The Gambler’s Fallacy

Have you ever walked past a roulette table and decided to place a bet on black because the last few spins had all been red? If you did this and believed it to be true, you were falling under the spell of the gambler’s fallacy. Out of all the “systems” and “superstitions” that people have about how to beat gambling games, the gambler’s fallacy is the one that is always on the guest list.

By definition, the gambler’s fallacy is the “mistaken belief that, if something happens more frequently than normal during some period, it will happen less frequently in the future, or that, if something happens less frequently than normal during some period, it will happen more frequently in the future.” Basically, it’s the belief that the natural laws of statistics are forced to even themselves out before your very eyes.

Learning by Example

If you’re confused still, that’s ok. This is another one of those things that is best learned through examples and analogies. Now, before we go into this example, we need to make one thing clear. The gambler’s fallacy only applies to situations that are completely random. We will give an example of this shortly as well. Let’s look at the famous example of flipping a coin.

A coin flip is completely random in that it will land on heads 50% of the time and land on tails 50% of the time in the long run. Imagine that we flip the coin once and it lands heads. We flip the coin again, and it’s heads again. We flip the coin again, and it’s heads. What are the chances that the coin will flip tails on the next toss? The answer is 50% or the exact same odds as the first toss. If you’re thinking that it is more likely to land on tails because it’s landed on heads so many times, you are falling victim to the gambler’s fallacy.

The point is this. Each toss is completely independent of the last one. There is absolutely no carryover and no reflection on the past history of the tosses. Coins do not have memories, and there is not some magical force or statistics police that will come correct any mathematical improbabilities.

Here’s why people get confused. For those that do not like math, we highly recommend skipping down to the next section. We will do our best to keep it simple.

The odds of the coin landing on heads is 50% or 1 in 2 which can be written as the fraction 1/2. To find the odds of two different things happening in sequence, you multiply the fractional odds of each together. So, for example, the odds that the coin will land on heads twice in a row are:

If we wanted to find the chance of the coin landing on heads 4 times in a row it would be:

So you’re probably saying to yourself, wait, so that fourth toss should be way less than a 50% chance it will be heads if it’s only a 6.25% chance it will land four times in a row. This is wrong. Each flip is still independent and has the exact same odds of landing on heads or tails. These odds only talk about the likelihood of the statistical variance landing this way.

Sometimes people struggle to wrap their heads around it when they think about it mathematically. The better way to think about it is to think logically. If we flip a coin and it lands on heads 10 times in a row and then bring the coin to you with that information, are you really going to think that coin is more likely to land on tails now? No, because you know that coins don’t have memories. This is just more challenging for people when there is no break in what they are seeing before their eyes.

The Clustering Illusion

The reasoning behind the gambler’s fallacy is something that is commonly referred to as the clustering illusion. The clustering illusion is the idea that people are prone to view streaks of random events as being non-random in the short term. The problem with this is that “streaks” or “runs” or “patterns” are much more likely to appear in small sample sizes than people might think they are. You may think that seeing heads four times in a row is crazy impossible, but it really isn’t. For every 16 times that you flip a coin four times, it’s going to hit heads four in a row.

The Most Famous Example – Monte Carlo

While the gambler’s fallacy pops up every single day in casinos all across the world and the web, the most famously noted example took place at a brick and mortar casino in Monte Carlo in 1913. The roulette ball at the casino fell on black an astounding 26 times in a row. As you might imagine, gamblers lost millions betting on red because they were sure that nature had to “even itself out.” With every spin of the wheel, gamblers were convinced that it “had to fall on red” the next spin because of statistics.

In reality, each spin had the exact same likelihood of red or black as the one before and would with the one after that. If you’re thinking that they should have started betting on black instead during the streak, you’re just as guilty but of the reverse gambler’s fallacy. This is the belief that streaks exist and will continue based on some magical power somewhere in the world.

Be Careful of Non-Randomness

In the previous example, we talked about the reverse fallacy where some might have thought you should start to bet on black because it was coming more often. While this will almost never be true in a regulated casino live or online, in other things in the world, it may be the sign of something else.

Let’s say we’re playing a gambling game where I drop a metal nail from the top of a 10-foot ladder and you can bet whether or not it will bounce to the left or to the right of the ladder when it hits the ground. We tell you as well that as expected the odds of the nail going to either side are 50%. We drop the nail 10 times, and 7 of the 10 times the nail goes to the left and the other 3 times it goes to the right.

Now, you being the educated reader of this article know that this does not affect the odds of the next drop, and you continue to bet on whichever side you like. The problem is that while we may have told you that the odds were 50/50, they actually are not. We are dropping the nail every time with our left hand which is causing it to fall to the left more often. The odds of each drop are not 50/50 but are in fact skewed to falling to the left.

You will never have this issue with casinos online, or brick and mortar as everything is extensively tested and regulated, but you may run into this in other areas of your life.

In our coin flip examples, the coins land truly random. In our nail example, though, the nail is more likely to fall to the left because of how we are dropping it. Don’t immediately jump to the conclusion of the gambler’s fallacy until you confirm that the action you are studying is truly random.