Sleeping beauty paradox
Poll
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| | 10 votes (33.33%) | ||
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| | 1 vote (3.33%) | ||
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| | 2 votes (6.66%) | ||
| | 1 vote (3.33%) | ||
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30 members have voted
Off-topic, of course, but I was taught that "who" is used correctly there. It is the nominative case subject of the verb "is", or perhaps "is being asked." (The object of the preposition "to" is the phrase "who is being asked.")Quote: RSJust spent more time than I should have trying to figure out if you should have used "whom" instead of "who". :( I'm pretty sure it's "whom", but the Internet can't verify that for me. It has failed me once again.
And now back to your regularly scheduled paradoxes.
Quote: AyecarumbaIn fact, if she was to answer the question before being put to sleep the first time, what would her answer be? This may be the "pair of ducks".
1/2.
1. There is no reason for the coin flipper to actually flip the coin until Tuesday morning -because Sleeping Beauty must be awakened on Monday in any case.
So, if Sleeping Beauty says on Monday that the coin flip will be Heads either 1/3 or 2/3 of the time, isn't it the same as saying it about a coin flip that has not yet occurred? And doesn't that defy randomness?
2. In fact, once Sleeping Beauty has been informed of the rules of the situation (before she goes to sleep) she is aware that she will be awakened twice if the outcome is Tails or only once if the outcome is Heads. She has 100% of the information that she will have when she is awakened on Monday or Tuesday. If Sleeping Beauty knows that she will be saying 1/3 or 2/3 on Monday and Tuesday, why shouldn't she be predicting a 1/3 or 2/3 probability of Heads before she is put to sleep? And, again, doesn't the prediction of 1/3 or 2/3 of the outcomes being Heads for a random coin flip in the future defy randomness?
End Rant: This problem is NOT about the mathematics of calculating probabilities - it is about defining the state of existence of an observer.
Quote: gordonm888Questions for the advocates of 1/3 or 2/3:
1. There is no reason for the coin flipper to actually flip the coin until Tuesday morning -because Sleeping Beauty must be awakened on Monday in any case.
So, if Sleeping Beauty says on Monday that the coin flip will be Heads either 1/3 or 2/3 of the time, isn't it the same as saying it about a coin flip that has not yet occurred? And doesn't that defy randomness?
I agree that Monday morning wakening could be automatic and then flip the coin on Tuesday morning. It wouldn't change the answer. On any awakening SB could figure out that the chances it was caused by a tails flip on Tuesday. This chance is 1/3.
To everyone who thinks a wakening doesn't tell us anything, you're wrong. Just the wakening in itself is information.
What if instead of being woken up once, she was woken a million times (we'll have to assume this would be in the afterlife). Then on any wakening it would be pretty easy to guess it was probably because a tails was flipped and it is one of the million wakenings. Certainly Occam's Razor would suggest that.
Your link doesn't like the ' in it! Use this one instead.Quote: Wizard...Occam's Razor...
This has lots of details and background and "simple wikipedia" has an overview.
Quote: WizardSleeping Beauty volunteers to be the subject of an experiment. She is told truthfully that on Sunday she will be put to sleep. Immediately after, a fair coin will be flipped. Here is what will happen on happen, according to the flip:
Heads: She will be awakened on Monday only.
Tails: She will be awakened on Monday and Tuesday.
On each awakening she will be asked "What is the probability the coin landed on heads?" On any awakening, she will not know what day it is. After answering the question, she will be put back to sleep with an amnesia-inducing drug that will cause her to forget the awakening.
On Wednesday, she will wake up naturally.
Sleeping Beauty is a perfect logician.
How should she answer the question upon being awakened?
SB1, SB2, SB3, and SB4 volunteer to be the subject of an experiment. They are all told truthfully that, on Sunday, they will isolated in separate rooms and put to sleep. Immediately after, a fair coin will be flipped. Here is what will then happen, according to the flip:
Heads: On Monday, all but SB1 will be awakened. On Tuesday, all but SB2 will be awakened.
Tails: On Monday, all but SB3 will be awakened. On Tuesday, all but SB4 will be awakened.
On each awakening, each volunteer will be asked "What is the probability that this is the only day you will be awakened?" On any awakening, none will know what day it is. After answering the question, each will be put back to sleep with an amnesia-inducing drug that will cause her to forget the awakening.
Note that SB2's predicament is identical to the original Sleeping Beauty's, and the others are in a functionally-equivalent one.
Each is a perfect logician. How should they answer the question upon being awakened?
The only possible answer is 1/3.
ChesterDog and gordonm888 make the best arguments, but they can be combined. ChesterDog's analysis says that:
Pr(H&Mon|awake) = 1/3,
Pr(T&Mon|awake) = 1/3,
Pr(H&Tue|awake) = 0,
Pr(T&Tue|awake) = 1/3,
... so Pr(H|awake) = Pr(H&Mon|awake) + Pr(H&Tue|awake) = 1/3 + 0 = 1/3.
