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Nassim Nicholas Taleb’s book The Black Swan introduced an old term and then, annoyingly, redefined it. For Sir Karl Popper, the black swan was an observation about logical quantification: if you assert “all swans are white” then the observation of a single black swan falsifies the assertion.

Taleb’s observation is different, although related: he’s observing that really unexpected events are unexpected: we have a model of the world that says “The US mainland is secure from attack” that seems perfectly plausible on 10 September 2001; we believe “Islamist terrorism is on the run” and then a bomb blows up in Boston.

(There’s a more sophisticated way to deal with all of these called Bayesian inference. We’ll leave the details for a science column, but in a few words, a Bayesian starts with an assumed a priori estimate of the probability of an event. After observation, they have a new a postieriori estimate that incorporates new experience.)

But there’s yet a third way to think about these that shows us how mathematics and probability can show us surprising things.

(Yes, this is a diet and exercise column, just a little further down.)