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The Gambler’s Fallacy – why ‘due’ is a lie

Posted by ThePOGG on Mar 24, 2021

This article was written by ThePOGG.

Have you ever played Roulette and seen red come up 10 times in a row? Did it make you think black has to happen soon? Have you ever seen a slot game being advertised as ‘hot’ because lots of players have won on it recently? Does it make you want to play that game? Or conversely, a game that is advertised as ‘cold’ because it has paid out less than it should have recently. Do you think that game has to be ‘overdue’? We are naturally constructed, evolved, to look for patterns in a chaotic world around us. This is part of what has made us such a successful species. Gambling offers fertile ground to look for patterns and the gambling industry often actively encourages us to do so. Sadly, what instinctually feels right and what is real are very different things and this is often exploited by hucksters and charlatans to dupe unwitting players into gambling more than they should or buying bogus systems. It is called the Gambler's Fallacy.

What is the Gambler’s Fallacy?

In its most simple terms the Gambler’s Fallacy is the erroneous belief that past events impact future outcomes. It is a form of Apophenia, which sees the subject rationalize gambling based on the erroneous assumption that the player has some degree of insight into what will happen next based on knowledge of the results that have already occurred.

The vast majority of gambling games are what is known as ‘statistically independent’. This is a complicated-sounding term, but it is really very simple. Here are two examples of a statistically independent event and how the Gambler’s Fallacy can come into play:

The Fair Coin

If we take a fair coin – that being a coin that has exactly a 50/50 chance of landing on heads and tails – and flip it, we know that on half of the occurrences that we will get heads and the other half of the time we will get tails.

But what happens if we have already flipped the coin and it has landed heads? If we flip the coin again shouldn’t tails be the expected result? We know that the coin has a 50/50 chance of each result so we look for the real-world results to align with this figure. And this is where the problem starts. Because of our expectation of how the results should end up, we introduce an unseen power that ‘forces’ the coin to do as we expect. Ask yourself these basic questions. Has the coin changed in any way because of landing on heads? Has the physical world around it, and ergo the forces acting on the coin, changed because the coin landed on heads? If we accept that the coin is fair and that the coin/world has not changed because it previously landed on heads, why would tails be more likely to occur on the next toss? The answer is obvious - tails is no more likely to occur on the second flip than it was on the first.

The Fair Dice

You are playing a game that uses a six-sided dice. A fair six-sided dice – meaning that each of the numbers is equally likely to occur. If you roll a 6 you win. You start the game and you roll the die 3 times in succession. Each roll results in a 6 showing.

What can you conclude from that? Should you retain this die because it is lucky, or ‘hot’. Do these results suggest that a 6 is more likely to occur on your next roll? Should you change to another die? This one has rolled 3 6’s, surely the other numbers have to start coming up more frequently to balance this out?

Again we have to ask ourselves the basic questions: has the die physically changed? Has the world changed? If nothing has changed, why would the die be more or less likely to roll a 6 on the next roll? Does the dice ‘know’ that it has rolled three 6’s and as such adjust itself to rebalance the outcomes?

The truth in the above examples is that if the coin and the die are truly fair, then the previous outcomes do not have any impact on the future outcomes. Common sense tells us that if there have been no physical changes to the device or the world, then there cannot be any change to the expected outcome of a toss/roll. We know that the device in question is not self-aware and cannot alter the outcome. But these physical truths conflict with the expectations that probabilities create in our mind and as such we often end up ignoring the physical truth in preference of what we perceive to be an insight into future events. Especially when knowledge of these future events could benefit our lives, as is the case with gambling.

In simple terms, the Gambler’s Fallacy means that were we to see heads come up a lot we conclude that tails must be more likely on future tosses to ‘balance out’ the excess heads.

How do streaks actually balance out?

So if a past event does not impact the future, how does a streak get balanced out?

A fair coin has been tossed 10 times and we’ve received 10 heads. This is a unlikely event. Surely we now have to receive additional tails to ‘balance out’ all those heads? To bring us back to that 50/50 probability we were expecting?

