In this article, we explain all you need to know about the Gambler’s Fallacy (also known as the Monte Carlo Fallacy). We also include a breakdown of fixed odds, probability, house edge, law of averages, random outcomes and more.

The Gambler’s Fallacy is essentially the mistaken belief that past results can or will influence future results. It is also known as the Monte Carlo Fallacy, the Finite Supply Fallacy and the Fallacy of the Maturity of Chances.

The Fallacy can apply to all sorts of scenarios, but it most commonly arises with gambling. The term ‘Monte Carlo Fallacy' refers to one of the most famous examples (which we'll come to shortly).

Fixed Odds

Of course, past performance can often be a good indicator of future performance. This applies to many fields and real-life situations, particularly in sport.

For example, when wagering on a football match with football betting sites, it would be prudent to take into account recent form when considering the odds and likely outcome of the match. However, the Gambler’s Fallacy in the context we are discussing here won't be about football, or any sport, with all its inherent variables.

For illustrative purposes, we will use two far more straightforward examples. Firstly, a simple coin-toss, and secondly, a roulette wheel landing on red or black.

Why these examples? Well, with any given coin flip or spin of the roulette wheel, we have a fair and equal 50/50 chance of heads or tails or red or black coming up, right? Wrong. And therein lies our first potential error.

With a ‘fair’ coin, then yes, the chances of any given flip landing on heads or tails is indeed 50/50. However, on a roulette wheel, as well as the numbers 1-36 – evenly split into 18 red and 18 black – you will also find a green ‘0’ on a European table, and a further green ‘00’ or double-zero on an American table. Both of these innocuous-looking additions are there to give a built-in ‘house edge’. We will touch on this again later, but first, let’s concentrate on a ‘fair’ 50/50 coin flip.

Odds & Probability

If we were to bet on the outcome of a ‘fair’ coin toss, we'd expect the betting odds to accurately reflect the statistical probability of either possible outcome. In this case, with a 50/50 chance of either heads or tails, true odds of even money (1/1 as a fraction and 2.0 as a decimal as per odds converter) should be offered. Betting £1 on heads returns £2, and therefore a £1 profit, if does indeed land on heads. Tails would result in the loss of your £1 stake.

No one is ever likely to offer you odds against (better than even money) on either of these outcomes. If they did, they would quickly go bust. You may, however, see both eventualities offered at odds-on prices, i.e. worse than even money.

For example, odds of 10/11 may be offered on both heads and tails which has implied probability of 52.4% (note: this is more than the actual probability of 50%). Add the implied probability of either heads or tails occurring in this scenario and you get a total of 104.8% – a statistical impossibility. The result of one coin toss will either be 100% heads and 0% tails, or vice versa. Neither can ever be over 100%.

So, how do you account for that extra 4.8%?

House Edge

We mentioned house edge earlier, also known as the bookies’ overround. Essentially, this is when bookmakers literally round out to a point above 100% to ensure their edge.

On a single coin-toss, if you win, the house having an edge isn't such a big deal. In this instance, you may well win at odds of 10/11 (1.91), and your pound would return you a 91p profit.

However, if you bet on heads and tails coming up, you lose your £1 stake. If you bet again and this time you are correct, you would have staked a total of £2 and seen a return of just £1.91, meaning an overall loss of £0.09. This, in very simple terms, is how a bookie or casino manufacturers its ‘edge’. By doing this, they ensure that they will always make a profit in the long term, no matter the outcome.

Law Of Averages & Large Numbers

Ignoring the house edge for a moment, let’s assume a ‘fair’ coin toss with a 50/50 chance of either heads or tails landing. This is true for any single coin flip. However, when you begin to consider multiple coin flips, the picture can become skewed.

The bettor begins to rely on the law of averages and/or the law of large numbers to get close to the same 1:1 ratio (50/50 chance) of either outcome.

Flipping a coin one million times is unlikely to achieve an outcome of exactly 1:1. It is likely to be extremely close, though, meaning that the difference would be negligible and in line with expected or standard deviation.

However, if you only flip the coin 100 times, you could easily see a 70:30 outcome either way before the results ‘normalise’, and get closer to 50:50 again if you continued to a very large number (the previous one million flips, for example).

