What is the Gambler’s Fallacy? – What it Means & How to Avoid It

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The classic definition of the gambler’s fallacy, also known as the Monte Carlo fallacy, is the mistaken belief that past events can influence future events although each individual event is independent of all others.

The purpose of this guide is to take an in-depth look at the gambler’s fallacy, provide examples, and explain how to avoid falling for it altogether.

what is gambler's fallacy
Image credit: Anna Shvets

What is the Gambler’s Fallacy?

The gambler’s fallacy is the popular but incorrect notion that if an event, whose occurrences are independent and identically distributed, has happened more often than expected, it’s less likely to happen in the future and vice versa.

The gambler’s fallacy is also called the Monte Carlo fallacy due to a situation back in 1913 at the Monte Carlo Casino where the roulette wheel spun black 26 times in a row, which only has a 1 in 66.6 million chance. This outlier caused bettors to lose millions of francs on the next spin as the high-rollers thought that red was due.

In reality, in a 50-50 situation played out over billions of simulations, streaks that are mathematically unlikely to happen will happen a certain number of times in the long-run. All of these individual events are independent of each other and previous events have absolutely no influence on the next.

While, in theory, billions of 50-50 events should finish around 50-50, in the short-term the results can have a lot of variance.

Example of the Gambler’s Fallacy

A classic example of the gambler’s fallacy is a simple coin flip.

For every individual coin flip, there is a 50% chance of the coin landing on heads and a 50% chance of the coin landing on tails. Each event is completely independent of the other. If a coin lands on heads 15 times in a row, the chance of the next coin landing on tails is still 50%.

Simply put, it is not “due” to land on tails just because it has landed on heads 15 times in a row. This is what people who have been caught out by the gambler’s fallacy would think.

We flipped a coin 1,000 times and it landed on heads 536 times (53.6%) and landed on tails 464 times (46.4%). During these 1,000 flips, there were several lengthy streaks of six or more heads or tails in a row, most notably heads happening 10 times in a row, a 1-in-1024 chance.

In this short-term simulation, there were more streaks of heads in a row than tails, sometimes being separated by just a single occurrence of tails.

There wasn’t a perfect counterbalance of tails at all, proof that the perceived law of small numbers does not exist. You will not achieve equilibrium in a short amount of time, no matter how much it may make sense. A 1,000-flip sample is not a significant sample size whatsoever.

In order to determine what a significant sample size is, we must use the central limit theorem (CLT). Luckily, this author has a mathematics degree and can explain the CLT in layman’s terms.

Central Limit Theorem Formula
Image: GeeksForGeeks

In probability theory, the CLT states that, under appropriate conditions, the distribution of sample means will approximate a normal distribution as the sample size gets larger. In non-nerd terms, the more times we flip a coin, the closer we will get to equilibrium.

In this example, we use the equation above to determine how many flips it would take to have a 95% chance (confidence interval) of both sides’ results being between 49.5% and 50.5% (a 0.5% standard error).

Using the formula above, it would take a whopping 40,000 coin flips to achieve an equilibrium within a 1% difference 95% of the time. So no, tails is not “due” to happen after a coin lands on heads 10 times in a row.

Games Where the Gambler’s Fallacy is Common

Roulette

You’ll often find hot and cold numbers posted near the roulette table at both the best online roulette casinos and at a brick-and-mortar places.

You’ll also see a display of a large number of the last spins, usually around the last 15 or so. This is done in order to encourage bettors who believe in the gambler’s fallacy to make wagers either for or against what’s already happened.

In reality, all this data that you’ve been given doesn’t matter at all as each spin is independent of the last. Each individual number has a 1:37 or 1:38 chance of hitting, depending on if you play European or American roulette, respectively.

The odds of black or red stay the same. The roulette wheel, or the random number generator if you’re playing online, has no memory. It does not matter what happened in the past.

Craps

Depending on what you bet and what stage of the game you’re in, rolling a seven can either be really good or really bad. Oftentimes, players go on a hot roll and don’t roll seven for a while, which is usually incredible for the large majority of the table.

In reality, each time a two-dice roll occurs, there’s a 1:6 (16.67%) chance that a seven will occur, meaning a seven won’t be rolled five out of six times.

In order to figure out the chances of back-to-back rolls without a seven, you multiply ⅚ * ⅚ which equals 25/36 (69.44%). You keep multiplying the result by ⅚ in order to see the chances of rolling consecutive non-sevens.

In the table below, you can see the odds of various levels of hot rolls. As you can see, the odds of rolling 15 times in a row without hitting a seven is 6.49%, which means, in the long run, it should happen about once every 15.4 times.