This seemingly logical conclusion is what leads so many players to make bad decisions.

The truth is that streaks of one result are not inevitably balanced out by other results. They are instead faded out by large numbers.

Take the example above of having received 10 heads tossing a fair coin. Knowing that the coin is fair and there is an equal chance of heads or tails coming up, the next 90 tosses would have the mathematically expected result of 45 heads and 45 tails. That means that over the 100 tosses we’ve received 55 heads and 45 tails. Over this particular set of 100 tosses, heads came up 55% of the time – a little more frequently than we would have expected from a fair coin. Already, we’ve moved from 100% heads (over the first 10 tosses) to only 55% heads.

Let’s extend the experiment again and instead of 100 tosses, keep going until we reach 1000. So we know we start with 10 heads. If the other 990 tosses result in the 50/50 distribution we expect we end up with 505 heads and 495 tails. Now we see that heads occurred only 50.5% of the time.

Moving up to 10000 tosses we end up with 5005 heads and 4995 tails, meaning that heads occurred 50.05% of the time.

What the above shows us is that while the initial 10 heads was unusual, there is no need for extra tails to ‘balance out’ this result. With a fair coin, the larger the number of tosses gets, the closer our results are to the expected 50/50 despite this short unusual streak. The larger the number of tosses, the less significant this short-term streak becomes.

Renowned professional gambler James Grosjean wrote a fantastic article on this – The Denominator – Where “Due” Happens – that was a large part of the influence for writing this piece.

Gambler’s Fallacy – Reverse Position

There is also the reverse position of the Gambler’s Fallacy, that being that where a player sees an unusual string of results – for instance, ten heads in a row – they conclude that the coin in question is not ‘fair’ and as such will continue to bet heads, under the belief that heads is more likely to occur.

This is an equally invalid conclusion that is potentially just as dangerous to players. While 10 heads in a row is an unlikely event – occurring once in every 1024 groups of 10 flips – it is not so unlikely that it will not occur with a fair coin. In most cases it is significantly more likely – with some variation depending on the regulatory standards at the gambling operator in question – that the player has witnessed an unlikely but possible event rather than that they have found an ‘unfair’ game. We discuss the barriers to unfair games in our article ‘Are Online Casinos Rigged?

This reverse position of the Gambler’s Fallacy again leads the player to conclude that they have information about the likelihood of upcoming events that is not based on fact.

The Gambler’s Fallacy in action

There are real-world examples of the Gambler’s Fallacy resulting in people en masse losing significant amounts of money.

Monte Carlo Fallacy

This is the most famous occurrence of the Gambler’s Fallacy and in fact the ‘Monte Carlo Fallacy’ has become another commonly used name for the Gambler’s Fallacy.

In 1913 a highly unusual pattern of results played out on one of the Monte Carlo Casino’s Roulette wheels. The ball kept landing on black. This went on for such an extended period of time that a large crowd reportedly gathered and started betting heavily on red. This group of people were clearly falling subject to the Gambler’s Fallacy, assuming that because black had occurred so many times, that red must be more likely, or “due”, to occur next. After 26 consecutive spins resulting in black, red finally landed. Those betting red on the 27th spin won, but not before the casino reportedly made millions from all the losing red bets on preceding spins.

The lack of reds on the first 26 spins did not make red more likely to occur. It was still an 18/37 chance of the ball landing on red each spins. Just as it was before the first spin.

Italian Lottery

In 2005 the number 53 had failed to appear in the Venice lottery for 152 consecutive draws (close to two years worth of draws). This resulted in Italians placing increasingly high volumes of bets on the next draw including the number 53 and reportedly resulted in 53 people losing their homes to bailiffs, a woman drowning herself and a man killing his wife and child before killing himself due to debts incurred from wagers placed on the lottery based on the incorrect belief that because 53 had not occurred recently, that made it more likely to occur in upcoming draws.

These incidents demonstrate the very real and very serious consequences that can come to pass as a result of people believing the Gambler’s Fallacy.

