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Everything about the Gambler’s Fallacy with Examples

Are we biased to our beliefs? Yes, we are. As much one denies it, there are very few times when humans keep emotions aside. The gambler’s fallacy is one such lesson in human behaviour where you believe in something which has no statistical basis. This has many applications in the field of investing and behavioural sciences that we shall unearth in this article.

Risk comes from not knowing what you are doing

Warren Buffett

Gambling and Investing are not cut from the same cloth. And yet, most investors tend to approach an investing problem like a gambling problem. They use the same bias, prejudices and far-fetched “logic & conviction” that we see in coin flips.

Movies like 21, The Hangover and Ocean’s Eleven have popularized the hopes of you-can-win-big-on-your-lucky-day. Or better still, you can devise a system that is your sure-shot way to success on the casino floor.

Gambling looks cool in movies. And gambler's fallacy or not, remember the hero always wins in the movies
Gambling looks cool in movies. And gambler’s fallacy or not, remember the hero always wins in the movies

What is Gambler’s Fallacy?

The gambler’s fallacy is a belief that if something happens more frequently (i.e. more often than the average) during a given period, it is less likely to happen in the future (and vice versa).

So, if the great Indian batsman, Virat Kohli were to score scores of 100 plus in all matches leading upto the final – the gambler’s fallacy makes one believe that he is more likely to fail in the final. Cricket commentators have a fancy phrase for it – “law of averages”.

In reality, the situations where the outcome is random or independent of previous trials, this belief turns out false.

What Virat Kohli scores in the final has no bearing on scores in matches leading up to the big day. This fallacy arises in many other situations but all the more in gambling.

The Gambler's fallacy at work with potentially disastrous consequences
The Gambler’s fallacy at work with potentially disastrous consequences

The Monte Carlo Fallacy

The gambler’s fallacy is also known as the Monte Carlo fallacy.

It gets this name because of the events that took place in the Monte Carlo Casino on August 18, 1913.

The Monte Carlo casino in 1913 where the gambler's fallacy took place
The Monte Carlo casino in 1913 where the gambler’s fallacy took place

The event happened on the roulette table. One of the gamblers noticed that the ball had fallen on black for a number of continuous instances.

This got people interested. The “gambler’s fallacy” kicked in with people thinking since the ball had fallen on black so many times, it is likely to fall on red sometime soon.

To derive “undue advantage of this certain-to-happen” belief, casino patrons started pushing money onto the table.

Oh well, did it fall on red?

Yes, the ball did fall on a red. But not until 26 spins of the wheel. Until then each spin saw a greater number of people pushing their chips over to red.

While the people who put money on the 27th spin won a lot of money, a lot more people lost their money due to the long streak of blacks.

Everyday Examples of the Gambler’s Fallacy

Studies have found that asylum judges, loan officers, baseball umpires and lotto players employ the gambler’s fallacy consistently in their decision-making.

The fallacy is more omnipresent as everyone have held the belief that a streak has to come to an end.

We see this most prominently in sports. People predict that the 4th shot in a penalty shootout will be saved because the last 3 went in.

Now we all know that the first, second or third penalty has no bearing on the fourth penalty. And yet the fallacy kicks in.

Similarly, if it hasn’t rained for 4 days during the rainy season, you think it will rain on the fifth day. This is inspite of no scientific evidence to suggest so.

We just can’t help thinking about the past in making future decisions. Even if there is no continuity in the process.

Gambler’s fallacy and the coin toss

The gambler’s fallacy can also be illustrated with the repeated toss of a fair coin.

Now, the outcomes of a single toss are independent. And the probability of getting a heads on the next toss is as much as getting a tails i.e. ½ for heads and ½ for tails.

Let’s toss the coin again and like the first toss, this one also lands on heads.

The probability of two heads in two coin toss is ½ x ½ = ¼ (i.e. 25% probability)

This is where the amateur investor starts to falter.

He tends to believe that the chance of a third heads on another toss is a still lower probability.

i.e. ½ x ½ x ½ = ⅛ (12.5% probability).

This 12.5% probability might be true if one were examining the probability of three heads before the start of the series. However, one has to account for the first and second toss to have already happened.

This means the probability of a heads or a tails on the third toss is still ½. And not ⅛ for heads and ⅞ for tails as the gambler’s fallacy might lead to

Got it?

We run this back to the Monte Carlo Casino game of 1913.

Let’s deduce the probabilities that gamblers might have assumed versus the real probabilities.

Here is how the gambler’s fallacy plays –

  • Spin 1 : There is a 50% probability of the ball landing on Black
  • On spin 2 : There is 25% (50% x 50%) probability of the ball landing on Black
  • Spin 3 : 12.5% probability (50% x 50% x 50%)
  • .. and so on ..
  • Spin 25 : 0.0000030%
  • Then, spin 26 : 0.0000015% or “15 in a billion” or “1 in 66 million” probability
  • Finally, spin 27 : 0.0000007% or “7 in a billion” or “1 in 133 millionprobability
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What was going on in the gambler’s head?

When the gamblers were done with Spin 25, they must have wondered statistically. They would have said to themselves: “Surely now with a 30 in a billion probability for it to come black, there is no way that the 26th spin will not land on a red”.

However that too did not happen.

Statistically, this thinking was flawed because the question was not if the next-spin-in-a-series-of-26-spins will fall on a red. The correct thinking should have been that the next spin too has a 50:50 chance of a black or red square.

Numerous experiments have been done on the gambler’s fallacy which seems to lend more weight to the idea.

