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.

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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 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.

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.

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 100,001^st 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.

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 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.

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.

Conclusion

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.

Introduction

The Gambler's Fallacy is the mistaken belief that if an independent event has not happened in a long time, then it becomes overdue and more likely. It is also equally incorrect that if an outcome has happened a disproportionate number of times lately, compared to statistical expectations, then it becomes overheated and less likely to occur the next time. An example of this fallacious thinking might be that if the number 23 hasn't been drawn in a 6-49 lottery the last 100 games, then it becomes more likely to be drawn during the next drawing.

Many worthless betting strategies and systems are based on belief in the Gambler's Fallacy. I got the idea for writing about this after reading an 888 online roulette article by Frank Scoblete entitled How to Take Advantage of Roulette Hot Spots. In that article, Scoblete recommends taking a count of each outcome for 3,700 spins in single-zero roulette and 3,800 spins in double-zero roulette in the hunt for 'hot numbers.' Never mind that this would take about 100 hours to make this many observations, assuming the industry standard of 38 spins per hour.

Before going further, let me say that I strongly believe modern roulette wheels made by top brands like Cammegh are extremely precise and any bias would be minuscule compared to the house advantage. Thus, testing a modern roulette for bias would be a total waste of time. Now, testing a 30-year-old hand-me-down wheel in a banana republic might be another story. However, you're on your own if you win a lot of money from said casino and try to leave with it.

That said, if you track 3,800 outcomes in single-zero roulette, the average number of times any number will hit is 3800/38=100. I ran a simulation of over 1.3 trillion spins, counting how many times each number was hit, sorting the outcomes to find the most frequent number and how many times it was observed, and keeping a count of how many times the most frequent number in each simulation was seen.

Hottest Number in 3,800 Spins of Double-Zero Roulette

As a former actuary, I hate to use a layman's term like the 'hottest number,' but that is how gamblers talk so will go with that. That said, following are the results of the count of the hottest number in millions of 3800-spin simulations.

Count of the Hottest Number in 3,800 Spins on Double-Zero Wheel

Here is what the table above means in plain simple English.

• The mean, or average, count of the hottest number is 122.02.
• The median count of the most frequent number is 121. This means that over 50% of time the most frequent number appeared 121 times or less, as well as 121 times or more. This is possible because the probability of 121 observations is in both groups.
• The mode, or most count of the hottest number is 120, which happens 8.29% of the time.
• The 90th percentile is the smallest number such that the probability the count of the hottest number is at least 90% .
• The 95th percentile is the smallest number such that the probability the count of the hottest number is at least 95%.
• The 99th percentile is the smallest number such that the probability the count of the hottest number is at least 99%.
• The 99.9th percentile is the smallest number such that the probability the count of the hottest number is at least 99.9%.

Hottest Number in 3,700 Spins of Single-Zero Roulette

The results are very similar with 3,700 spins tracked on a single-zero wheel. Following is a summary of the results.

How Much Do You Win If Your Number Hits In Roulette Game

Count of the Hottest Number in 3,700 Spins on Single-Zero Wheel

The following table shows the full results of the simulation on both wheels. The two commulative columns show the probability that the count of the hottest number is the number on the left column or more. For example, the probability the hottest number in 3,700 spins of single-zero roulette is 130 or more is 0.072044.

Summary of the Count of the Hottest Number in 3,700 Spins of Single-Zero Roulette and 3,800 spins of Double-Zero Roulette

Count of the Hottest Numbers in 300 Spins in Double-Zero Roulette

What if you don't want to spend 100 hours gathering data on a single wheel? Some casinos are kind enough to give you, on a silver platter, the number of times in the last 300 spins the four 'hottest' and 'coolest' numbers occurred. The image at the top of the page shows an example taken on a double-zero wheel at the Venetian.

In 300 spins, the average number of wins on a double-zero wheel for any number is 300/38=7.9. As you can see from the image above, the four hottest numbers were 20, 5, 29, and 2, which occurred 15, 14, 13, and 12 times respectively. Is this unusual? No. In a simulation of over 80 billion spins, the most frequent number, in 300-spin experiments, appeared most frequently at 14 times with a probability of 27.4%. The most likely total of the second, third, and fourth most frequent numbers was 13, 12, and 12 times respectively, with probabilities of 37.9%, 46.5%, and 45.8%. So the results of the 'hottest' numbers in the image above were a little more flat than average.