How the gambling industry manipulates players into believing the Gambler’s Fallacy

Sadly, alongside the Gambler’s Fallacy seeming to be intuitively correct, there are also many individuals and businesses that actively promote facets of the Gambler’s Fallacy as truth. They do this for direct financial gain. Players who hold to this mistaken belief are far more likely to gamble recklessly, pursuing ‘due’ outcomes. Here are some examples of how the industry fosters these invalid beliefs:

Big winner advertisements

You will see this regularly – a player hits a jackpot while playing in the casino, or wins the lottery and news stories pop-up all over the internet talking about this big win. The problem with this practice is not that the win is being advertised, but that it is being advertised selectively. What is not being conveyed is how much has been lost – not just by this player, but by all players – before this big win was hit. By selectively only reporting about the winning player, a distorted image is created of the likelihood of players receiving a life-changing payout, encouraging players to believe – despite minuet odds of winning – that they could be the next person to receive a huge payout. Nothing exemplifies this better the original UK National Lottery slogan: “It Could Be You”, which was retired in 2020.

This business model has recently been significantly expanded by the invention of online video streaming/sharing services. This is now one of the biggest growing markets in online gambling advertisement. This is also a major medium used to convince players that their chances of winning are far higher than they really are, by showcasing players winning to convince other players that they can win.

To be clear – the results of these professional streamers are in no way reflective of the results you will experience. These events are mathematically independent!

What happens is that the professional streamer engages in a live playing session, often with hundreds or thousands of players signing in to watch them playing. They play at stakes that are unsustainably high. They can do this because, in all likelihood, they are being heavily subsidized by the gambling operator they are playing at. This is discussed in our article – Are Slots Streamers Playing with Fake Money.

Why do these big stakes make a difference? Because the reality is that the majority of gamblers play with the dream of winning a life-changing amount of money. Logically these sorts of wins occur very infrequently when you are playing at low stakes. Most people, presented with these videos, are not good at contextualizing the amount being risked against the size of the win. They see these streamers playing and winning big amounts and it encourages them to think that if they play at these stakes, they could win too.

Where you see a losing session played by a streamer, it is unremarkable and soon forgotten. But the winning sessions have a far more significant emotional impact.

Add to this that the professional streamer is almost certainly playing with vastly reduced risk because they are almost unquestionably being funded by the operator and you have a situation that significantly misrepresents the likely real world experience.

This effect is then compounded. The streamer takes the videos of their playing sessions and selectively cuts them to showcase the big wins. They then post these 'big winner' videos all over streaming services like YouTube. And this is where the Gambler’s Fallacy really comes into play. By editing out long sessions of losses, players who watch these videos are enticed to believe that wins occur more frequently than they do and to engage with the world view that the historic results of these professional streamers in some way indicates the results that they are likely to experience. In short ‘if they won that much, why can’t I?’.

In our opinion this practice is extremely dangerous for players, and given the high levels of competition that is seen amongst operators for the traffic coming from these professional streamers, it seems fair to guess that the traffic being sent by these services is lucrative to the gambling operators.

Derren Brown – The System [SPOILER ALERT]

A really good example of how people can be manipulated by this type of marketing was put together by mentalist and illusionist Derren Brown in his special ‘The System’:

***Spoiler Alert – if you read past this point we will discuss how the outcomes presented in this show were achieved. For entertainment purposes – as it is a really enjoyable watch – we would recommend watching the show first.***

The show is set-up to present the idea that Derren Brown has a “100 percent guaranteed” system for winning on horse races and follows a person – Khadisha - that has been selected to apply this system through 6 different rounds of wagering. Each round of wagering Khadisha is encouraged to bet more. This results with them placing a £4k bet on the 6th and final race.

