Imagine yourself as a spectator in a coin flipping tournament. You notice that in one of the plays, the coin has landed on heads for 5 consecutive flips. If you were given an opportunity to bet on the next flip, would you bet on heads or tails?

The basic concepts of probability tell us that for random events like outcomes of coin flipping, both the head or tail are equal likely. In other words, the probability of a head and a tail are both 1/2 (0.5). So using your elementary knowledge of probability, you could reason that the universe will try to balance out the too many heads. When you use this argument to put a bet on tails, you are falling for the bias known as Gambler’s Fallacy.

The Gambler’s Fallacy is the mistaken belief that, if something happens more frequently than normal during some period, it will happen less frequently in the future, or that, if something happens less frequently than normal during some period, it will happen more frequently in the future (presumably as a means of balancing nature). It is called Gambler's Fallacy because it rampant among gamblers and speculators.

With independent events (the gender of kids, result of toss using a fair coin, etc.) there is no harmonizing force at work. The coin doesn’t know that it had landed heads in the last 5 tosses.

The most famous example of the gambler’s fallacy occurred in a game of roulette at the Monte Carlo Casino in 1913 when the ball fell in black 26 times in a row. This was an extremely uncommon occurrence, although no more or less common than any of the other 67,108,863 sequences of 26 red or black. Gamblers lost millions of francs betting against black, reasoning incorrectly that the streak was causing an “imbalance” in the randomness of the wheel, and that it had to be followed by a long streak of red.

If a coin (a fair one) has equal probability (50:50) of turning heads or tails, then why is it fallacious to expect a tail after 5 consecutive heads? In his book, Thinking Fast and Slow, Daniel Kahneman writes:

People expect that a sequence of events generated by a random process [coin toss] will represent the essential characteristics [equal probability of head and tail] of that process even when the sequence is short [few tosses]. In considering tosses of a coin for heads or tails, for example, people regard the sequence H-T-H- T-T-H to be more likely than the sequence H-H-H-T-T-T, which does not appear random, and also more likely than the sequence H-H-H-H-T-H, which does not represent the fairness of the coin. Thus, people expect that the essential characteristics of the process will be represented, not only globally in the entire sequence, but also locally in each of its parts. A locally representative sequence [the sequence of 5 heads which you observed], however, deviates systematically from chance expectation: it contains too many alternations and too few runs.

Another consequence of the belief in local representativeness is the well-known gambler’s fallacy. After observing a long run of red on the roulette wheel, for example, most people erroneously believe that black is now due, presumably because the occurrence of black will result in a more representative sequence than the occurrence of an additional red. Chance is commonly viewed as a self-correcting process in which a deviation in one direction induces a deviation in the opposite direction to restore the equilibrium. In fact, deviations are not “corrected” as a chance process unfolds, they are merely diluted.

Kahneman explains this further using an anecdote about famous economist Paul Samuelson.

The great Paul Samuelson—a giant among the economists of the twentieth century—famously asked a friend whether he would accept a gamble on the toss of a coin in which he could lose $100 or win $200. His friend responded, “I won’t bet because I would feel the $100 loss more than the $200 gain. But I’ll take you on if you promise to let me make 100 such bets.” Unless you are a decision theorist, you probably share the intuition of Samuelson’s friend, that playing a very favorable but risky gamble multiple times reduces the subjective risk.

Samuelson’s friend was pretty smart. He understood that Gambler’s fallacy arises out of a belief in a law of small numbers, or the erroneous belief that small samples must be representative of the larger population. Hence he was willing to bet on the aggregate outcome of bigger sample size than a single outcome.

Please note that in the above example, we're assuming that it is a fair coin. A fair coin ensures that coin tosses are independent events, a pre-requisite for gambler's fallacy to hold true.

If I told you that in the same coin tossing tournament, there is play where the coin has turned heads for 50 consecutive tosses. What would you bet on for the 51st coin toss? The probability of that happening, using a fair coin, is 1 in a 1100 trillion but there’s also a chance that the coin is biased. A biased coin or an imperfect roulette wheel can also lead to outcomes like 5 consecutive heads but for totally different reasons. In these

cases, the gambler’s fallacy might superficially seem to apply, when it actually does not. This is the reverse of gambler’s fallacy, also known as **hot hand fallacy**. It originates from basketball where players who scored several times in a row are believed to have a hot hand, i.e. are more likely to score at their next attempt.

The idea of gambler’s fallacy might look paradoxical to the mental model of mean reversion but it’s not. In his book, The Art of Thinking Clearly, Rolf Dobelli explains the subtle difference between the two.

…if you are experiencing record cold where you live, it is likely that the temperature will return to normal values over the next few days, If the weather functioned like a casino, there would be 50% chance that the temperature would rise and a 50% chance that it would drop. But the weather is not like a casino. Complex feedback mechanisms in the atmosphere ensures that extremes balance themselves out.

So, take a closer look at the independent and interdependent events around you. Purely independent events really only exist at the casino, in the lottery and in theory. In real life, in the financial markets and in business, with the weather and your health, events are often interrelated.

This means a domain prone to truly random and independent events, like a casino, is more prone to gambler’s fallacy. However, in systems which behave like complex adaptive systems that are prone to self-correction, both mean reversion (negative feedback loops) and extreme outcomes (positive feedback loops) are a possibility.

Intuitively. we find it hard to deal with randomness. As a result, we tend to put a tremendous amount of weight on previous events, believing that they’ll somehow influence future outcomes. Gambler’s fallacy dictates that with independent events, there is no balancing or harmonizing force at work.