People sometimes use the term ‘law of averages’ to mean that in a set of steps, the results will average out over time. Putting it in simpler terms, if we believe that the law of averages is real, tossing a coin in the air a million times will result in the coin landing heads-up 50% of the time and tails-up 50% of the time. This isn’t true, though, and the law of averages has no actual meaning.

Indeed, this is the reason so many people lose so much money by gambling in casinos and other gambling establishments. The way the notion works is that “Joe” has lost 10 times in a row, so he thinks that it is time for his ‘luck to change’. Professional gamblers know better, but this is what most people might think. Joe will firmly believe that 10 losses in a row increase his chances of winning the eleventh time. Why isn’t it the case?

### The fallacy of law of averages

The law of averages isn’t actually a scientific law. It is a factual fallacy. It is an attempt by everyday people to understand a scientific law called the Law of Large Numbers. The law of large numbers says that given a large enough sampling, any deviation from the expected probability will average out. This may be a little confusing, but the important thing is that it is **NOT** the same as the law of averages.

The biggest problem with the law of averages is that it assumes that the results will even out with a relatively small number of attempts, mostly because of the belief that each attempt changes the odds of having the desired result. Thus, the assumption is that if you flip a coin 10 times and it lands heads-up all 10 times, that the next flip should result in a tails up result because that result is somehow overdue.

### Why law of averages doesn’t work

There is a problem that should be apparent. It is this: The belief is that assuming that the coin has an identical chance of landing heads up or tails up, which gives a ratio of 1:1, if the coin is flipped 11 times and lands heads up the first 10 times, the chances are 11 to 1 that it will land tails up on the eleventh flip.

Did you see the mistake in this line of thinking? If the ratio of heads to tails is 1:1, that ratio doesn’t change, no matter how many times the coin is flipped. This means that even though the result was ‘heads’ 10 times in a row, there is still a 50% chance that it will land heads up on the next flip. The flips aren’t additive, they are independent of one another. This means that what happened the previous 10 flips has absolutely no bearing on what the result is going to be on the next flip.

Put in a slightly different but valid way, the coin has no memory of what happened before.

This is one of the problems with trying to apply probability laws and theories with things that aren’t governed by probability. Also, the more complex the set of steps, the less merit that the law of averages has.

Here are a couple of examples that show how worthless the law of averages is:

Let’s say that you live in a place that gets an average of 300 days of sunshine per year. It has rained all day for the past 65 days. The law of averages would indicate that the next day, it should be sunny, right?. However, this would obviously be an error if the notion was based solely on the numbers.

How about if the statistics show that 10% of a town of 1,000 are sick with the flu. Let’s say that you talk to nine people and every one of them is well and healthy. Law of averages says that the next one you talk to will have the flu. In reality, though, assuming that the statistic is correct, you could actually talk to 900 people before speaking to one who had the flu.

The law of averages really doesn’t mean anything. It is a layman’s term to try to understand an actual scientific law and it is almost invariably applied to far too small a sampling to have any meaning or to things that are totally not governed by probability.

Unfortunately, a vast number of people try to use the law of averages on things that have nothing at all to do with statistical probability. They might not call it the law of averages and they might not even think of it that way, but that is exactly the principle they are attempting to use. Saying something like, “It averages out in the end” is an expression of the law of averages.

This is no joke. Have you ever wondered why weather forecasts are so often in error? It is because the forecast is based largely upon the law of averages. If they say that there is a 10% chance of rain, it means that in similar weather systems, there has been rain 10% of the time. That is all that it means. Thus, forecasters can be caught saying that there is only a slight chance of rain in the middle of a deluge. That is only one example. There are many more that are used daily by people in their everyday lives and by governments making policies.

Suppose you flip a coin nine times in a row, getting heads every time. Of course, the tenth flip will have the same odds as before, but the probability of getting heads ten times in a row is quite low. Even though the tenth flip is still even odds, it is still unlikely that you will flip ten heads in a row. The individual flip has the same odds as always but the probability of getting N in a row will continue to drop

It still isn’t the law of averages. It *starts* toward the path of the law of large numbers, simply because the more flips there are, the more certain that the averages will begin to even out. There have been people who have ‘broken the bank’ in Vegas by not playing the odds. Again, playing the odds would be the law of averages and since people have won enormous sums by going against the odds, it is an indicator that the law of averages has no meaning. If it actually had meaning, the chances against breaking the bank would be so astronomical that it would virtually never happen. It wouldn’t be impossible, but it would be extremely improbable. Yet, it has happened a number of times.

Sure, I’m just bringing up that even though the odds of every coin toss remain the same that the aggregate probability remains a factor. It doesn’t effect any given coin toss, simply describes the most likely overall trend. I wasn’t trying to argue for the law of averages (which isn’t even a law) and I really liked your article. It just felt like you left that part out…

That would actually be a great point to write about to explain the law of big numbers. In fact, it does a really good job of explaining what the law of big numbers is. You’re right, that probability remains a factor and the larger the sampling, the greater that factor becomes. Statistically, the odds against 10 heads-up tosses in a row aren’t enormously high. The odds against 10,000 heads-up tosses in a row are so great that they are very close to impossible.

…and yet the odds of each toss in that series remains 50/50. I expect most folks believe in the law of averages just because it is easier to (mis)understand

You have given me a different perspective about law of averages. Thanks!

I’m very glad to hear it. Way to many people use the term without realizing what it is they are saying.