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Author Topic:   The Sleeping Beauty Problem
Percy
Member
Posts: 22901
From: New Hampshire
Joined: 12-23-2000
Member Rating: 6.8


Message 1 of 14 (910731)
05-10-2023 8:35 AM


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

Replies to this message:
 Message 2 by PaulK, posted 05-10-2023 9:05 AM Percy has not replied
 Message 3 by AZPaul3, posted 05-10-2023 12:20 PM Percy has replied

  
Percy
Member
Posts: 22901
From: New Hampshire
Joined: 12-23-2000
Member Rating: 6.8


Message 7 of 14 (910738)
05-10-2023 2:55 PM
Reply to: Message 3 by AZPaul3
05-10-2023 12:20 PM


AZPaul3 writes:
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.
That was my feeling, too.
--Percy

This message is a reply to:
 Message 3 by AZPaul3, posted 05-10-2023 12:20 PM AZPaul3 has not replied

  
Percy
Member
Posts: 22901
From: New Hampshire
Joined: 12-23-2000
Member Rating: 6.8


Message 10 of 14 (910741)
05-10-2023 3:21 PM
Reply to: Message 6 by PaulK
05-10-2023 1:50 PM


PaulK writes:
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.
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

This message is a reply to:
 Message 6 by PaulK, posted 05-10-2023 1:50 PM PaulK has replied

Replies to this message:
 Message 11 by PaulK, posted 05-10-2023 3:53 PM Percy has replied
 Message 12 by AZPaul3, posted 05-10-2023 3:56 PM Percy has not replied

  
Percy
Member
Posts: 22901
From: New Hampshire
Joined: 12-23-2000
Member Rating: 6.8


Message 13 of 14 (910744)
05-10-2023 4:55 PM
Reply to: Message 11 by PaulK
05-10-2023 3:53 PM


PaulK writes:
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.
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

This message is a reply to:
 Message 11 by PaulK, posted 05-10-2023 3:53 PM PaulK has not replied

  
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