Before the results of the final race have been decided it is then revealed how the results are achieved. Khadisha was not the only person that was placing wagers on these events. In fact, the show started with 7776 players. These players were split into 6 even groups. Each group was asked to wager on a different horse in the upcoming race (there were 6 horses in each race). So after round one, there were 1296 winning players. These again were split into 6 groups, each of which were asked to wager on one horse in the upcoming race, leaving 216 who won both the first and second races. This was repeated on the 3rd (leaving 36 winning players), the fourth (leaving 6 winning players) and the 5th rounds, leaving only Khadisha from the starting 7776 players who had won on all 5 races.

By selectively editing what was shown to remove the losing examples and only show the single person who won in all 5 races, the show created the misconception that there was a genuine winning system in exactly the same way that by editing videos of playing sessions down to solely showcase big wins, professional streamers mis-represent to players the chances of winning.

While I’m not going to discuss the final outcome of the show, it should be noted that everyone who participated in the show was refunded all losses they experienced. The show was framed as an educational tool to help the public understand how ‘guaranteed winning systems’ were used to exploit them, a job that I personally feel it achieved very successfully.

Hot and Cold games

A far more understated strategy to find being used to encourage players to engage with the Gambler’s Fallacy is the ‘hot’ or ‘cold’ categorization of games (mostly slots games).

The way games are generally categorized – there will be variation between parties advertising games in this manner – is that if a game has either returned more than its expected Return to Player (RTP) over the preceding short period of time it is categorized as ‘hot’, telling players that this game has been paying out. Likewise, if the game pays out less than its expected RTP it would be listed as ‘cold’.

The ‘hot’ and ‘cold’ labels are entirely meaningless. They encourage players to engage with the belief that historic results give some indication of likely future results. And by tying these to math-based terminology, like RTP and percentages, they reinforce this view as mathematically credible. It is not.

The RTP figures being used to categorize games in this fashion are what is known as ‘Real RTP’ (rRTP) figures. Despite the use of the word ‘real’ these are not figures that are useful to anyone other than the gambling operator.

When advertising the actual, or ‘theoretical’, Return to Player (tRTP) of any slots game, the figure has to be derived from millions of rounds of play. This very large sample size ensures that all potential results have been considered. So those rare big wins that occur in slots games have all been triggered multiple times and those rare very extended losing streaks have also occurred. The tRTP ensures that we actually know what the average result is and as such accurately represents the ‘cost’ of the game. We discuss this in greater detail in our article Online Slots RTP Explained.

rRTP figures are derived from a short snapshot of play, meaning that rare occurrences either good or bad for the player may result in the game deviating from the long-term expectation. These figures are then presented with the game tagged as ‘hot’ or ‘cold’. Simply because a player/s has experienced a rare event on this game recently, gives no indication that the event is likely, or unlikely, to reoccur.

The hot and cold labels encourage players to either play a game because it is ‘due’ to payout (cold games – the Gambler’s Fallacy) or because it has been paying other players (hot games – the Reverse Position of the Gambler’s Fallacy).

Progressive Jackpots – the exception to the hot and cold fallacy

There is one notable quasi-exception to the hot/cold game fallacy – progressive jackpot games.

A progressive jackpot game is a game where a small piece of every wager is placed into the jackpot pool. The more bets that are placed the larger the jackpot prize becomes (for a more detailed explanation of Progressive Jackpot games see our article Online Slots Payouts Explained. In this manner it is similar to rollovers when playing the lottery.

And it is this dynamic that means that progressive jackpot games can in fact be hot or cold though not in exactly the same way as described above. The higher the progressive jackpot total the better the wager is for the player. So a progressive jackpot game that has not been hit for a long time is in fact better for the player (in most cases) than one that has been won recently. This means that a ‘cold’ progressive jackpot game is a better choice for the player.

However, a distinction should be drawn here between cold progressive jackpot games and other games that gambling operators describe as ‘cold’. For normal games, the label of ‘cold’ is given because players who have played the game recently have won less than expected. But as the games have no memory, the fact there has been a recent spell of lower than average payouts does not give any indication of what will happen with further play. These games are not, as the ‘cold’ label suggests, overdue.

