A key aspect of risk intelligence is recognizing the limits to your knowledge. People's judgment of risks is deeply compromised by psychological biases. One of the most pervasive of these is a phenomenon called the favourite long-shot bias, first observed by the American Psychologist Richard Griffith in 1949. Since then, numerous studies have found evidence of the bias at racetracks and other sports betting markets all around the world. Indeed, it is probably the most discussed empirical regularity in sports gambling markets, and the literature documenting it now runs to well over a hundred scientific papers.

Daniel Kahneman and Amos Tversky provide perhaps the best theoretical framework in which to understand the phenomenon. In the famous 1979 paper in which they presented their Prospect Theory, they noted that people's ability to perceive differences between extreme probabilities was far greater than their ability to notice differences between intermediate ones. The difference between 0% and 1%, for example, is much more salient than that between 10% and 11%, and the difference between 99% and 100% looms much larger than that between 89% and 90%.

As a result, we tend to overreact to small changes in extreme probabilities and underreact to changes in intermediate probabilities. We will pay far more for a medical operation that increases our chance of surviving from 0% to 1% than one that increases it from 10% to 11%. We will also pay more for a lottery ticket that increases our chance of winning from 99% to 100% than one that increases it from 89% to 90%.

This oversensitivity to small changes in likelihood at both ends of the probability spectrum gives rise to an interesting phenomenon in gambling on horse racing; punters tend to value long shots more than they should, given how rarely they win, while valuing favorites too little given how often they win. The result is that punters make bigger losses over the long run when they bet on long-shots than they do when betting on favorites (they still make losses when betting on favorites because the racetrack takes a percentage of each bet, but the losses are smaller). This is why bookies rejoice when a long-shot wins; that's when they make their biggest profits.

According to data published by Erik Snowberg and Justin Wolfers, in their article Explaining the Favorite-Long Shot Bias: Is it Risk-Love or Misperceptions?, there's an oversensitivity to small changes in likelihood at both ends of the probability spectrum — for very short and very long odds. Betting on horses with odds between 4/1 and 9/1 has an approximately constant rate of return (at minus 18%), which implies that bettors are quite good at distinguishing between probabilities over this range. It is only when punters bet on favorites (with odds shorter than 4/1) or long shots (with odds longer than 9/1) that they get into difficulties.

What can we do to compensate for the difficulty we have when dealing with small differences in probabilities? The best option is to turn to mathematics. Remember: a key aspect of risk intelligence is recognizing the limits to your knowledge, and this includes recognizing the limits of risk intelligence itself. The financial crisis of 2007-2008 was partly due to an over-reliance on mathematical models and a corresponding failure to exercise judgment. The opposite mistake — ignoring mathematical models and relying entirely on subjective estimates — can be equally dangerous when distinctions of less than 1% matter.

Such distinctions are particularly important when it comes to extremely low probabilities of less than 1%. It is impossible to feel the difference between, say, a probability of 0.01% (1 in 10,000) and 0.001 (1 in 100,000), and yet the first is ten times greater than the second. In this territory, you must therefore abandon all recourse to epistemic feelings, and rely completely on your calculator.