Sleeping Beauty

[dethwing]

hmmm i dont know any more what i think.
I used to be completely 1/2....now....grrr who came up with this darn puzzle anyway !!

[araya]

Well I just read the minotaur's solution, and although I am highly unsatisfied, I guess that it was a given that the solution was not going to be concrete. I wonder if the minotaur had a more solid solution in mind before reading all of the many objections and points of view on the problem. Anyways, I think it's safe to say that we all did an excellent job of solving the problem - many different opinions were discussed, and a hell of a lot of thought experiments were performed, all of which didn't really get us anywhere except to discover that the answer is highly ambigous, which is evidently the 'correct' answer. Good work, people. Even you lowly 1/2-ers

[Ghost Post]

Agreed, Araya. I have to admit that although I was never convinced of the 1/3 argument, I can see how some weak-minded, misguided soul could be convinced to put there trust in the 1/3 argument (jk). But it was pretty fun to have the discussion, even though toward the end, we were not arguing over the same thing but over our own (two different) viewpoints.

Cheers.

[Ghost Post]

I tried to follow this discussion and found all I ended up understanding is that my head REALLY hurts!!

[Griffin]

Minotaur said in his solution that had SB been cloned instead of awaken twice, the situation would be the same (Which is an arguement that supports the people who say 1/2). I disagree, though I'm not sure exactly why they're different. I do know that in the original SB problem, if she guessed that the coin had been heads upon being awoken, she would be right 1/3 of the time, while in the cloning experiment, if she guessed heads she would be right 1/2 of the time.

[extropalopakettle]

I've yet to hear any 1/3'er offer a satisfactory explanation to the following:

Suppose that on either Monday afternoon (for heads) or Tuesday afternoon (for tails), when the experiment as it was described is over, they will say to her "The experiment is now over. What is the probability that the coin landed heads?" Furthermore, she knows they will do this. This is before she knows whether it's Monday or Tuesday.

What will she answer then? If the experiment were repeated many times, half the time it would be over on Monday (heads), and half the time it would be over on Tuesday (tails). Therefore, by the logic used by either 1/3'ers or 1/2'ers, at this point she would answer 1/2.

Let's digress a minute. If I know I am going to be twice asked the probability that a coin landed heads - once before I know anything about the result, and once after I know the result, then I know that the first time I will answer 1/2, and the second time I will answer either 0 or 1. No problem here. I know that later when I have more information, I will revise my estimate. But I don't know WHAT I will revise it to. Thats important. If somehow I knew that later (after receiving more information) I would say 0, I would in fact know now that it's 0.

On the other hand, in the case I present above, when SB is awakened and is supposedly claiming a 1/3 probability, she knows, NO MATTER WHAT, that she will later say 1/2, and furthermore, she knows that she will be claiming 1/2 knowing at least as much about the outcome as she does now. If she knows that later, knowing as much as, or more than, she knows now, she will say 1/2 - how can she justify saying 1/3 now?

It's analogous to saying "I don't have enough information to take a good geuss at Sleeping Beauty's name, but tomorrow I will have enough information to believe her name is Jane". This makes no sense. If I now know that tommorrow I will have enough information to believe her name is Jane, that in itself is information (that I have NOW) to lead me to believe her name is Jane.

It's altogether different, and acceptable, to say something like "I don't have enough information to take a good geuss at Sleeping Beauty's name, but tomorrow I will have enough information to form some belief about what her name is". I might know that tomorrow I will have more information, but not know what that information will be. But in the former case I'm saying specifically WHAT I will believe her name to be tomorrow, based on the information I'll have then, while claiming I have no clue today.

Does anyone NOT see the problem here?

And the problem does not go away by calling it a paradox or "ambiguox" (I like the term).

There are no true paradoxes (I believe). If you start with true assumptions, and use sound logic, you can't derive contradictions (by definition of "sound logic"). When it seems that you can, as in a "paradox", there is either a false assumption, or an unsound deduction (or a bad formalization).

Regarding the "ambiguox" (which may overlap, at the least, the "bad formalization" problem), if there are different interpretations about how to define (hence estimate) probability, in this case I can stick to just one of them (the one that derives 1/3) and still get the contradiction (because that interpretation still derives that she'll say 1/2 when told the experiment is over).

The contradiction, again, being that now she concludes it's 1/3, although she knows that later, knowing at least as much as she does now, and using the same logic, she will conclude it's 1/2. If she knows that later she'll have information that, by her present way of reasoning, will lead her to conclude 1/2, then that in itself is information enough to conclude 1/2 now.

Please help me out on this one.

[mwf]

I loved the problem and found the posts quite entertaining. It was fun to see people arguing over a problem with no right answer, well the answer they gave was the right one for them. I think most of the people forgot to read the puzzle or assumed that there must be a right answer. There were a few post that said that there was probably no right answer. Good job.

As for 1 in 2 or 1 in 3 or any other answer be more right than any other. I will say it will depend on how you see the problem. I realy hate to call it a problem since there was no goal or prize. The only thing you may have gotten from it was a lose of one day of your life. I must say if someone would like me to play S.B. for the experiment, they will have to pay a lot of money to do it.

I now think I will chang my answer if I was S.B. too 1 in 2 on Monday and 1 in 3 on Tuesday. *grin*

[PairOfDocks]

I do not think this is a paradox, merely a probability problem.

The question really is: "What is the probability of SB saying 'tails' and remembering this as the correct answer?"

In order to for this to happen, SB has to toss tails on two tosses. The probability of two tails on two tosses is .5 times .5 equals .25

Therefore, if SB says "heads" when she wakes up for the last time (whether this is the first trial or the second trial) she has a .75 probability of being correct. If she says "tails" she will only be correct .25 percent of the time.

The 1 in 3 is just incorrect. However, 1 to 3 would be correct.

Enjoyed the puzzle, especially someone's contribution of a new word -- I believe it was "ambidox". I apologize to the author if I misspelled it; perhaps he/she would be good enough to correct me.

Later,
PairOfDocks