Studying the coin toss problem with different grade students

A study was conducted by Fischbein and Schnarch in 1997. They administered a questionnaire to five student groups from grades 5, 7, 9, 11, and college students. None of the participants had received any prior education regarding probability.

The question asked was – “Ronni flipped a coin three times and in all cases heads came up. Ronni intends to flip the coin again. What is the chance of getting heads the fourth time?

Now, we know that the answer to that is 50%, but here is how the students answered –

  • Grade 5 – 35% times it will come heads
  • Similarly Grade 7 also replied 35%
  • Grade 9 – 20%
  • And grade 11 went lower at 10%
  • Finally, college students – 0%

Interestingly, the gambler’s fallacy played the most with the college students and none of them gave any chance to the coin landing on heads.

Inverse Gambler’s Fallacy

The Inverse Gambler’s Fallacy is where after a series of events of a similar kind, the gambler believes that the series is bound to continue and is the more likely outcome.

In our coin toss example, the gambler might see a streak of heads. This becomes a precursor to what he thinks is likely to come next – another head.

This too is a fallacy. Here the gambler presumes that the next coin toss carries a memory of past results which will have a bearing on the future outcomes.

Ian Hacking has described the inverse gambler’s fallacy as a situation where the gambler entering the room sees a person rolling a double six and erroneously concludes that the person must be rolling the dice for some time.

Hacking says that the gambler feels it is very unlikely for someone to get a double six in their first attempt.

Now, we know the probability of getting a double six is low irrespective of whether it is the first or the hundredth attempt. The fallacy here is the incorrect belief that the player has been rolling dice for some time.

Retrospective Gambler’s Fallacy

The retrospective gambler’s fallacy is a situation where the gambler observes multiple successive “heads” on a coin toss. And concludes that the previous unknown flip would have been a “tails”.

Let’s understand with an example.

In his 1796 work “A Philosophical Essay on Probabilities”, Pierre-Simon Laplace wrote on the ways in which men calculate the probability of having sons.

The chances of having a boy or a girl child is pretty much the same. Yet, these men judged that if they have a boys already born to them, the more probable next child will be a girl.

The expectant fathers also feared that if more sons were born in the surrounding community, then they themselves would be more likely to have a daughter.

It’s as if saying, “I am due the other”.

We see this fallacy in many expecting parents who after having multiple children of the same sex believe that they are due having a child of the opposite sex.

Scenarios where the Gambler’s fallacy does not apply

The gambler’s fallacy does not apply in scenarios where the probabilities of different events are not independent or mutually exclusive.

For example – in a deck of cards, if you draw the first card as the King of Spades and do not put back this card in the deck, the probability of the next card being a King is not the same as a Queen being drawn.

The probability of the next card being a King is 3 out of 51 (5.88% probability) while that of it being a Queen is 4 out of 51 (7.84%).

This effect is particularly used in card counting systems like in blackjack. Many books on Blackjack have talked widely on this & many other strategies

Lessons from the Gambler’s Fallacy on Investing

1. Selective reporting

Statistics are often used to make content more impressive and herein lies the problem. When a public speaker says “the GDP is down this quarter”, it is safe to assume that the GDP was up the previous quarter.

This same problem persists in investing where amateur investors look at the most recent reported data and conclude on investing decisions.

Like, reporting that the crude prices are forecasted to go up by 20% without mentioning that the prices were beaten down by 60% over the rest of the year.

2. Assuming small samples are representative of the larger population

The gambler’s fallacy arises from the belief that a small sample represents the larger whole.

Daniel Kahneman (the author of Thinking, Fast and Slow) and Amos Tversky have studied this for years. They have come to interpret that people believe short sequences of random events should be representative of longer ones.

This means if you were to see a bunch of reds at point x and after a few randomness, you see another red streak – one tends to believe that the population is largely red with some small streaks of black thrown into the mix.

Investing entirely works on this.

Often we see investing made on the premise. One thinks anything can be bought because the macro-economic picture of the country is on a high. And hence, your stock will also go up.

This is far away from the truth with a number of stocks currently lingering at their 52-week low even as the Indian Nifty and Sensex continues to touch new heights of 12,000 points and 40,000 points respectively.

3. The outcome is a result of the gambler’s skill

At some point in time, you would have had a streak of six when rolling dice. Notice how in your next roll, you will turn your body as if to have figured out the exact movement of the body, hand, speed, distance and revolutions you require to get another six on the roll.

This is the gambler’s fallacy where the gambler believes that the outcome is the result of one’s own skill. This mistaken belief is also called the internal locus of control.

How to Avoid the Gambler’s Fallacy?

1. View every event as a beginning

The best way to avoid the gambler’s fallacy is by treating each event as if it is a beginning and not continuation of previous events.

This would prevent people from gambling when they are losing. It would help them avoid the mistaken-thinking that their chances of winning increases in the next hand as they have been losing in the previous events.

We see this in investing aswell where investors purchase stocks and mutual funds which have been beaten down. This is not on analysis but on the hope that these would again rise up to their former glories. It is not uncommon to see fervent trading activity on stocks which are fallen angels or penny stocks.

2. Reduce your illusion of control of being able to predict events

The Gambler’s fallacy gets accelerated by an individual’s belief on one’s perceived ability to predict random events.

In all likelihood, it is not possible to predict these truly random events. But some people who believe that have this ability to predict support the concept of them having an illusion of control.

This is very common in investing where investors taunt their stock-picking skills. They do this purely on the basis of being able to “feel” that a company is going to do very well or not in the near future. This is not entirely random as these stock pickers tend to offer loose arguments supporting their argument.

A useful tip here. You will do very well to not predict events without having adequate data to support your arguments.

Additional Resources

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