Likewise, when playing a progressive jackpot game where the jackpot is larger than average is not ‘overdue’. It is no more likely to payout the progressive jackpot than the same game immediately after payout out the progressive jackpot. As with a normal game, past events do NOT impact future events. What makes the lack of recent winners good for players is the larger jackpot prize, not an increased likelihood of winning the progressive jackpot.

Lucky/Unlucky numbers

Another subset of the Gambler’s Fallacy that is regularly used by gambling operators is to display lucky/unlucky results. This is seen most commonly in lottery-style games or roulette. The operator provides a list of numbers that have recently been hit, or that have not come up recently. This aligns with the hot/cold games that were discussed above, with the intention being to encourage players to buy into the line of thinking that some numbers are ‘overdue’ or that others are lucky beyond the natural probabilities of the game.

Examples of this are as follows:

Roulette: Many Roulette tables have a display board showing recent results. Here’s an example from a Daily Mail article notable specifically because red 19 had come up 7 consecutive times.

It should be noted that the article in question incorrectly gives the odds of this happening as 114 billion to one. The odds are in fact closer to 3 billion to one. This error comes about as the journalist has approached the problem by of asking the question ‘what are the odds of a specific number [19] coming up 7 times in a row on roulette?’ when if fact this sequence could have occurred with any number. The first number in the sequence defined what number the sequence would be. So the question should have been ‘what are the odds of any number coming up 7 times in a row on roulette?’ giving the answer 1 in ~3 billion.

Lottery: In 2016 after 13 consecutive rollovers of the UK National Lottery, resulting in a jackpot pool of over £50 million, Camelot – the company licensed to run the lottery – issued press detailing the most frequently drawn lottery numbers.

This data was expanded on in 2018 to include the ‘least drawn lottery numbers’ and ‘most overdue lottery numbers’. Where there is nothing that is factually accurate about this information, it is very clearly issued with the knowledge that players want this information to try to draw conclusions about the possible upcoming results. As the operator are well aware that there is no way to glean knowledge about future results from this information – and given the inclusion of the word ‘overdue’ – the operator are at the very least indirectly fostering belief in the Gambler’s Fallacy amongst their players.

Progression Betting systems

And now we get to the bottom feeders of the gambling industry. The snake oil salesmen. The charlatans, hucksters and conmen who make a living selling the unwitting information that will in all likelihood cause them significant financial damage. The progression betting system salespeople.

A progression betting system can take many forms, but all of them are based on the Gambler’s Fallacy. There are two basic types:

Negative progression: This is where the player is encouraged to increase their bet after a loss, with the idea that with each successive loss the chance of a win increases (the Gambler’s Fallacy).

Positive progression: This is where the player is encouraged to increase their bet after a win, with the idea that the player is ‘hot’ and more likely to win the next round (the Reverse Position of the Gambler’s Fallacy).

There are many tweaks and combinations on these systems, and these confidence tricksters will work very hard to convince you their system is differet. It isn't.

The most famous of the progression betting systems is the Martingale negative progression betting system. This system would see the player double their bet on any loss. By continuously doubling the bet after any loss, the theory goes that the player will always end up 1 betting unit ahead as sooner or later they are going to win.

It is pretty straightforward to demonstrate why this system does not work. The record for the highest number of consecutive red spins at roulette is 32. If we start with a bet of 0.01 on black the progression for this event would be as follows (assuming an American Roulette double zero wheel):