The following table shows the probabilities of the four hottest numbers in 300 spins of double-zero roulette. For example, the probability the third most frequent number happens 15 times is 0.009210.

Count of the Hottest Four Numbers in 300 Spins on a Double-Zero Wheel

The next table shows the mean, median, and mode for the count of the first, second, third, and fourth hottest numbers in millions of 300-spin simulations of double-zero roulette.

Summary of the Count of the Four Most Frequent Numbers in 300 Spins of Double-Zero Wheel

Count of the Coolest Numbers in 300 Spins in Double-Zero Roulette

The next table shows the probability of each count of the four collest numbers in 300 spins of double-zero roulette.

Count of the Coolest Four Numbers in 300 Spins on a Double-Zero Wheel

The next table shows the mean, median, and mode for the count of the first, second, third, and fourth coolest numbers in the 300-spin simulations of double-zero roulette.

Summary of the count of the Four Least Frequent Numbers on a Double-Zero Wheel

Count of the Hottest Numbers in 300 Spins of Single-Zero Roulette

In 300 spins, the average number of wins on a single-zero wheel for any number is 300/37=8.11. The next table shows the probability of each count of the four coolest numbers in 300 spins of double-zero roulette. For example, the probability the third most frequent number happens 15 times is 0.015727.

Count of the Hottest Four Numbers in 300 Spins on a Single-Zero Wheel

The next table shows the mean, median, and mode for the count of the first, second, third, and fourth hottest numbers in millions of 300-spin simulations of double-zero roulette.

Summary — Count of the Four Hottest Numbers — Double-Zero Wheel

Count of the Coolest Numbers in 300 Spins in Single-Zero Roulette

The next table shows the probability of each count of the four coolest numbers in 300 spins of double-zero roulette. For example, the probability the third coolest numbers will be observed five times is 0.287435.

Count of the Coolest Four Numbers in 300 Spins on a Double-Zero Wheel

The next table shows the mean, median, and mode for the count of the first, second, third, and fourth coolest numbers in the 300-spin simulations of single-zero roulette.

Summary of the count of the Four Least Frequent Numbers on a Single-Zero Wheel

How Much Do You Win If Your Number Hits In Roulette Online

The least I hope you have learned from this article is it is to be expected that certain numbers will come up more than others. To put it in other words, it is natural that some numbers will be 'hot' and some 'cool.' In fact, such differences from the mean are highly predictable. Unfortunately, for roulette players, we don't know which numbers will be 'hot,' just that some of them almost certainly will be. I would also like to emphasize, contrary to the Gambler's Fallacy, that on a fair roulette wheel that every number is equally likely every spin and it makes no difference what has happened in the past.

Finally, it should not be interpreted that we give an endorsement to the 888 Casino, which we linked to earlier. I am very bothered by this rule in their rule 6.2.B. Before getting to that, let me preface with a quote from rule 6.1, which I'm fine with.

'If we reasonably determine that you are engaging in or have engaged in fraudulent or unlawful activity or conducted any prohibited transaction (including money laundering) under the laws of any jurisdiction that applies to you (examples of which are set out at section 6.2 below), any such act will be considered as a material breach of this User Agreement by you. In such case we may close your account and terminate the User Agreement in accordance with section 14 below and we are under no obligation to refund to you any deposits, winnings or funds in your account.' -- Rule 6.1

Let's go further now:

The following are some examples of 'fraudulent or unlawful activity' -- Rule 6.2

Next, here is one of many examples listed as rule 6.2.B

'Unfair Betting Techniques: Utilising any recognised betting techniques to circumvent the standard house edge in our games, which includes but is not limited to martingale betting strategies, card counting as well as low risk betting in roulette such as betting on red/black in equal amounts.' -- Rule 6.2.B

Let me make it perfectly clear that all betting systems, including the Martingale, not only can't circumvent the house edge, they can't even dent it. It is very mathematically ignorant on the part of the casino to fear any betting system. Why would any player trust this casino when the casino can seize all their money under the reason that the player was using a betting system? Any form of betting could be called a betting system, including flat betting. Casino 888 normally has a pretty good reputation, so I'm surprised they would lower themselves to this kind of rogue rule.

How Much Do You Win If Your Number Hits In Roulette Games

Written by: Michael Shackleford