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Anatoly_Vorobey comments on Probability puzzle: Coins in envelopes - Less Wrong Discussion
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Well, not everything - it isn't Laplace's rule of succession, but if you correct the mistake, you've pretty much solved part 2. Instead of a fixed value you get an equation you can solve for m.
That's true. It also invalidates my answer for part 1, which is a bit trickier to correct, because you no longer have the nice symmetry.