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philh comments on A (very) tentative refutation of Pascal's mugging - Less Wrong
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A word of caution - Solomonoff induction applies to things like the laws of physics, not to all hypotheses. Otherwise, if you flipped a coin 100 times, you would expect to see 100 heads much more often than average, and we don't.
If you flip a coin 15 times, this result:
HHHHHHHHHHHHHHH
is far more probable than this:
HTHTTHTHTTTHHTH
That's because some coins are rigged, and it's much easier to rig a coin to conform the first pattern than the second.
First reaction: I don't know about "far" more probable. What's the prior that a coin is rigged? I would have said less than 1/32768, but low confidence on that.
According to this, you can't rig a coin to do that, which increases my confidence.
But you can rig your tossing, even by mistake; if it lands heads, and you balance it to flip with heads up again, then it's slightly more likely to land heads. I remember hearing a figure of 51% for that; in which case H*15 has probability 1/24331 instead of 1/32768; about a third more probable. But that scenario (fifteen times) is itself unlikely... if we estimate P(next is heads | last was heads) = 0.505 (corresponding to keeping the same side up 3/4 of the time, I still feel that's an overestimate), we get 1/28204, 16% more likely.
If we switched to dice, I would agree that 666666666666666 is far more probable than 136112642345553.