A similar analysis that supports gordonm888's result would have to say:
Pr(H&Mon|awake) = 1/2,
Pr(T&Mon|awake) = 1/4,
Pr(H&Tue|awake) = 0,
Pr(T&Tue|awake) = 1/4,
... so Pr(H|awake) = Pr(H&Mon|awake) + Pr(H&Tue|awake) = 1/2 + 0 = 1/2.
Now, say that after SB tells you Pr(H|awake)=1/2, you tell her what day it is. This does give her "new information" about the flip. If you tell her it is Monday, she can eliminate the events with "Tuesday" in them:
Pr(H|awake&Mon) = Pr(H&Mon|awake)/[Pr(H&Mon|awake)+Pr(T&Mon|awake)] = (1/2)/(1/2+1/4) = 2/3.
The problem is, that gordonm888 was right to point out that the coin doesn't have to be flipped until Monday Night. If 1/2 is correct, after she is told it is Monday she must predict that tonight's coin flip has a 2/3 chance to result in Heads.
+++++
The apparent paradox is caused by a change in how we need to define probabilities. In normal problems, every independent sampling is counted once. In this problem, it may be counted once or twice.
Betting arguments, even if they would get the right answer, are invalid because the number of bets placed depends on the result you are betting on.
But the "no new information" argument is invalid for the same reason, because you do have information about whether there going to be one, or two samples; that is, there will be one if the result is Heads, and two if there it is Tails.
The point of my alternative problem was to remove that ambiguity.
https://wizardofvegas.com/forum/questions-and-answers/math/22498-sleeping-beauty-problem/
The twist to the question I found on the paradox was pretty interesting.
Quote: z2newtonI was reading a blog where it talked about this probability puzzle. I'd never heard of the Sleeping Beauty problem before so I thought I'd post it here to see what the more math inclined have to say about it. From what I can gather there is not a consensus on the answer.
[http://en.wikipedia.org/wiki/Sleeping_beauty_problem]
Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake: if the coin comes up heads, Beauty will be awakened and interviewed on Monday only. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday. In either case, she will be awakened on Wednesday without interview and the experiment ends.
Any time Sleeping Beauty is awakened and interviewed, she is asked, "What is your belief now for the proposition that the coin landed heads?"
I guess they knock her out before she can ask what day it is.
[edit]
Since this is a gambling forum here is an interesting twist I found. The experiment is the same except Sleeping Beauty is given the chance to make a bet before the experiment begins. She is given 3 to 2 odds that the coin will come up heads. Instead of the original question, when she is awakened she is asked if she wants to cancel her bet.
To make it interesting lets say she bets $1000 on heads.
Quote: Wizard
“What is the probability the coin landed on heads?"
Sleeping Beauty is a perfect logician.
How should she answer the question upon being awakened?
Late to the party and armed with a string of fire-crackers in the guise of there is no stricture limiting the form of her reply:
“If you asked me this question yesterday 0% otherwise 50%”.
I’m the guy who replies to the question “is the cup half-full or half-empty?” by finding the 3rd option: it’s too big.
Steven
EDIT: replaced negative form (“If you didn’t”) of SB’s answer.
Sleeping Beauty consents to the following experiment and is informed of what will happen. On Sunday night she will be put to sleep. Then a 6-sided die will be rolled. If a 6 is rolled, she will be woken up once a day for the next 100 days. Upon each awakening, she will be asked to predict the coin toss. If she is right, she wins $1000. Then she will be put back to sleep and any memory of the awakening will be removed. If anything else is rolled, she will be woken up once once, after 100 days, and asked the same thing. Then on day 101 she will be woken up, asked nothing, given all money won and she goes on with her life.
What should Sleeping Beauty predict on each awakening?
Quote: unJonThanks for resurrecting this. I don’t see the paradox. Of course it was 1/3 in the first scenario. Otherwise you could AP the situation with whomever wants to make book on 50%.
link to original post
I agree. You can see that the 1/2 camp outnumbers the 1/3 camp in the poll by 10 to 8.
Here is the YouTube video I like on the topic.
Direct: https://www.youtube.com/watch?v=XeSu9fBJ2sI
He is also in the 1/3 camp, but tries to explain both sides. Here it seems the 1/2 camp argues the coin was fair and you're not given any information when awakened, so there is no justification to alter the 50/50 chance of the original coin.
https://anthropic-principle.com/preprints/beauty/synthesis.pdf
Quote: WizardQuote: unJonThanks for resurrecting this. I don’t see the paradox. Of course it was 1/3 in the first scenario. Otherwise you could AP the situation with whomever wants to make book on 50%.
link to original post
I agree. You can see that the 1/2 camp outnumbers the 1/3 camp in the poll by 10 to 8.
Here is the YouTube video I like on the topic.
Direct: https://www.youtube.com/watch?v=XeSu9fBJ2sI
He is also in the 1/3 camp, but tries to explain both sides. Here it seems the 1/2 camp argues the coin was fair and you're not given any information when awakened, so there is no justification to alter the 50/50 chance of the original coin.
link to original post
There is a response to the Veritasium video which addresses the aspect of tails' Monday and Tuesday (starting at about 03:10) being dependent, not independent, and therefore not giving reason for a 1/3er's position.
Your opinion?
