Let’s start with a small riddle.
A man, who is a statistician by profession, believes that his next child will be a girl since his wife has already borne him three sons.
Do you find his argument convincing?
The argument intuitively doesn’t feel right. But why?
We’ll circle back to this puzzle but before that let me indulge you in another interesting thought experiment.
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?
I know you’re a value investor and don’t believe in speculating or gambling away your hard earned money on frivolous coin flipping tournaments, but this being a thought experiment I would request you to play along.
So what’s your answer?
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).
Chapter 12 - Gambler’s Fallacy | Mental Models, Investing, And You
So using my elementary knowledge of probability, I would reason that the universe will try to balance out the too many heads. When I use this argument to put my bet on tails, I am falling for a bias called 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). (source: Wikipedia)
That explains why our statistician friend’s argument is flawed. The gender of the fourth child is causally unrelated to any preceding chance events or series of such events. His chances of having a daughter are no better than 1 in 2 i.e., 50-50.
With independent events (the gender of kids, result of toss using a fair coin, etc.) there is no harmonising 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. (source: Wikipedia)
So why is it called a gambler’s fallacy? Because it’s rampant among gamblers and speculators.
Chapter 12 - Gambler’s Fallacy | Mental Models, Investing, And You
There was a guy who claimed that he had a scientific way of playing the lottery. He diligently maintained a spreadsheet of winning numbers and would bet on those numbers which had appeared the least. Alas! another victim of Gambler’s fallacy.
Now 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? That’s a fair question to ask.
To answer that, let me take help from Daniel Kahneman. In his book, Thinking Fast and Slow, Danny 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
Chapter 12 - Gambler’s Fallacy | Mental Models, Investing, And You
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’s insights are remarkable. So if you didn’t understand the above two paragraphs, please read them slowly and then re-read them.
Now if you were given an opportunity to bet on many such tosses, say 100 tosses, what would be your strategy for betting? Again the assumption being that it’s a fair coin (with no specific bias for either head or tail) and with the knowledge that probabilities are still 50:50.
Kahneman explains this using an anecdote about famous economist Paul Samuelson. He writes-
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
Chapter 12 - Gambler’s Fallacy | Mental Models, Investing, And You
be representative of the larger population. Hence he was willing to bet on the aggregate outcome of bigger sample size than a single outcome.
You probably noticed that I have been mentioning the use of a fair coin for our tosses. A fair coin ensures the pre-condition for a gambler’s fallacy to hold true, i.e. independent events.