The Sleeping Beauty Problem

The May issue of Scientific American describes the Sleeping Beauty Problem. On Sunday Sleeping Beauty is put to sleep and a coin is tossed. If it comes up heads she is awakened on Monday, asked what was the probability of heads, then is put back to sleep, which causes her to forget she was ever awakened.If it comes up tails she is again awaked on Monday, asked what was the probability of heads, then put back to sleep forgetting she was ever awakened, then awakened again on Tuesday and asked the same question, then put back to sleep.Whether the result of the coin toss was heads or tails, Sleeping Beauty is awakened on Wednesday.What should Sleep Beauty have answered when awakened on Monday and Tuesday when asked about the results of the coin toss.I'll give my answer in a later post. I don't know why this isn't trivially simple, but many mathematicians apparently think it isn't.--Percy

The generalised case is more interesting, but I think it can be trivially solved by enumeration.So yes, count me in with those who don’t see why it’s interesting mathematically (although it is interesting psychologically).

The question the experimenters ask is "what is the probability of heads?". There is only one answer she could give.The paper then goes on to change the question to "what are the probabilities of each of the three scenarios?". That is not the same question at all.I don't understand the problem in having two different answers. There is no problem. There is no comparison. They are two different questions from two separate frames of reference. I don't understand the consternation SciAm is hyping here.

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The paper then goes on to change the question to "what are the probabilities of each of the three scenarios?". That is not the same question at all.

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For Sleeping Beauty the probability that the coin came up heads is the probability that she is in the scenario where the coin came up heads.Like the Monty Hall problem, information affects the relevant probabilities.

I can see that. But, the probability of heads/tails is .5 no matter how many scenarios you attach to each outcome and no matter what the girl may or may not know when.The problem is surreptitiously transitioned from a question of the outcome of a binary state to a question of probabilities of multiple states. They are not the same.So what is being asked? What is the probability of tails? or what are the probabilities of various scenarios from the girl's reference frame? They are not the same question.What am I missing?

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I can see that. But, the probability of heads/tails is .5 no matter how many scenarios you attach to each outcome and no matter what the girl may or may not know when

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The a priori probability doesn’t change, but this is an after the fact probability with relevant knowledge. To take a very simple example if you can see that the coin has come up tails the probability that it came up heads is zero.

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So what is being asked? What is the probability of tails? or what are the probabilities of scenarios? They are not the same question

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It is the probability of heads to the girl in her situation.Or to put it another way, it’s the probability that she would be right if she guesses that the coin came up heads.If she guesses heads she will be right in one of the three equally likely (to her) scenarios. If she guesses tails she will be right in two of them.So if she guesses heads she will be right 1/3 of the time. Therefore the probability that the coin came up heads - to her in that situation - is 1/3.

That was my feeling, too.--Percy

Yes, I know all that ... if ... the question was for this smart girl to give her perspective and not the physical reality of a coin flip.I didn't take the question to be that though I can see where that question arises.Excellent explanation, PaulK.

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Yes, I know all that ... if ... the question was for this smart girl to give her perspective and not the physical reality of a coin flip.

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Which it actually is. Nobody is asking her for the a priori probability of a coin flip coming up heads or tails. If they were, then nobody would be talking about the problem

But SB also knows that the probability of tails at the outset was 1/2, so the probability of being in one of the tail scenarios is 1/2, or 1/4 for each.What if we change the scenario. What if the scenario for heads remains the same, but the scenario for tails is that SB is awakened 99 times that she never remembers. Is the probability of heads now 1/100 from SB's point of view? That seems absurd and feels like proof that that's the wrong answer. SB knows she has a 50% probability of being in the heads scenario, and a 50% probability in being in one of the tails scenarios. That there are 99 tails scenarios (or 2, doesn't matter how many) might be just one of those irrelevant distractions puzzle makers like to throw in to throw you off.--Percy

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But SB also knows that the probability of tails at the outset was 1/2, so the probability of being in one of the tail scenarios is 1/2, or 1/4 for each.
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That is obviously incorrect. When Sleeping Beauty is woken up on Monday the probability isn’t 1/4 that the coin is tails. There is no perspective which makes sense of that.

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What if we change the scenario. What if the scenario for heads remains the same, but the scenario for tails is that SB is awakened 99 times that she never remembers. Is the probability of heads now 1/100 from SB's point of view? That seems absurd and feels like proof that that's the wrong answer.

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And yet it is correct. If you run both cases, Sleeping Beauty will be right only once out of a hundred times if she guesses heads.

The operative words being "from SB's point of view".She doesn't know if this is her first or 69th awakening. All she knows is she is awake and there are 99 ways for that to happen if the coin was tails and only one way for that to happen if the coin was heads. Yeah, 99 to 1 for tails. In her mind, flipping a heads is the least probable event explaining her being awake.Interesting.

You're right. I went back to the original problem statement and see that I wasn't recalling it correctly. I'd forgotten that she's asked each and every time she awakens.--Percy

Isn't this just a version of the Monty Hall problem?Monty Hall problem - WikipediaI say 'just' that one messes with my head too.