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TheOtherDave comments on Yes, Virginia, You Can Be 99.99% (Or More!) Certain That 53 Is Prime - Less Wrong
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Well, what's most interestingly improbable here is the prediction of a 200-coin sequence, not the sequence itself.
I suspect what's going on with such "extraordinary" sequences is a kind of hindsight bias... the sequence seems so simple and easy to understand that, upon revealing it, we feel like "we knew it all along." That is, we feel like we could have predicted it... and since such a prediction is extraordinarily unlikely, we feel like something extraordinarily unlikely just happened.