SpinBetTotal WinFrequency of Occurrence (approx.)
10.01-0.011 in 2
20.02-0.031 in 4
30.04-0.071 in 9
40.08-0.151 in 20
50.16-0.311 in 42
60.32-0.631 in 89
70.64-1.271 in 187
81.28-2.551 in 395
92.56-5.111 in 833
105.12-10.231 in 1,758
1110.24-20.471 in 3,712
1220.48-40.951 in 7,837
1340.96-81.911 in 16,544
1481.92-163.831 in 34,926
15163.84-327.671 in 73,734
16327.68-655.351 in 155,660
17655.36-1310.711 in 328,616
181310.72-2621.431 in 693,744
192621.44-5245.871 in 1,464,572
205245.88-10491.751 in 3,091,874
2110491.76-20983.511 in 6,527,290
2220983.52-41967.031 in 13,779,835
2341967.04-83934.071 in 29,090,763
2483934.08-167868.151 in 61,413,831
25167868.16-335736.311 in 129,651,423
26335736.32-671472.631 in 273,708,559
27671472.64-1342945.271 in 577,829,181
281342945.28-2685890.551 in 1,219,861,606
292685890.56-5371781.111 in 2,575,263,390
305371781.12-10743562.231 in 5,436,667,156
3110743562.24-21487124.471 in 11,447,408,440
3221487124.48-42974248.951 in 24,230,084,490

As the above shows to survive this streak of losing black spins, starting from the smallest possible wager, we would need a bankroll of over 85 million (to cover both the loss we’d experienced by the 32nd spin and the wager we would have to place on the 33rd). And all of this risk for a total of 0.01 profit.

In all fairness, we have chosen an extreme example to demonstrate how quickly a progression betting system escalates out of control. There is less than a 1 in 24 billion chance of 32 consecutive losses occurring in this manner. But this example favors the progression betting system in other ways. How many roulette wheels will allow a bet as low as 0.01? How many wheels would accept a single bet of a million-plus? Or even 100k? How many wheels would allow a minimum bet that low and a maximum bet that high? You may have guessed “none” and you would be right. The reality is that you are likely to have to start with a higher minimum bet and you will be restricted to a far lower maximum bet. Realistically it will be unusual to encounter a roulette wheel with a minimum lower than 0.10 and more likely 0.50 or 1. Your maximum will likely be capped at a few thousand at most, and less per individual bet. With this type of limit, you will be cut off on the above table at around spin 11 or 12. 12 losing spins in a row is around a 1 in 8k chance. To put this in perspective, hitting a royal flush when playing Jacks or Better Video Poker is approximately a 1 in 40k chance. Any serious Video Poker player will hit multiple royal flushes in their playing lifetime. Really serious players will hit these with relative frequency. Or thought of another way, for every 8 thousand progressions you start, you’re going to lose 12 times in a row – finding yourself unable to place the next bet due to table limits – once.

The truth is that progression betting systems do work – sooner or later your colour will come up - but only under very specific conditions. That being that you have no table limits and you have an infinite bankroll. If you can find these conditions, you would win in the long run using progression betting systems. But if you had an infinite bankroll, why would you be gambling? For that matter, if you had 85 million, where would the excitement be in starting a progression with the hopes of winning a whopping 0.01? Outside of these artificial conditions, progression betting systems fail to overcome the house edge. They may result in frequent small wins, but they tend to also result in infrequent, but comparatively catastrophic, losses.

You will find YouTube and similar video sharing sites all littered with people promoting their ‘winning strategies’ – all of which want you to buy their secret and each of which are as bogus as the last. They use words like “probability” to convince you of their authority. They will show you charts and graphs of streaks in roulette to make their presentations look credible. They will say things like – and I quote from a video I watched while writing this article – “because the 3rd column came up last spin, the first two columns are more likely to come up on the next spin”, directly encouraging people to believe in the Gambler’s Fallacy (which is what all of these systems rely on). These system sellers know that what they are selling is bogus. But if they can con you out of 29.95, they see no problem with that.

But do not fall into the trap of believing that it is only small-scale con merchants that are engaged in propagating progression betting systems. We are aware of major software providers who build progression betting options into their slots games and a least one of the biggest names in the business host a blog that regularly features a writer who promotes progression betting systems as good for Roulette players.

Eliot Jacobson - Ph. D. Mathematics

Michael Shackleford – Adjunct Professor of Casino Maths at the University of Las Vegas – previously offered a bounty.

LVBear – Professional player.

Norm Wattenberger – Creator of Casino Verite Blackjack software.

Michael Bluejay – former pro player – currently offers a bounty

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