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Author Topic:   The Sleeping Beauty Problem
PaulK
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Posts: 17894
Joined: 01-10-2003
Member Rating: 8.2


Message 2 of 14 (910733)
05-10-2023 9:05 AM
Reply to: Message 1 by Percy
05-10-2023 8:35 AM


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).

This message is a reply to:
 Message 1 by Percy, posted 05-10-2023 8:35 AM Percy has not replied

  
PaulK
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Posts: 17894
Joined: 01-10-2003
Member Rating: 8.2


Message 4 of 14 (910735)
05-10-2023 12:30 PM
Reply to: Message 3 by AZPaul3
05-10-2023 12:20 PM


quote:
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.
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.

This message is a reply to:
 Message 3 by AZPaul3, posted 05-10-2023 12:20 PM AZPaul3 has replied

Replies to this message:
 Message 5 by AZPaul3, posted 05-10-2023 1:34 PM PaulK has replied

  
PaulK
Member
Posts: 17894
Joined: 01-10-2003
Member Rating: 8.2


(1)
Message 6 of 14 (910737)
05-10-2023 1:50 PM
Reply to: Message 5 by AZPaul3
05-10-2023 1:34 PM


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

This message is a reply to:
 Message 5 by AZPaul3, posted 05-10-2023 1:34 PM AZPaul3 has replied

Replies to this message:
 Message 8 by AZPaul3, posted 05-10-2023 3:00 PM PaulK has replied
 Message 10 by Percy, posted 05-10-2023 3:21 PM PaulK has replied

  
PaulK
Member
Posts: 17894
Joined: 01-10-2003
Member Rating: 8.2


Message 9 of 14 (910740)
05-10-2023 3:08 PM
Reply to: Message 8 by AZPaul3
05-10-2023 3:00 PM


quote:
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.
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

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 Message 8 by AZPaul3, posted 05-10-2023 3:00 PM AZPaul3 has not replied

  
PaulK
Member
Posts: 17894
Joined: 01-10-2003
Member Rating: 8.2


Message 11 of 14 (910742)
05-10-2023 3:53 PM
Reply to: Message 10 by Percy
05-10-2023 3:21 PM


quote:
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.

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.
quote:
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.
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.

This message is a reply to:
 Message 10 by Percy, posted 05-10-2023 3:21 PM Percy has replied

Replies to this message:
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