The law of large numbers vs the law of small numbers

Toss a (balanced) coin 10 times and you might observe 7 heads, but toss it 1 zillion times and you’ll most likely get very near 50 percent, which is closer to the underlying probability.

The phrase law of large numbers is employed because, it concerns the way results reflect underlying probabilities when we make a large number of observations.

In real-life situations we often make the opposite error: we assume that a sample or a series of trials is representative of the underlying situation when it is actually far too small to be reliable.

The misconception—or the mistaken intuition—that a small sample accurately reflects underlying probabilities is so widespread that Kahneman and Tversky gave it a name: the law of small numbers.  The law of small numbers is not really a law. It is a sarcastic name describing the misguided attempt to apply the law of large numbers when the numbers aren’t large.

When people observe the handful of more successful or less successful years achieved by the Sherry Lansings and Mark Cantons of the world, they assume that their past performance accurately predicts their future performance.

Consider a situation in which two companies compete head-to-head or two employees within a company compete. Think now of the CEOs of the Fortune 500 companies. Let’s assume that, based on their knowledge and abilities, each CEO has a certain probability of success each year (however his or her company may define that). And to make things simple, let’s assume that for these CEOs successful years occur with the same frequency as the white pebbles or the mayor’s supporters: 60 percent. (Whether the true number is a little higher or a little lower doesn’t affect the thrust of this argument.) Does that mean we should expect, in a given five-year period, that a CEO will have precisely three good years? 

No. Even if the CEOs all have a nice cut-and-dried 60 percent success rate, the chances that in a given five-year period a particular CEO’s performance will reflect that underlying rate are only 1 in 3! Translated to the Fortune 500, that means that over the past five years about 333 of the CEOs would have exhibited performance that did not reflect their true ability. Moreover, we should expect, by chance alone, about 1 in 10 of the CEOs to have five winning or losing years in a row. 

What does this tell us? It is more reliable to judge people by analyzing their abilities than by glancing at the Scoreboard. Or as Bernoulli put it, “One should not appraise human action on the basis of its results.

Going against the law of small numbers requires character. For while anyone can sit back and point to the bottom line as justification, assessing instead a person’s actual knowledge and actual ability takes confidence, thought, good judgment, and, well, guts.

Executives’ winning years are attributed to their brilliance, explained retroactively through incisive hindsight. And when people don’t succeed, we often assume the failure accurately reflects their talents and abilities. 

Full Disclaimer: I am not a financial planner. The views expressed in this post are all mine and they may or may not suit your needs. Please do you own due diligence. I do not make money on any of the products suggested in this post. 

References:

• The Drunkard's Walk: How Randomness Rules Our Lives