Over fewer occurrences still, let’s say 10, you might get 5 heads and 5 tails, but you could easily see 9 heads and 1 tails, or even 0 heads and 10 tails. (Who hasn’t tried best of 3, best of 5 and so on, until you get a favourable outcome?)

Again, you would expect this to regress to the mean over time. The issue is that, in gambling, you don’t generally have unlimited time or, more importantly, funds to benefit from the law of averages or the law of large numbers. This is where the Gambler’s Fallacy can become extremely problematic, and can cause punters to lose large sums of money.

The Gambler’s Fallacy is rooted in pure applied mathematics. It deals with the law of averages and the law of large numbers.

If you're worried that this article might get a little too technical, fear not. The aim here is to try and explain, in practical terms, what the Gambler’s Fallacy is, and how to avoid falling foul of it while betting.

Let’s say you bet on heads for each of the first 10 coin flips. You see 5 heads and 5 tails. At odds of Evens, you would have won as much as you have lost. Thus, you are exactly as you started – at break-even point.

So, you continue for another ten flips. This time your heads comes up 6 times and tails just 4. The result is that your 6 heads have won you £6 and your 4 tails have lost you £4. This leaves you with a £2 profit. So far, so good.

Now, what happens if you come up against a streak of tails? If the next 10 flips all land on tails, you are not only down £10, but your own cognitive bias would likely tell you that by the law of averages, you must be due a heads. This is the Gambler’s Fallacy in action.

What You Are Actually Doing

You are attempting to load probabilities associated with the law of large numbers onto a singular event that carries no such bias. Furthermore, you are falsely assuming that all those preceding tails results will influence the next coin flip. In reality, they do no such thing.

The probability of the coin landing on heads or tails remains 50/50, just as it was on the very first coin flip and just as it will be for every future coin flip, no matter what the past outcomes have been.

Think of it this way: if you have 10 coin flips and the first 5 have all been tails, if you were expecting a 50/50 split at the outset, you now require the next 5 to all be heads. You are therefore assigning heads a probability of 100% for each of the next five flips.

Of course, heads retains its original 50% chance for each individual coin flip. So you can see how you have then fallen into the trap of wildly overestimating your chances of seeing heads in any of the next 5 coin flips, based purely on past results that we have shown to be irrelevant.

The reverse is also true. Attributing a run of 5 tails to a ‘hot’ or lucky streak may see you win in the short term. However, thinking that future outcomes are more likely to be tails based on a past trend would be wrong.

Random Outcomes

The event itself, unlike you, has no memory of any hot or cold streak that may have gone before. Each subsequent coin flip is just like the first. It is reset and can turn no better than that 50/50 binary outcome. It will either be heads or tails (1 or 0 in binary terms) and remains completely random.

By assigning any other parameters based on past events as potential influences for future coin flips, your own (understandable) cognitive bias has led you to fall into the trap known as the Gambler’s Fallacy. If it’s any consolation, you wouldn’t be the first, and you most certainly wouldn’t be the last – but it's always best to be aware.

August 18th, 1913 – Monte Carlo

The term ‘The Monte Carlo Fallacy' was coined after a fateful night in a casino on the French Riviera. It was August 18th 1913, to be precise.

Upon noticing an ever-increasing number of black outcomes on the roulette wheel, people started pushing more and more chips onto red based on the mistaken notion that the probability of landing on a red was increasing after every black.

Of course, a red did eventually land, but only after 26 blacks. Anyone left standing and still betting on red obviously won on that last spin. Meanwhile, many unwitting bettors had already fallen foul of the Gambler’s Fallacy. They were left penniless and cursing their horrendous luck.

The inverse Gambler’s Fallacy (also called the hot hand phenomenon) is closely linked to the Gambler’s Fallacy. This Fallacy makes us expect the same phenomenon to happen, based on a past string of events.

For example, in a game of roulette, the ball has just landed on red five times in a row. This leads you to believe that it will probably land on red again, so you increase your stake and bet on red. The false belief underlying this phenomenon is that past events are somehow influencing current events, even though, in the case of a roulette table, each spin is completely independent of the previous. Of course, the odds of getting red are exactly the same as black, just like in all of the previous spins.

We can sum up the two fallacies as follows:

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    In the Gambler’s Fallacy, we falsely expect a reversal of outcomes, based on a previous streak.

