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OrphanWilde comments on What's wrong with this picture? - Less Wrong
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All sequences, both written and flipped, are equally improbable. The difference is in treating the cases where the two sequences are identical as logically distinct from all other possible combinations of sequences. They're not nearly as distinct as you might think; imagine if she's off by one. Still pretty improbable, just not -as- improbable. Off by three, a little less probably still. Equivalent using a Caeser Cipher using blocks of 8? Equivalent using a hashing algorithm? Equivalent using a different hashing algorithm?
Which is to say: There is always going to be a relationship that can be found between the predicted sequence and the flipped sequence. Two points make a line, after all.
I agree with you that the probability of Alice's sequence being a sequence will always be the same, but the reason Alice's correct prediction is a difference in the two mentioned situations is because the probability of her randomly guessing correctly is so low—and may indicate something about Alice and her actions (that is, given a complete set of information regarding Alice, the probability of her correctly guessing the sequence of coin flips might be much higher).
Am I misunderstanding the point you're making w/ this example?
Which seems more unlikely: The sequences exactly matching, or the envelope sequence, converted to a number, being exactly 1649271 plus the flipped sequence converted to a number?
They're equally likely, but, unless Alice chose 1649271 specifically, I'm not quite sure what that question is supposed to show me, or how it relates to what I mentioned above.
Maybe let me put it this way: We play a dice game; if I roll 3, I win some of your money. If you roll an even number, you win some of my money. Whenever I roll, I roll a 3, always. Do you keep playing (because my chances of rolling 3-3-3-3-3-3 are exactly the same as my chances of rolling 1-3-4-2-5-6, or any other specific 6-numbered sequence) or do you quit?