The Gambler’s Fallacy is when cognitive biases cloud judgment and give inaccurate expectations of the likelihood of an outcome. Understanding it requires us to uncover what lurks in the gap between maths and feelings. So what is the Gambler’s Fallacy?
You may not be acquainted with the term, but you’ll almost certainly have experienced it: that familiar feeling that takes hold when a sequence of events happens more frequently than usual in a given timeframe and leads you to believe that, because of that, it is less likely to happen again in the future.
But is it, really?
Example of Gambler's Fallacy
To illustrate that, imagine you get heads ten times in a row in a coin flipping match. How likely is it that the next flip will again result in heads? After such a streak, is it more likely that the coin will carry on landing on heads, or is it even more likely to break the trend and land on tails?
For now let's put a pin on that and delve into the history books to see where the term comes from.
Gambling the fallacy
Gambler's Fallacy is also known as "The Monte Carlo fallacy", named after an unlikely event at Le Grande Casino in Monte Carlo, on the night of August 18, 1913. At the roulette wheel, the colour black came up 29 times in a row - a probability that David Darling has calculated as 1 in 136,823,184 in his 2004 book, "The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes".
This night entered gambling hall of fame due to the huge amount of money that was lost. After the wheel came up black the tenth time, patrons began placing ever larger bets on red, under the assumption that black could not possibly come up again, lest reality would break and the universe collapse unto itself.
If those gamblers were to explain their logic behind repeatedly betting against black in that situation, it might even make sense. Or at least common sense, which is notoriously more common than it has sense.
They could state that it's only logical, since probability is calculated as the number of favourable outcomes divided by the number of possible outcomes, that such large sequence would lead to increasingly reduced odds. So it would surely follow that, as the number of spins of the roulette wheel increases, the chances of black coming up again diminishes exponentially.
However, the universe is still here, and by the end of that night Le Grande's owners were substantially richer. So what happened?
The law of large numbers
At this point another disillusioned gambler might bring up a staple of probability theory, the law of large numbers. According to this law, "the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed."
In layman’s terms, this means that after a lot of spins, odds to each outcome tend to balance out. So, after that many blacks, it surely seems like time for a red, right? Because math?
Well, not quite. Though there is no exact definition of what ‘a lot of spins’ means, the law of large numbers dictates that the more times you play a game, the closer the proportion of each possible outcome relative to all results should be to the mathematical probability of it happening for a single play. And 29 is far from being a large enough number - we’re talking infinity-level stuff here. If anything, the law of large numbers disproves this idea, as given a large enough sample even longer streaks could (and eventually should) happen.
We are not saying the law of large numbers doesn’t apply to gambling; on the contrary, the gambling industry exists exactly because it does. That’s why, even if a sometimes a player can squeeze a whole Pacific island’s worth out of a casino, in the long run the house always wins.
But why does it still feel so counter-intuitive?
It all boils down to patterns and how we humans are hardwired to look for them in everything around us. And patterns we find - even where there aren’t any.
In that fateful Monte Carlo soiree, players confused the odds for a single black spin (1-in-2) with those for an unbroken sequence of 29 black spins in a row (1-in-136,823,184) because such a beautifully symmetric sequence is like catnip to our lusty cognitive biases. And while we chase after them like moths to a campfire, we are blindsided by the gut-sized flaw in our reasoning.
Mathematical probability, as expressed by Darling’s 1-in-136,823,184 calculation, only applies when treating the whole 29 spins combined into a single event and the outcome is measured as a combination of all previous results. But that was not the case, as players were betting (and losing) one spin at a time - and the outcome was measured after each bet, not all of them.
As it was, with all their fluids flushing towards a single magic pattern, players were oblivious to the fact that probability is agnostic, and the odds for a 29-all-black sequence are exactly the same as those for a 28-black-1-red.
Don’t believe us? Think about it this way: the ball has a 1-in-2 chance of landing in black. But this also means that red has the same 1-in-2 chance. So, a sequence comprised of black-black-black is likely to happen once for every eight 3-spin-sequence played (½ x ½ x ½ = ⅛). But guess what? So is black-black-red, or red-black-red, or any other combination - all of them have the same 1-in-8 chance of happening.
It’s also worth mentioning that the probability for any such specific sequence to happen is waaaay smaller, before the first spin, than for it not to happen (as not happening includes the sum of all remaining possible combinations). This also means that each ball landing on black would actually increase the odds each time, but only when analysed against the remaining spins. However, as our players weren’t playing against 29-spin odds, this was also not the case.
Games that depend on skill, endurance or experience can have their odds offset by what happened in their previous instance: an injured Cristiano Ronaldo would certainly put his team at a loss, while a losing card player may turn the tables by learning their opponents’ strategy.
Alas, roulette balls aren’t the brightest cookie in the jar. Unlike a poker player or a football team, they certainly don’t learn from their previous “matches” against the roulette wheel, as they have exactly zero memory.
They also care very little for math, so their landing upon black or red certainly doesn’t happen out of sheer probabilistic love. With no memory and no particular variable to motivate them, each spin is completely unrelated to the one before - it’s like every spin is the first one.
Heads or tails?
Getting back to that coin toss, the odds for either head or tails do not decrease irrespective of how many flips end up on heads in a row. Over 100 tosses, for instance, there is no reason why the first 50 should not all come up heads while the remaining tosses all land on tails. Chances are exactly the same.
So, unbelievable as it may seem, it's better to fight your instincts and not go all-in on that tails: odds are it might just as well land on heads again.
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