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AmagicalFishy comments on What's wrong with this picture? - Less Wrong Discussion
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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?