Coin Tossing and Investing Fallacies

Consider a coin that has returned 95 heads in 100 tosses. If you are asked to predict the result of the next toss, you can think in either of the following three ways.

  1. You remember and believe in the “Coin toss is an independent event” theory i.e. the outcome of the next toss doesn’t depend on the previous outcomes. For you, both heads and tails are equally likely to occur.
  2. You believe that the coin has returned more heads than it deserves. Logically, there should have been only 50 heads. Hence, you are due to get more tails on subsequent tosses. You would happily bet on tails for the next toss.
  3. You start thinking that the coin likes heads more than tails. Perhaps it’s biased, or the person tossing the coin is playing tricks. Anyways, you predict that the next flip would return heads, with reasonably high probability.


Case 1 is how most academicians would respond. People stick to long-held assumptions even when there’s a hint of disconfirming evidence. 

The difference between the first and the third case was highlighted by Nassim Taleb in the following thought experiment in his book The Black Swan. 

Two people are involved:

Dr. John who is regarded as a man of science and logical thinking

Fat Tony who is regarded as a man who lives by his wits

A third party asks them to “assume that a coin is fair, i.e., has an equal probability of coming up heads or tails when flipped. I flip it ninety-nine times and get heads each time. What are the odds of my getting tails on my next throw?”

Dr. John says that the odds are not affected by the previous outcomes so the odds must still be 50:50.

Fat Tony says that the odds of the coin coming up heads 99 times in a row are so low that the initial assumption that the coin had a 50:50 chance of coming up heads is most likely incorrect. “The coin gotta be loaded. It can’t be a fair game.”

This is known as the ludic fallacy, the tendency to believe that results from hypothetical games or experiments would apply in real life.

How do such assumptions translate into investing fallacies?

  1. Consider the Efficient Market Hypothesis which states that markets have everything priced in, stocks always trade at fair values, and consistently generating superior returns is impossible. Yet, we frequently see bubbles and bursts; many businesses have floated for less than their cash reserves on stock markets. People like Warren Buffett have exploited the same and consistently beaten the market for decades. (Check Buffett’s views on the same topic)
  2. Not challenging your assumptions and staying invested in a company even when facts change, can be detrimental. Particularly applicable to legacy companies. Retail investors keep holding on to worshipped conglomerates or erstwhile glorified brands in the hope of a turn-around. However, big-bang bankruptcies – common every 5-7 years – prove that no company lives forever. On that note, the average lifespan of companies listed on the S&P 500 is less than 18 years today.

In short, proactively look for disconfirming evidence and incorporate it into your assumptions. In Keynes’ words, “When the facts change, I change my mind. What do you do, sir?”


Case 2 is the classic gambler’s fallacy. People believe that if an event has occurred more than normal in the past, it should happen less frequently in the future. 

Gambler’s fallacy is applicable in many aspects of life, but it has been most destructive for investors and traders. Often, it’s the result of misapplication of regression towards means.

Regression towards mean has two characteristics:

  1. It occurs more frequently in nature e.g. rainfall, height distribution. Man-made systems are less prone to it.  
  2. The time period (or sample size) considered should be fairly large. Though precise estimation is impossible. Here’s an interesting article on the same, in the context of coin-toss.

Options trading is the center-stage of gambler’s fallacy. Once a stock goes in the opposite direction, traders (especially novice ones) double down on their original bet, expecting reversion to mean which seldom happens. Nithin Kamath explains the same here:

“Averaging down” is when you buy a stock at 100 and buy more as the price falls, say, to 90, some more at 85, and so on; buying more in hopes that a bounce will help recover the losses faster. Unfortunately, hope isn’t really a trading strategy. Stock prices tend to trend (go up or down for long periods), and buying more as it goes down may work out once in a while, but is generally a losing strategy in the long run. Buying more of the falling stock is essentially you trying to fix a trading mistake, which could have been avoided by having a stop loss. What makes it worse is that retail traders typically end up selling stocks which are profitable to average down on ones which are losing (also called the Disposition effect).

At least when you average down on stocks, you can afford to give it time to bounce. But when you buy stock or index options that have a limited time to expire, this strategy of averaging down is a sure-shot recipe for disaster.

It’s also applicable to long-term investing.

  1. Selling winners early. (“It has gone up too much, it can’t increase further”)
  2. Holding on to losers. (“It has fallen so much, it’s due for a return now.” or “It can’t go below that” or “I will sell once it rebounds to abc”)


Case 3 seems like the most logical choice. But without knowing enough about the coin and other conditions, it can simply be a case of type 2 gambler fallacy or hot-hands fallacy. This line of thinking helps build bubbles. 

  1. This stock/mutual fund must be good as it has beaten the market in last 2/5/10 years
  2. It has increased 10 times in three months, everyone is buying it. It must be a good stock

Strangely, some people become victims of the original gambler’s fallacy in these scenarios. Once you recognize a bubble, it seems logical to short the stock. History suggests that it’s not a prudent strategy. 

Even bigshots have lost money by publicly taking short positions on stocks they deem inflated (hint: TSLAQ). Eventually all bubbles burst, but the size and duration are totally unpredictable. 

Likewise, if a new product (especially food, apparel, and products where consumer tastes are demographically different) becomes successful in one region, wait and watch to confirm if it can be replicated elsewhere before buying the stock. Don’t assume that something will keep happening the way it previously has, especially if the sample size is inadequate.

Variability is high when the sample size is small.


Note: The purpose of this article is to highlight that even obvious things warrant deeper analysis, not to draw parallels between methodical investing and randomized coin tossing. 

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