Number of Rolls Chances of No Sevens Occurring
1 83.33%
2 69.44%
3 57.87%
4 48.23%
5 40.19%
6 33.49%
7 27.91%
8 23.26%
9 19.38%
10 16.15%
11 13.46%
12 11.22%
13 9.35%
14 7.79%
15 6.49%

We rolled a pair of dice 100 times and then 10,000 times (through a simulator, of course).

In our 100-roll same, a seven occurred a whopping 22 times (22%), an incredible 32% more than the expected result of 16.67%. But after 10,000 rolls (graph below), results were much more standardly distributed and we were closer to equilibrium, with a seven occurring 1,704 times (17.04%).

Dice Roll Simulator
Image: GeoGebra

Slot Machines

Each slot machine has a designated return-to-player (RTP), which is the percentage of money wagered a player should expect to get back when they play for an infinite amount of time.

For example, a slot with a 95% RTP will theoretically pay you back $95 for every $100 you wager. However, slots are volatile and this percentage is based on continuous play and includes jackpots.

When you play at the real money slots, the volatility is often given for each different game. This, as the name suggests, is a rough measure of variation in payouts, both above and below the RTP, that players should expect.

Just like many casino games, every spin of the reels is an independent event and has no impact on the next spin. Whether you believe it or not, the truth is you have 10 losing spins in a row, you don’t have better odds of a winning spin on spin number 11.

However, there is a little bit of an exception when it comes to certain slots, which is called advantage play…

Advantage play is exploiting a certain aspect of a slot machine in order to gain a mathematical advantage. This has nothing to do with betting systems, rather noticing when a machine is much more optimal to play than normal.

For example, if a bonus triggers when you collect 15 of a certain symbol over time and a previous player has left after collecting 14 symbols, then it could be advantageous to sit down and go after the bonus.

Of course, nothing is guaranteed, and you could still end up going on a dry spell or collecting a lower-than-average bonus when you finally trigger it.

How to Take Advantage of the Gambler’s Fallacy

Simply by recognizing that the gambler’s fallacy exists, you can begin to make better decisions both in the casino and in real life. If you understand that the odds of a certain outcome are going to be the same no matter what previously happened, you won’t be as tempted to bet on red at the roulette wheel after the last six spins were black.

Gaining a deep understanding of the gambler’s fallacy will allow you to budget and cut your losses at a slot machine instead of continuing to chase thinking that you’re due for a big hit since you’ve been losing for over an hour.

In the stock market, the gambler’s fallacy exists as most people end up selling winners way too early and holding onto losers for way too long, thinking that it’s “due” to change course.

Oftentimes, a steady stock will be either a continuous winner in the long-run or a continuous loser. While the existence of the gambler’s fallacy and those who act on it could temporarily alter the price, being able to recognize when a winner will stay a winner and a loser will stay a loser can help you achieve success in that arena.

When is the Gambler’s Fallacy Not the Gambler’s Fallacy

The gambler’s fallacy does not exist when it comes to non-independent events. Card counters in blackjack are able to profit because the deck is not the same without the previous cards in it.

For example, if you are playing with an eight-deck shoe and six of the 16 aces are dealt during the first round, the odds of an ace occurring in future rounds drops significantly.

A basic card counting system assigns a -1 value to tens through aces, a 0 value for 7s, 8s, and 9s, and a +1 value to 2s through 6s. When the value gets high enough or low enough, the player will deviate from their base bet by either betting larger or betting smaller based on their perceived edge or lack thereof.

A key lesson when learning how to play Texas Hold’em poker is recognizing when cards give you a “blocker effect”. Put simply, holding certain cards reduces the chances of your opponent holding them, thus giving you the opportunity to significantly alter your play.

For example, you may decide to make a huge river bluff with a pair of jacks on a K-T-5-7-A board if your betting line makes sense because the odds of your opponent having the nut straight with QJ goes down significantly as you hold two of the jacks.

Conclusion

When you go to the casino, looking for trends in games such as roulette and craps is going to get you absolutely nowhere. Making wagers based on the gambler’s fallacy is not smart and you should not be increasing the size of your bets simply because you believe a certain outcome is due.

If random and independent events cannot influence future outcomes in the game you are playing, it’s best to absolutely ignore all previous events.

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Blaise Bourgeois
Poker and Gambling Expert
Blaise Bourgeois
Poker and Gambling Expert

Blaise is an Expert Gambling Writer and a professional poker player in Brazil. He has played and traveled throughout Latin America for the last four-and-a-half years and recently won his first WSOP Circuit ring! He received his Master's in Sport Management and Sports Analytics from St. John's University. Blaise also holds a Mathematics and Computer Science degree from SUNY Purchase, where he still holds the school's Men's Soccer record for goals in a season. Blaise has worked for Catena Media, OddsSeeker, WSOP, PokerNews, and Poker.Org in various capacities. He has a passion for extensive research and aims to provide accurate…