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gwern comments on Confidence levels inside and outside an argument - Less Wrong
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That's an important and non-obvious assumption to make.
So, in short, the 10^-90 figure is based on the explicit assumption that the election is not balanced?
That's why the two methods you mention produce such wildy different figures; they base their calculations on different basic assumptions. One can argue back and forth about the validity or lack thereof of a given set of assumptions, of course...
That is why I posted it as a comment on this particular post, after all. It's clear that our subjective probability of casting a tie-breaking vote is going to be far less extreme than 10^-90 because our belief in the binomial idealization being correct puts a much less extreme bound on the tie-breaking vote probability than just taking 10^-90 at face value.