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Gambler’s Fallacy

To put it another way, it is the incorrect assumption that if a random outcome occurs more often than expected over a period of time then it is less likely to happen in the future. That’s simply not how randomness works.

Understanding the gambler’s fallacy, and the problems it can cause, is useful if you plan on being a successful gambler. It will prevent you from making the kind of mistakes that have led to many a gambler going broke.

On this page we have explained the fallacy in more detail, with examples. We’ve also covered why it result in gamblers losing money.

Understanding the Gambler’s Fallacy

The simplest way to understand the gambler’s fallacy is to consider the toss of a coin.Assuming a normal coin is being tossed, with no way of manipulating the result, the outcome is completely random. There are two possible results – heads or tails – and both results have an equal chance of happening.


As such, if a coin was tossed ten times in a row then you might logically expect to see it land on heads five times and on tails five times. Indeed, there is every chance that such a sequence would happen. However there is also every chance that the coin would land on heads ten times in a row, or tails ten times in a row.

There are many people that believe that if a coin did land on heads ten times in a row, then it would be more likely to land on tails on the next toss. While it’s not too difficult to see why people would believe that, the belief is wrong; the coin is still just as likely to land on heads as it is on tails on the next toss.

On every single toss of the coin the likelihood of each outcome is exactly 50% regardless of what has heppened previously.

The mistaken belief that because heads has come up ten times in a row it’s somehow less likely to come up on the next toss is basically what the gambler’s fallacy is. There’s actually no reason whatsoever why the next toss is less likely to be a heads. To understand why, you need to consider probability and the concept of independent events.

Probability & Independent Events

If you gamble in any form whatsoever, whether seriously or just for a bit of fun, you should ideally have at least a basic understanding of probability. Let’s start by looking at the dictionary definition of the word.

Probability (noun)
  • The quality or state of being probable; the extent to which something is like to happen or be the case.

So, simply put, probability describes how likely something is to happen.

Probability plays a role in all forms of gambling in one way or another. We won’t get into the details here, but suffice to say the more you understand about probability the more likely you are to be successful when gambling.

Expressing Probability

The correct way to express probability is as a number between 0 and 1. A probability of 0 means something cannot possibly happen, and a probability of 1 means something will definitely happen. A probability of 0.5 means that there is an equal chance of it happening or not happening.

Probability is also often expressed as a percentage, as this makes things easier to understand. To express it as a probability you simply multiply the relevant number by 100. So a probability of 0.5 becomes 50%, and a probability of 1.0 becomes 100%.

Probability of a Coin Toss

To demonstrate probability “in action”, it’s easiest to use an example. Let’s use a coin toss again. We’ve already established that the toss of a coin has
two possible outcomes, and each outcome is equally likely as the other. The probability of it being heads, therefore, is 0.5. The chances of it being heads
are exactly the same as it not being heads. Converted to a percentage, that’s 50%.

This proves the point we were making earlier. The chances of a coin toss resulting in heads is always 50% - regardless of how many times it has already
resulted in heads. This because each toss of the coin is an independent event.

What’s an Independent Event?
An independent event is one where the outcome is always random, and not affected in any way by what has happened previously. If heads does come up ten times in a row, the coin doesn’t “know” that and it is just as likely to be heads on the next toss as it tails.

Expectation & Variance

If you tossed a coin enough times, the expectation is that the number of times it comes up heads and the number of times it comes up tails should be approximately equal. This is because of probability. If there’s a 50% chance of each happening, then it stands to reason that both will happen the same number of times. However, probability is only the likelihood of something happening. Nothing is definite, unless the probability is 0 or 1.

You could toss a coin 100 times and it could land on heads 80 times and tails only 20.

Although you’d expect to see 50% of coin tosses land on heads, and 50% of coin tosses to land on tails, this won’t always be the case. This is because of variance, which is basically how much something can vary from what is expected to happen on average. The outcomes of random events won’t necessarily follow the average over any given period.

The more times you toss a coin, the more effect probability has and the less effect variance has. So if you tossed a coin one million times you could expect to see close to a 50/50 split of heads or tails. However if you only tossed a coin 10 times, you could easily see an 80/20 split, or a 90/10 split, or even a 100/0 split.

Once you understand all this it should be easy to understand why the gambler’s fallacy is indeed a fallacy. We will now explain why it is known as the gambler’s fallacy and why it can cause problems.

How the Gambler’s Fallacy Causes Problems

The reason the gambler’s fallacy is so named is it because thinking that the outcome of a random event is somehow affected by the outcome of a
previous random event, or events, is something that gamblers are liable to do. Without ever being aware of the term or what it means, many gamblers
have fallen into the trap of the gambler’s fallacy.

This happens most commonly in the casino and, in particular, at the roulette table. Assuming a fair game and an unbiased wheel, the outcome of a spin
of a roulette wheel is a random event. There are 37 numbers on the wheel (38 for American roulette) and each number has a 1 in 37 (or 1 in 38) chance
of coming up. This is true regardless of what numbers have been spun previously.

Where roulette players can get in trouble is on the even money bets such as red or black. 18 numbers are red and 18 are black, meaning that the
chances of a red number being spun are the same as the chances of a black number being spun. The chances are not exactly 50/50 because of the green
zero (and green double zero for American roulette) but the important point is that they are equal.

Again, those chances are equal regardless of what has happened previously. However, if you have ever played roulette for any length of time in a casino
then it is almost a certaintythat you will have heard someone ignore the basic rules of probability.

If RED has come up 8 times in a row,
then it’s must be BLACK next. Right?

In the above scenario, you shouldn’t be at all surprised to see a player place a sizeable wager on black. They’ll have convinced themselves that it simply
has to be black next due to the long sequence of reds.

As we have already shown, though, what has happened previously has no bearing on what is going to happen subsequently. It could have been red 20
times in a row and the chances of it being red on the next spin would still be exactly the same as it being black. This does not stop some roulette players
from thinking otherwise though, and basically falling for the gambler’s fallacy.

Casino players who fall for the gambler’s fallacy invariably go broke sooner rather than later.

The Gambler’s Fallacy and Betting Systems

The gambler’s fallacy is the biggest reason why people use negativeprogression betting systems. These involve increasing the stakes after losses.

The most famous example of such a system is the Martingale system. This works by placing even money wagers (on something such as red at the roulette table) and doubling the stake every time a wager loses. The idea is that a wager will win eventually and all the previous losses will be recovered plus a profit of the original stake.

Negative progression betting systems are ultimately doomed to fail though, because there is no guarantee that a wager will win before the stakes get too high. A roulette wheel could easily spin black 10 or more times in a row, and if that happened then anyone using the Martingale system would likely run out of money or reach the table betting limits.

This is why you need to be very careful when using negative betting systems. We do not believe that you categorically should not use such systems – as
they can be fun and they can help you win a bit in the short term – but you absolutely need to be aware of the gambler’s fallacy and understand that
they are not foolproof systems that guarantee profits.