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Decius comments on Thoughts on a possible solution to Pascal's Mugging - Less Wrong
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Fair enough. How many different sequences can be included in a given 'called set'? If I said "99 heads out of 100", then I'm identifying 100 different sequences.
In the end, though, I'm trying to set a ceiling: What's the most likely prediction I could make which would cause you to reevaluate the math behind odds? So far the lower limit is 1/2^100. Would you accept the call that at least 80% of the coin flips would be heads? My powers of telekinetic manipulation of coin flips are limited, you see, and both exhausting and unreliable.
It depends a lot.
For example, if I approached you and offered you a bet that you could not predict the flip of a series of coins, and you got it right three times in a row, that wouldn't particularly shake my confidence that each coin-flip could be modeled as an independent binary choice with equal chances on both sides.
OTOH if you approached me and offered me a bet that you could not predict the flip of a series of coins, and you got it right three times in a row, that would indeed shake my confidence.
But if I leave all the real-world stuff out of it, sure, a coin that comes up heads 80% of the time on, say, 100 flips would certainly make me suspicious.
Here's a bet then- Flip the coin nearest to you 100 times, and report the results. If you get 79 or fewer heads, then I will donate $20US to the cause of your choice (which may be you, personally). If you get 80 or more heads, then consider the possibility that I have the ability to alter the results of coin flips in a way which is unexplained by modern physics.
Or maybe I'm willing to gamble $20US on a very small chance (~half of six standard deviations, if I have the math right) that I can mindscrew you.
I used to date a girl who had a favorite card trick: she would hand you a deck of cards, ask you to pick one, and say "is it the three of clubs?"
Her theory was that she'd be wrong most of the time, but when she was right it would be really impressive.
It would be a lot more than $50 more impressive (the few times it works) if she said "I bet you a dollar that it's the three of clubs."
I was also considering the 'cider in my ear' angle. Just because you don't see any possible way that I could rig the bet, the fact that I proposed it is an indication above baseline that I might have.
See also.
I instantly thought about that, too.