Here’s a profound question about beards: is the number of acrobats with beards the same as the number of bearded people who are acrobats? Go with your gut instinct. It’s not a trick question. If you answer ‘yes’, then you’ve understood the central idea behind Bayes’s theorem. If you’re one of those people who likes to titter about how bad you are at mathematics, stop it. Retake your GCSE, learn how to pin this obviousness down in symbols, and you can produce artificial intelligence, forecast stock market collapses and understand this:
P(A|B) = (P(B|A)∙P(A))/(P(B))
This is Bayes’s equation, the formula which, as Tom Chivers insists in this remarkable, bold book, ‘explains the world’.
Rich mathematics begins by spotting an oddity in the obvious. Although the number of bearded people who are acrobats is the same as the number of acrobats who are bearded people, the two ways of saying the same thing reach the answer by different routes. They are like two paths circling up a mountain to arrive at the identical view. One path starts in the land of bearded people (P(B), in the previous equation, then meanders off to find out how many are acrobats, P(A|B). The other starts alongside the circus tent of tumblers and balancers, P(A), then heads on to how many are bearded, P(B|A).
Imagine you’re told someone is mild-mannered, loves books and wears glasses. Is this person more likely to be a librarian or a farmer? If you (you Sherlock!) diagnosed librarian, you’re wrong, because you’ve forgotten that there are vastly more farmers in the world than librarians.
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