But the probability before the coin flip is 50% since there are only two possibilities. The simple cause of the difference is how much information SB has. If SB received information about that fourth possibility it would be back to 50%. In her world, based on her information at the time thirder is correct. But to others with full information the halfes are correct. It’s all relative.

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]]>“What is the probability you would assign that the coin came up tails?”

But for SB to come to the “thirders” conclusion something like this should have been the question IMO:

“What likelihood should SB assign to tails if she is told her sole objective is to guess correctly the most number of times?”

From this perspective SB might make a different choice than with the original question.

It also seems to me that the “thirders” position is not explained well above. In the SB scenario the coin is flipped only once. If she would guess heads for each interview and the coin flip came up heads, she would make two correct guesses, one on Monday and another correct guess on Tuesday. But if it comes up tails then she will be guessing wrong only once on Monday. So she is twice as likely to guess right if she guesses heads. This is the basis of the “thirder” position, but the wording of the question should have been different (similar to the “likelihood” question shown above IMO) for her to more likely come to this conclusion.

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]]>This post has solidified my impression that MWI is not an interpretation of quantum mechanics, but is rather a way of thinking about probability in the context of a single event. It doesn’t seem to say anything about where the probabilities come from.

The minimal interpretation seems to be the ensemble interpretation. Given that science is in the business of describing general rules, why would we ever care about the “probability” of a unique nonrepeated event? And for repeated events, all these puzzles go away, no?

Once you already have probabilities, what does it add to postulate an amplitude equal to their square roots? “The same logic that says that probabilities are proportional to the amplitudes squared also says you should be a thirder” seems trivially true, in the sense that if you have established probabilities by any means whatsoever, you should be a thirder.

I’m a layman whose opinion doesn’t matter. But Sean might want to know that, for one reader anyway, the effect these posts are producing is the opposite of their goal. (Though they are certainly serving their larger goal, of being interesting and informative!)

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]]>P(Flip) = 1/2

P(Waking up) =1

If she has knowledge of what day it is that changes the semantics of the problem meaning that the mutually exclusive event of “coin flip” and the following event of “waking up” are conditionally dependent allowing her to assign different probability values (especially if it is a Tuesday). This appears to be a poorly worded problem. The problem most likely leaves sleeping beauty’s knowledge ambiguous to get people to ask the following question: What does sleeping beauty know and will she go out on a date with me (please check the box yes or no)? I could be wrong and what I love about probability theory is how it is deceptively awesome . Do I have any volunteers for this experiment?

P.S. Did you remove my earlier post? Because I could have sworn I posted here before.

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]]>So an experiment with Sleeping Beauty is more important than an experiment by Alain Aspect?

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]]>SB’s goal is to assign probabilities to the possibilities that 1) she just woke up, and the coin flipped at the beginning of the trial was heads, and 2) she just woke up, and the coin flipped at the beginning of the trial was tails.

Since the experimental procedure clearly states that every time the coin comes up heads she will be woken up twice, and every time she comes up tails she will be woken up once, then it is obvious that over time the ratio of “woken and heads” to “woken and tails” will be 2:1. This is the same as saying the probabilities are 2/3 and 1/3. Literally the definition of probability.

So while she knows that the odds of a fair coin flip in complete isolation is 50:50, *that’s not the case here*, and the result of the flip has consequences, and those consequences mean that 2/3rds of the time she awakes in a universe where a fair coin landed heads. Therefore that is the correct probability to give. And repeated trials would bear it out.

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]]>You should also add:

3. the wave function should describe all the particles that form an observer and its memory

4. you can deduce “relative states” of the model by deducing what the modeled observer has measured by examing its memory

Including the observer and its memory is how Everett avoided the collapse.

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]]>You can disprove my claims by specifying what SB is trying to achieve in a clear enough fashion and getting someone to bet against you despite this goal being clear to both parties.

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]]>You’re adding a random game on top of the original question and assuming SB’s answer to “What is the probability you would assign that the coin came up tails?” is the one that minimizes her own profit in this invented game. Any other answer would let her turn a profit by picking her guesses on whether it was heads or tails. You’re assuming the one who answers the question is playing _against_ SB when the problem statement clearly says _she_ is he one answering the question.

The key, of course, is the question “What is the probability you would assign that the coin came up tails?”. What is SB trying to achieve here? Any clear definition of that gives a clear answer, but my opinion is that any invention of games that doesn’t have SB answer honestly, as the problem states initially, that the probability is 50-50 _or_ that she can answer whatever she wants because nothing gives her an incentive to _lie_, is just plain stupid.

You can invent additional casinos or whatever that define what would optimize her expected value in iterated SB problems, but saying that 1/3 is the correct answer is just plain wrong unless you define a different game for which the answer is 1/3.

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