When the financial crisis broke in August 2007, David Viniar, chief financial officer of Goldman Sachs, famously commented that 25-standard deviation events had occurred on several successive days. If you marked your position to market every day for a million years, there would still be a less than one in a million chance of experiencing a 25-standard deviation event. None had occurred. What had happened was that the models Goldman used to manage risk failed to describe the world in which it operated.
If the water in your glass turns to wine, you should consider more prosaic explanations before announcing a miracle. If your coin comes up heads 10 times in a row – a one in a thousand probability – it may be your lucky day. But the more likely reason is that the coin is biased, or the person who flips the penny or reports the result is cheating. The source of most extreme outcomes is not the fulfilment of possible but improbable predictions within models, but events that are outside the scope of these models.
Sixty years ago, a French economist described the Allais paradox, based on the discovery that most people treat very high probabilities quite differently from certainties. Not only do normal people think this way, but they are right to do so. There are no 99 per cent probabilities in the real world. Very high and very low probabilities are artifices of models, and the probability that any model perfectly describes the world is much less than one. Once you compound the probabilities delivered by the model with the unknown but large probability of model failure, the reassurance you crave disappears.