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    In the reverse Gambler’s Fallacy, we falsely expect an outcome to continue, based on a previous streak.

We know that the Gambler’s Fallacy applies to pure gambling. But how does it apply to sports betting? While each spin on a roulette wheel has no recollection of the previous spins, and is not influenced by it in any way, sports do not follow the same pattern of statistical independence.

Game outcomes

Each player and team has specific characteristics that make them more or less likely to perform well in a variety of circumstances. They have individual talents, feelings and emotions that govern the way they generally behave. When looking at the career of a football player, over the course of a season, certain statistics seem to stick out. A player is likely to make the same mistake more than once, or have the same success more often.

This is where statistics come in. Punters, as well as bookies, rely on data such as current form, home and away records, and dozens of other metrics to predict the most likely outcome of a game. While each spin is independent of the previous one in a game of roulette, football games are not completely independent from previous games.

A coin toss has no memory of the previous toss, leading to a consistent probability of 50%. As any punter knows, odds in football or any other sport are rarely, if ever, 50%. Even in a close matchup, the bookies will slightly favour one over the other.

Betting Success

The Gambler’s Fallacy does not apply to the outcomes of sports events. However, does it apply to your success as a sports punter? Of course, if you have no predictive model of your own for the sport you are betting on, and are just following blind luck, the Gambler’s Fallacy does apply to you, at least for the success of your bets.

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    You may assume that because you have lost a few bets in a row, it will be more likely for you to win again.

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    You may assume that because you have won a few bets in a row, it will be more likely for you to lose again.

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    You may equally assume that because you have won a few bets in a row, you are more likely to win again.

The first two are an example of the Gambler’s Fallacy in action. The fact that you have lost bets will not make it more likely for you to win other bets.

The third is slightly more complex. Of course, choosing the right bets is the job of any intelligent punter, but it is important to look at the causation between the events. Did you win a few bets simply because of luck? Or do you have the right skills to predict game outcomes slightly better than the bookies do? In any case, it is important to be wary of the hot hand fallacy.

The Gambler’s Fallacy Can Actually Create Hot Hands

Psychology plays a huge role in the way we bet. How we perceive win or loss streak probability is largely determined by past outcomes. This also affects our choices of bets. A 2014 study of 565,915 sports bets made by 776 bettors demonstrated the following:

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    Sports bettors on a loss streak were more likely to bet on riskier odds after a loss. This often led to an even longer losing streak.

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    Sports bettors on a win streak were more likely to choose safer odds than before. This often led to an even longer winning streak.

The researchers interpret win streaks as follows:

Players on a win streak are afraid their luck will not continue (the Gambler’s Fallacy in action), which causes them to choose safer odds than they did before. By doing this, they actually create win streaks, since safer odds make it more likely to win. By believing the Gambler’s Fallacy, they actually manage to create their own hot hands.

The Gambler’s Fallacy is also commonly referred to as the Monte Carlo Fallacy. It is a logically incorrect belief that a sequence of past outcomes can or will influence the probability of future outcomes.

If only those in French Riviera casino on August 18th 1913 had understood that with each subsequent spin, the red that they were banking on in fact retained the same 50/50 chance that it had always had. Had they done so, perhaps they wouldn’t have been so quick to risk it all. However, hindsight is everything, as the saying goes, and their (monetary) loss is our gain, allowing us to learn from their mistakes and avoid the pitfalls of the Gambler’s Fallacy.

Knowing what the Gambler’s Fallacy is helps us look more critically at our betting strategy. Are our bets just lucky, or is our chosen methodology (if any) a successful one?

The Gambler’s Fallacy is the false belief that something that has a fixed probability will be have a different probability based on the past outcomes. In this case, people falsely believe that past events are affecting the future.

The Gambler’s Fallacy is incorrect because, in specific circumstances, such as roulette or a coin toss, each event should be considered independent from past occurrences.

The Gambler’s Fallacy is real and true when applied to games of pure chance, where each game is independent of the previous. It is false to believe that an outcome of a die toss will affect the next toss.

In order to stop the Gambler’s Fallacy, you first need to recognise that you are applying it. You need to understand that simply because an event happened before another, in a game of chance, it will not affect it. © 2006-2024