The Gambler's Fallacy and the Misuse of the Law of Large Numbers


If you have ever spent any time in a casino, it is likely that you have seen supposedly random events that appeared to be non-random. Perhaps the roulette wheel came up red seven times in a row, for example. If your reaction was to place a large bet on black because black was “due” or to bet on red because red was “hot”, then you fell into the trap commonly known as the gambler’s fallacy (or the reverse gambler’s fallacy in the latter case), which is the mistake of thinking that the outcome of a random event is influenced by outcomes of prior events when in reality, each event stands on its own.

The gambler’s fallacy can sometimes lead to the gambler’s ruin, as seen in this BBC article about several Italians who financially destroyed themselves by betting their life savings on #53 in the Venice lottery because it had failed to be drawn for an abnormally long time. The tragic consequences of these bets could so easily have been avoided, especially in light of the fact that these bets resulted from a misunderstanding of the Law of Large Numbers, which states that 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.

The key point of this mathematical law is that the observed proportion of trials meeting a given criterion (e.g., coin flips coming up heads) converges on its expected value as the number of trials increases. This is not the same thing as saying that the number of heads converges on 50% of the number of flips. For example, if a coin is flipped 100 times, it would not be unusual to have 55 heads and 45 tails (this would be a one standard deviation event which has a 32% probability of occurrence). However, if the coin is flipped 1,000 times, the chances of having 550 heads and 450 tails are quite remote (this would be a three standard deviation event which has a 0.3% probability of occurrence). If this event were to happen, it does not mean that in the next set of 1,000 flips we expect more tails than heads to even the score. If we simply had the expected number of 500 heads, then the overall percentage of heads in the 2,000 flips would drop to 52.5%, in accordance with what we would expect from the law of large numbers.

A beautiful explanation of the contrast between the gambler’s fallacy and the law of large numbers is found in Wikipedia.

“The gambler's fallacy arises out of a belief in the law of small numbers, or the erroneous belief that small samples must be representative of the larger population. According to the fallacy, "streaks" must eventually even out in order to be representative…The gambler's fallacy can also be attributed to the mistaken belief that gambling (or even chance itself) is a fair process that can correct itself in the event of streaks, otherwise known as the just-world hypothesis.”

A mathematical concept (but not a law) that is closely related to the gambler’s fallacy is reversion to the mean. The gambler who expects to see “average” results grossly underestimates how large a sample size is needed for convergence to the average. The gambler’s fallacy arises from the mistaken belief that mean reversion happens much quicker than would be predicted from the law of large numbers. One area of investing where mean reversion is prevalent is the performance of active managers relative to their benchmarks, especially if they currently have a “hot hand”. John Bogle has repeatedly pointed out this problem and has received a great deal of criticism from the active management industry because reversion to the mean negates the possibility of skill being the explanation for an active manager beating his or her benchmark. Interestingly, one of Bogle’s most vociferous critics, hedge fund manager Andrew Feinberg had a change of heart when he was forced to explain his own lackluster performance in 2011 when he said, “Mutual fund managers who beat the market for a time have a nasty habit of reverting to the mean.”

At Index Fund Advisors, our approach to investing is that every day has the same expected return, which, in contrast to betting on a coin flip, should be positive in order to compensate investors for bearing risk, and historically the S&P 500 Index return has been about 0.03%/day with a 50/50 split around the average or a 51/49 positive to negative expected outcome. The law of large numbers can work to our advantage in two ways, or what we call double diversification. This can be accomplished by maximizing the number of securities held (asset diversification) and maximizing the number of days of market exposure (time diversification). Regardless of how many anomalous periods we see of consecutive up or down days, we do not waiver from the most basic principle of every day having the same expected return. If there is a better approach to investing, we have yet to see it.