How to Disprove the Gambler’s Fallacy

Thinking Woman With Question Marks and Roulette Table Background

The Gambler’s Fallacy has multiple names. It’s also described as the “Fallacy of the Maturity of Chances.” I’ve also seen it called “The Monte Carlo Fallacy.”

No matter what you call it, gamblers love it.

The Gambler’s Fallacy is the belief that a random event affects subsequent random events in a game of independent events.

It shouldn’t be confused with the understanding that some games do have a memory. As the composition of a deck of cards changes in blackjack, so do the odds.

But in a game like roulette, where each spin of the wheel is an independent event, past events have no bearing on the probability of future events.

In this post, I’m going to show you how to disprove the Gambler’s Fallacy.

Random Events Don’t Become Overdue

I took a trip to the Winstar Casino once with a lovely woman. On the way there, she explained to me that she didn’t just play slot machines. She played slot machines with a strategy.

And, her strategy was simplicity itself:

She just made sure to play the same slot machine every time she visited. The longer she played that one machine, the likelier she was to eventually hit a win on it when it came “due.”

I tried to explain to her that every spin of the reels on a slot machine is a discrete, independent event, but she wasn’t hearing it. She was absolutely convinced that if she just stuck with that machine, the longer she played, the likelier she would be to win.

I tried to explain to her that she could switch from machine to machine and probably have the same increased probability of eventually winning, but she just wouldn’t hear it.

She believed in the Gambler’s Fallacy.

How a Slot Machine Works

Suppose you have a simple slot machine with three reels and 10 symbols on each reel. And you should also suppose that each of those symbols has a 1/10 probability of coming up.

To get the probability of a specific symbol coming up on all three reels at the same time on the payline, you just multiply the probabilities together. When dealing with multiple events and wanting them all to happen at the same time, you multiply the probabilities together.

Your results are 1/10 X 1/10 X 1/10, or 1/1000.

When you spin the reels and hope to win, you have a 1/1000 probability of hitting that combination.

Casino Gambling Floor

If you miss it and spin the reels again, you have the same number of symbols on each reel with the same probability of coming up.

The formula doesn’t change. The slot machine game doesn’t remember what happened before. The results are entirely random and, most of all, the results are independent of each other.

People think that real money slots pay out less after a winning spin to catch up with their theoretically predicted payback percentage, but that’s not even necessary. The difference between the payout odds and the odds of winning take care of that over the long run.

This phenomenon is called The Law of Large Numbers.

What About the Law of Large Numbers?

The Law of Large Numbers suggests that the more trials you have, the close your results will get to the mathematically predicted results. This would seem to contradict the Gambler’s Fallacy, but the truth is more complicated than that.

Yes, the Law of Large Numbers suggests that your results will PROBABLY get closer to the predicted results, but the large numbers in question are SO large that the result of the next spin has a minimal effect on the averages.

For example, if you make 100 spins on a slot machine, you’re still playing in the short term. The long run hasn’t even come close to getting there. The outcome of the next spin can heavily skew the average results per spin.

But once you’ve made 100,000 spins, the results are probably starting to get closer to the average.

If you win 1000 to 1 on the 100,001st spin, though, it doesn’t affect the average that much. The number has gotten too big for an individual outcome to affect it much.

So, even though the odds don’t change as you play, the Gambler’s Fallacy still isn’t true.

What About the Gambler’s Fallacy and the Game of Roulette?

Roulette is perfect for understanding how to disprove the Gambler’s Fallacy. In fact, most roulette betting systems are products of the Gambler’s Fallacy.

When you bet on a single number in roulette, you can easily calculate the probability of winning that bet. You just compare the number of ways to win with the total number of possible outcomes.

A roulette wheel has 38 numbers on it, and every number has an equal probability of coming up. This makes the probability of winning a single-number bet just 1/38.

If you bet on the number 17 and hit the 17, what is the probability that the 17 will hit on your next spin?

The formula doesn’t change based on your previous result. You still have 38 numbers on the wheel, and only one of them is numbered 17.

Roulette Dealer Collecting Chips

The probability remains 1/38.

Frank Scoblete would like you to think that the numbers get “hot” at the roulette table. He would have you look at the historical results on the board and find a number that has hit more than once in the last hour to bet on.

His assumption that one of these numbers has gotten “hot” is just as erroneous as thinking it’s gotten cold.

The probability is the same – 1/38.

If the 17 got removed from the wheel after hitting, that WOULD change the probability of every outcome on the table.

But that 17 is still there, and it’s still just one number out of 38 numbers.

How Do Betting Systems Try to Use the Gambler’s Fallacy?

I bring up the Martingale System pretty often here. It’s the classic betting system where you double the size of your bet after every loss. The idea is that eventually the worm has to turn, and when it does, you’ll win back the money you lose on the previous bets.

The idea is that when a bet hits several times in a row, it’s less likely to hit again. In the Martingale System, you assume that if black has hit seven times in a row, it’s less likely to hit on the next spin because of how unlikely it is that you’ll have the ball land on black eight times in a row.

The trouble is, you’re not betting on the ball landing on black eight times in a row.

You’re betting that it will land on black on the next spin.

Since you have 38 numbers, and 18 of them are black, the probability remains 18/38, or 47.37%.

The conclusion is that eventually you’ll have a losing streak that lasts long enough that you won’t be able to afford the next bet. Or, even if you can afford it, the casino won’t let you place the bet because of their maximum betting limits.

But the Martingale isn’t the same betting system that relies on believing in the Gambler’s Fallacy.

The Paroli System

The almost direct opposite of the Martingale System is the Paroli System. It doesn’t work, either, but it’s illustrative of the diametrically opposite approach working at least some of the time.

In the Paroli System, instead of doubling the size of your bet after a loss, you double the size of your bet after a win. Most of the time, you have a win goal in mind where you reset to your initial bet size.

The idea behind the Paroli System is that sometimes outcomes get hot, and when they do, you can take advantage of it by letting your bet ride.

For example, you set a goal of winning $40.

Roulette Marker and Stack of Casino Chips

You start by betting $5 on black. You win, so now you bet $10 on black. You win again, and so you now bet $20 on black. This time when you win, you have your $40 win goal. So you start over again by betting $5 on black.

After a loss, you just start with your initial betting unit again.

Like the Martingale System, the Paroli System doesn’t work, because the fantasy that numbers get hot or cold in any kind of predictable way is just not how reality works.

If it were this easy to guarantee yourself a win at a casino, the casinos would go out of business.

And I’ve never seen a player at a roulette table get backed off for using any kind of betting system.


Believing in the Gambler’s Fallacy is one of the major mistakes that gamblers make, so how can you disprove it?

It’s easy.

Just spend a little time using one of the many betting systems that assume that past events have some kind of influence over future results.

It won’t take long before your system fails – disproving the Gambler’s Fallacy.