A has happened at least once.

The law of large numbers is a way of saying that given enough attempts, results will tend towards expected averages.

Fire example: if you have a 100 sided dice and roll it once. You expect every number to have a 1-in-100 chance of occurring, but after 1 roll only one value will come up… say 61. While every other value doesn’t get rolled.

Roll the dice 1,000,000 times however (a large number) and now each value will get rolled about 10,000 times +/- a couple.

This can be really valuable in say insurance where a company wants to insure thousands and thousands of people/cars/houses so that what actually happens is close to what you would expect for long term averages. If they only insured a single house then claims would be all over the place and much harder to predict.

The Law of Large Numbers is what people mistakenly refer to as the “Law of Averages”.

That’s all you need to know about it.

Flip a coin 2 times. There’s a 50% chance it’s heads. But it’s totally likely you will slip tails twice in a row. As the number gets bigger, the more likely it is for the number of coin flips to reflect the true statistics. For 100 it could be 60 tails. For 1000? 550. 10000? 5001.

It depends how mathematically precise you want to be. There is a weak version, and a strong version. The ELI5 version probably isn’t precise enough to make a distinction between these.

The ELI5 version is that the average (mean) of many samples of the same random process will tend towards the true average (mean), as you take more samples.

To be more precise, for the weak law of large numbers, if we have a sequence of i.i.d.r.vs (independent and identically distributed random variables) X_i, each of which have expectation E(X)=mu, then the mean of the first n variables X_i, which is typically denoted by a bar over X_n, is a random variable which converges in probability to the expected value. By definition, this means that for any given distance epsilon from the true mean, the probability that the mean of the first n random variables is within this range of the true mean tends to 1 as n tends to infinity. So in other words, you can pick any small (but nonzero) range around the true mean, and any probability close to (but not equal to) 1, and I will be able to find a number N such that N or more copies of the random variable will have a mean within this range of the true mean with probability greater than your given value. In other other words, with enough samples, the sample mean will be arbitrarily close to the true mean with arbitrarily high probability.

The strong law of large numbers is more difficult to express in words, while conveying it’s true meaning. It basically says that if you took a sample mean for every sample size n as n tends to infinity, then this sequence of sample means will almost surely converge to the true mean: there is 0 probability that it will do anything else (converge to a different value, or diverge). I don’t know what the ELI5 version of this is. Imagine taking a sample of size 1, then a sample of 2, then 3, etc. Then you’ll get a “better” average each time. The law states that your sequence of averages will converge to the true mean 100% of the time (so if you did this many times, the proportion that did anything else would have “measure” 0; almost all of your sequences would converge to the true mean).