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## Coincidences and the Lottery

Coincidences happen surprisingly often. Yet they are often not meaningful, i.e. they are “just a coincidence” and do not imply that we should change our worldview. For example, suppose there are a million people in contention for a lottery, and John Smith is found to win. Before knowing this, our probability for it is $10^{-6}$:

$P(\textnormal{John Smith wins} | \textnormal{fair lottery}) = 10^{-6}$

People often get afraid of this tiny probability, and proclaim something like “it’s not the probability of John Smith winning the lottery that is relevant, but the probability that someone wins”. However, this is anti-Bayesian nonsense. This tiny probability is, by Bayes’ rule, relevant for getting a posterior probability for $\textnormal{fair lottery}$. So how is it that we often still believe in the fair lottery (or that a coincidence is not meaningful)?

The answer is quite simple: the likelihood for the alternative, $\textnormal{unfair lottery}$ hypothesis, is just as small:
$P(\textnormal{John Smith wins} | \textnormal{unfair lottery}) = 10^{-6}$.
The reason is that before we knew who won, we had no reason to single out John Smith, and had to spread the total probability (1) over a million minus one alternatives (that the lottery was rigged in favor of one of the other entrants). Using analogous reasoning, yes, coincidences have tiny probability, but they also have tiny probability given the hypothesis of a mysterious force operating, because before the coincidence happened we didn’t know which of the multitude of coincidences were going to occur.

For more on this topic, you may be interested in this paper (by myself and Matt).