An Interesting Problem

[iago]

My philosophy prof referred to this as the "Sleeping Beauty" problem.  Anyway, there are two possible answers, and I know a pretty good counterargument to either, so I hope this will make for a good discussion.

The Setup:
Ok, you volenteer for a scientific assignment where the first flip a coin, then put you to sleep until monday.  Depending on the coin:
Heads - you wake up monday, and you get to go home
Tails - you wake up monday, you're put back to sleep, and you wake up tuesday but can't remember that you were ever woken up.

Now, before you go to sleep, you're asked which side you think the coin landed on; clearly, it's 50% heads, 50% tails.  

So you wake up; you don't know whether you've already been woken up once and it's tuesday, or if this is the first time/only time, and it's monday.  They ask you which side the coin probably landed on.  What is your answer?

[Adron]

Now, assuming that you know that you will be asked which way you think the coin flipped every time you wake up, and that you want to make the right guess as often as possible, consider either always saying heads, or always saying tails.

If you always say head, then you'll either score one correct answer or two wrong answers, with equal probabilities. If you always say tail, you'll either score one wrong or two correct answers.

Conclusion: For the best possible score over a large number of repeats of the experiment, lacking any other mechanism for determining the day or result of the toss, you should always guess tails, for an average of 66.7% correct answers.

[Thing]

I disagree with your conclusion Adron with the following observations:

Assuming you only get two chances to guess during the experiment (One before and one when you wake up the final time) then ...

Before you go to sleep you have to choose one of two choices, either heads or tails.  Regardless of what you choose, there is a 50% chance that you will be correct.
Upon awakening, you are asked to choose again.  You still have a 50% chance of being correct.  Therefore, your odds are 50/50.

If you are given a chance to guess on Monday even though you will be put back to sleep until Tuesday then ...

Your odds are still the same, you just get three chances to guess.  The odds of being correct on each guess are always the same.

If you are going to repeat the experiment a number of times, accumulate statistics, and have a guess on the Mondays that you are put back to sleep, then your idea of picking tails all the time makes sence.
Let's say you run the experiment 100 times and 50 times it lands heads, 50 times tails.  You will have 250 guesses total.  If you guess tails every time, you will be wrong 100 times and correct 150 times.  That makes 60% correct.

[Adron]

Oh ok, I didn't take the initial guess into consideration since that's an obvious 50% chance. The conclusion is still the same, if your goal is to have as large a percentage as possible of correct guesses over an extended number of tests, you should keep guessing tails.

[iago]

So the conclusion is that there's a greater chance of the coin landing on tails.  

To support this:
Say you do it for a month, 4 times.  Twice, it'll be heads.  Twice, it'll be tails.  However, when it's tails, you are woken up monday, tues, mon, tues, and if it's heads, you're woken up mon, mon.  Therefore, there's a 2/6 chance that it's heads and a 4/6 chance that it's tails.

Now, realistically, you are asked the odds before going to sleep.  You say 50%.  You're put to sleep, you wake up, not knowing what day it is.  He asks you what is probably landed on.  Why would you change?   It's still the same coin, and it still has a 50% chance of being on either side, so why would it be any differnet?

Those are arguments for each side of the case.  Which do you think is better?

[Adron]

So the conclusion is that there's a greater chance of the coin landing on tails.  

That's not the conclusion.

The conclusion is that when tails is the correct answer, your answers are given more weight, and when heads is the correct answer, your answers are given less weight.

The problem is then similar to: Alice and Lenore are going to a friend who lives 20 km away. If Alice drives 20 km at 20 km/h and Lenore drives 10 km at 10 km/h + 10 km at 30 km/h, who gets there fastest?

[iago]

But when you're woken up, there's a greater chance that the coin had landed on tails.

That's my professor's stance, anyway.  And he related it to a gambler's dilemma.

This was my response to him:
It's like a game of roulette.  The only difference is, there's a (say) 1/36 chance of landing on any number, usually.  BUT our table is rigged so that 1/3 of the time it will land on 17, and the other 2/3 it's completely random.  The odds are just a little more than 1/3 that it will for sure be 17, and the odds are a little less than 1/36 that it will be *something* else.  At that point he said his brain hurt and moved on.



And to Yoni: sorry, I had meant to leave your post, just delete the others.  I like your idea of quantum states, *except* that the person who is doing the experiment had already observed the value and therefore it wasn't unknown :PO