Monte Carlo Fallacy: Don’t Look for Patterns Where They Don’t Exist

Casino de Monte-Carlo, August 18, 1913. The crowds, sitting around tables and having their drinks in common rooms, are getting more and more intrigued. Like a swarm of bees clustering around a hive, they are watching spinning roulettes with pure excitement. It seems, these are not ordinary hours in the lavish casino this evening. The gamblers’ attention is especially focused on one particular roulette wheel, in which the ball is falling into the black pockets again and again, eighteen times in a row by now. Every time the croupier collects the accumulating fortune – piles of chips laying on the red zone – more and even greater sums land on the table, as if tonight the players had unlimited cash in their pockets.

That day, the ball eventually fell into a red pocket after twenty-six blacks in a row. In the meanwhile, gamblers have lost millions of francs in just a few hours.

In the following weeks, this highly unlikely occurrence has remained in the highlights; it has been the topic of discussion at all the banquets and gleaming dinner tables: how could such an unlikely event happen? Well, if we dive into the mystery, even with a very basic knowledge of probabilities, we can understand that what happened is nothing supernatural; in fact, we should not even be surprised. The chances of having a 26-turn-long streak of red or black outcome is around 1 in 67 million – which is low indeed – comparable to the likelihood of winning the jackpot with a single lottery ticket.

However, what is even more interesting than the occurrence of a lot of subsequent blacks, is the behavior of the gamblers that night.

As the streak of blacks was persisting, more and more gamblers were betting against black in the deep conviction that the outcome of the next spin needed to be red – holding the belief that temporary imbalances must always equalize in the long run.

This is a famous example of the gambler’s fallacy, or to be consistent, the Monte Carlo fallacy: a misbelief that if something happens more often than expected during a given period, it will happen less in the future”.

The fallacy arises from the false interpretation of the properties of a roulette. Every time the croupier spins the wheel, the previous outcomes become completely irrelevant as they have absolutely no impact on the present turn. To phrase it more technically, each turn is independent of one another, and there is no such thing as ‘equalizing imbalances’.

Then why are gamblers so easily fooled by historical data? Our brain seems to struggle when dealing with independent events, since the majority of events in our lives can be closely related and are seen as a sequence of connected episodes. Being good at recognizing patterns in this complex world has always been the competitive edge for us in evolution, so it is no surprise that we often desperately look for patterns, even where they do not exist. This phenomenon is commonly known as apophenia.

Amos Tversky and Daniel Kahneman proposed that the gambler’s fallacy is a cognitive bias produced by the representativeness heuristic, which states that people evaluate the probability of a certain event by assessing how similar it is to their prior experiences. According to this view, “after observing a long run of red on the roulette wheel, for example, most people erroneously believe that black will result in a more representative sequence than the occurrence of an additional red”. People expect a short sequence of random outcomes to share properties of a long run sequence, believing that deviations from the average should balance out.

However, this expectation is quite wrong. Take a look at the table below to better understand the meaning of relative and absolute deviations from the average. The table represents the results of a simulation of 10, 100 and 1000 spinning with the roulette wheel 5 times, where the most extreme outcomes of each sample size have been highlighted.

Imagine you always bet on black. In the first pillar, you‘ve got 7 blacks – 70% of all the outcomes – but you’ve got only 4 more gains than losses in absolute terms. On the other hand, you’ve got 449 blacks according to the third pillar, which is 44.9% of all the outcomes. Nevertheless, since you have much fewer gains than losses, you will most probably be disappointed. As the sample size is increasing, imbalances are indeed equalizing in relative terms (percentages of black and red outcomes are closer to the theoretical expected value – 48.65%), but the absolute deviation is getting bigger and bigger – and this is what eventually counts when you have to pay the bills.

Afterword. Roulette is a game, where falling into the trap of gambler’s fallacy is actually not a handicap. The crowds, who were so persistently and irrationally betting against the black, had the same chances of doubling their fortunes than losing everything! As we know, the probability of all the outcomes are equal, regardless of what historical data suggests. Therefore, it turns out that the biggest mistake was not favouring the red to the black or vice versa, but taking part in a game with negative expected value.