The Gambler's Fallacy

Updated: September 27, 2026

This definition is from the Oxford Dictionary of Current English. I use this dictionary because it is the one Google uses for its definition service. I am not sure that even this user-friendly definition does justice to the Gambler's Fallacy. We should call it a fallacious way of thinking about statistics. It just happens to be one that is very common (or at least most commonly expressed) among gamblers. The basis of this way of thinking is the belief that a random outcome is either more or less likely to occur after another random event or series of events has occurred. The reason it is a "flaw in reasoning" is that past events do not affect the probability of future events. In fact, the worst possible mindset for casino gamblers - the idea that you simply have to -win back- any losses you experience - is a direct result of the widespread belief in this fallacy. For this reason, and because any knowledge of casino mathematics is a weapon against the casino's edge, I want to break down the Gambler's Fallacy and explain why it does not work.

A Brief History of The Gambler's Fallacy

This strange way of thinking is sometimes called the Monte Carlo Fallacy because of a famous example of widespread use of this flawed logic during a series of roulette outcomes at Monte Carlo. According to several reputable sources, the infamous roulette sequence occurred around August 18, 1913. It was an incredible event witnessed by many.

The outcome began to come up -€œblack-€ several times in a row.

In the publication How to Take a Chance by Darrell Huff & Irving Geis, players began doubling and tripling their bets on red after the fifteenth black result. An interesting pattern emerged during this period.

By the time the twenty-sixth black result appeared, bettors had lost -€œmillions of francs-€ betting against black.

Their logic? -€“ The -€œstreak-€ of black outcomes had to end, because it wasn't a random series.

Statistical Independence

A fancy phrase exists that's relevant to this discussion: statistical independence.

Statistically-independent events are two that have no statistical effect on one another.

For example, I can think of two things that happened to me this morning as I was getting ready for work. I had a new blend of tea for breakfast and I saw the first hummingbird of the season at my window feeder.

These events display statistical independence, because their independent occurrences had no impact on each other.

If they did, I could have the same cup of tea and watch a hummingbird at my window any time I wanted. Or maybe every time I saw a hummingbird, I would get an overwhelming urge to drink tea.

That's just not the case.

Statistical independence is not the same as randomness. This is geeky math stuff, but it is important to understand.

A sequence is random when its parts are statistically-independent of one another.

The perfect example is the venerable coin-flip.

Coin flips are truly random because the possible outcomes (heads or tails) are statistically independent. You cannot predict (better than the 50/50 chance given by the existence of only two options) what the next outcome will be based on knowledge of the previous outcomes.

If you want to understand a little more about statistical independence, try this little experiment I've devised.

Complete both lists of numbers with the number you think will appear next in the sequence.

This set of numbers is probably familiar to you.

These are the first five -odd numbers- from the number line, and it is obvious that the next number should be -11.-

The only reason we can predict the next number is because we have seen -€œbehind the curtain.-€

We know what to expect because we know the system the sequence is making a reference to.

Now try to predict the next number in this series:

You have almost no chance of picking the next number in this sequence because I have randomly selected these numbers from a list of personal contacts on my phone.

In order to know the next number in the sequence, you would have to sit here with me while I go through the list.

You would have to have inside information to guess the next outcome, because each outcome is independent of the others.

Believe it or not, this simple random number line is similar to the way random number generators produce mathematically "random" results to simulate real-world random events.

Common Use of The Gambler's Fallacy

You will often hear players who have fallen victim to this line of thinking use the word "due," as in "certain outcome is due to happen." They say this because the recent results of a game are different from what they expected.

You can see two basic applications of this classic misunderstanding of gambling mathematics. They have a great deal in common - in both, bettors make bad assumptions about future outcomes based on the past.

The first common form of The Gambler's Fallacy takes us back to the coin-flip example.

Understand that each time the coin is tossed, it has a 50% chance of landing heads and a 50% chance of landing tails.

If a person flips a coin five times and sees the same result each time, he may decide that the next flip of the coin will produce a "heads" result because "heads" are "tails"; in reality, the sixth flip has the same chance of coming up heads or tails as the previous five. The other common form of this thinking involves events that are not statistically independent. I have found that this type of fallacy is far more common among regular gamblers than the first, but it is just as bad a way to think about math.

In this case, the fact that the outcome of one game can actually affect the outcome of another. Because these events are interdependent, the fallacy falls apart.

Why Do We Fall For The Gambler's Fallacy?

My favorite description of this line of thinking is that it confuses the long-term with the short-term.

People know that a coin flip should produce an even number of heads and tails, so when the results (over the course of just a few flips) are not random, they try to find patterns.

This is a deep-seated thing that probably has something to do with the human brain.

Scientists call these ingrained human patterns of thought and behavior "cognitive biases." Like any bias, they are difficult to address.

The best thing you can do to avoid feeling the sting of this and other errors is to educate yourself.

You're already doing that by learning everything you can about the math behind gambling and this fallacy itself.

However, simple education about the nature of gambling math does not seem to work well on a large scale to combat this kind of thinking.

Beach & Swensson Study

A scientific study (Beach and Swensson, Journal of American Psychology, 1967) demonstrated this. The researchers showed participants a shuffled deck of cards with simple shapes and asked them to guess which shape would appear next in the sequence.

One group received no preparation at all. The other group was given a short lesson in gambling math beforehand and was instructed NOT to rely on fallacies during the test.

Both groups performed identically, proving that the experimental group was still relying on the Gambler's Fallacy, or some version of it, to make their number series predictions.

Is there no hope for man? Are we destined to believe in this nonsense until we give all our money to the casino?

Here is another study that indicates that this isn't quite the case:

Fischbein and Schnarch Study

Researchers Fischbein and Schnarch published the results of a questionnaire in the Journal for Research in Mathematics in 1997, showing that we become less susceptible to these logical hiccups as we age.

They gave their questionnaire to five different groups of students in different grades - 5th, 7th, 9th, 11th, and a group of college-level students trained in higher mathematics.

They were asked the following question: -Ronnie has flipped a coin three times and all three times it has come up heads. Ronnie wants to flip the coin again. What are the odds of getting heads the fourth time?

Obviously, the correct answer is 50%. Each flip of the coin is an independent event, so both outcomes have a 50% chance of occurring.

According to the study, as students got older, they stopped using fallacious logic to answer incorrectly and started giving the right answer.

This study was important not only to gamblers, but also to neuroscientists and psychologists. It is seen as proof that cognitive biases can be overcome with age, experience and education.

Are All Attempts at Predicting Outcomes Wrong?

It is possible to make educated predictions about events, independent or not.

Any evidence-based prediction is likely to be a good one, and it is possible to make evidence-based assessments of all betting propositions, no matter how simple or complex.

Here's an example -€“ imagine again that you tossed five -€œheads-€ in a row.

It is perfectly reasonable for you to assume that your probability of making another series of five heads is less than your probability of making another series of two heads. This is a good statistic, and while it

might not be all that valuable to the average bettor, it makes logical sense.

Being able to make a logical prediction requires a basic understanding of the laws of probability. This is also a good way to avoid falling prey to the Gambler's Fallacy.

One of the nice side effects of learning about these and other fallacies is that you will be less of a sucker on the casino floor or at online gambling sites. After all, gambling systems are all based on this or some other logical fallacy.

Instead of wasting your money trying to beat a random roulette wheel or dice roll at the craps table, you can spend a few minutes doing your own research and see how these fallacies work for yourself.

Trying to predict future outcomes based on nothing but knowledge of past events is a recipe for failure.