Ok, so you think the coin is 50/50 in this spot.

Let's say she gets an unplanned visual clue that tells her it's actually day 1. What now?

For your answer to make sense, you would have to say it's overwhelmingly heads.

Unfortunately, if she's told it's day 1, that's the most obvious 50/50 spot possible.

I discussed this problem in post 92. If she is randomly told the day, and it happens to be day 1, then it is very likely heads. If she is always told it is day 1 when it is day 1, then it is 50/50 for her. It isn't inconsistent.

Nvm.... I am still in the 50% camp but now I see it basically depends on what you are asking about. So thirders are right too.

1) P(H | awake) = 1/3
2) P(awake) = 1
3) This implies that unconditionally, P(H) = 1/3

Still waiting for you to address the absurdity that arises from your conclusion. Which of 1) or 2) is wrong?

This is not analogous to the theism/atheism debate so you should leave that out of this.

The answer in the OP is clearly 1/3 heads. At this point I would only recommend that if you still think it is 50/50 you should stay away from gambling for any significant amount of money, at least until you understand why 50/50 is wrong.

The OP asks for your probability that the coin toss was heads. It is a combination of the coin toss probability (50/50) and the information that you gain by being awakened. It is not a question of the coin toss alone. Thus in the bold case the answer is 0% for the heads result even though the coin is 50/50. If you analyze that case mathematically and then insert the numbers from the OP case you will get 1/3 heads. Your intuition is misleading you. You should trust the math. Always trust the math over your intuition.

I have never gambled significantly but I have always wondered how gamblers manage to make a living doing it. I suspect correctly analyzing situations like this might have something to do with it. It is interesting that Sklansky hasn't commented on this.

If you reworded the question she was asked when awakened to this:

"How much would you wager for the opportunity to win $1 if the coin toss was heads?" that is the same question in the OP but put into monetary terms. Now what is the breakeven number for her wager, $0.50 or $0.33?

Let's assume that the first 5 minutes when she wakes up she will not be able to find her glasses so she will not be able to determine whether she is still in the experimentroom or not, and let's give her an amnesia pill after she goes to bed on monday as well whether it was heads or tails(this should not change the experiment in any way, because she is not asked the question on (heads & tuesday)). So she has no information about whether she is still in the experiment, about what day it is(monday or tuesday), and about the outcome of the coinflip.
So, before she finds her glasses:

P(monday) = 0,5
P(tuesday) = 0,5
P(heads) = 0,5
P(tails) = 0,5

At this point these are still independant probabilities, if we somehow slip the information to her it is tuesday this should not influence the probability's regarding the coinflip (as long as she believes this to be an accident, that is, she must not believe that this means the experiment has ended, because that would imply the coinflip was heads), but let's not do this yet. So:

P(monday & heads) = 0,25 =
P(monday & tails) = 0,25 =
P(tuesday & heads) = 0,25 =
P(tuesday & tails) = 0,25

Anybody disagree?


P(monday & heads) => still in experiment
P(monday & tails) => still in experiment
P(tuesday & heads) => not in experiment = P=0,25
P(tuesday & tails) => still in experiment

Now she must conclude from this there is a 25% chance the experiment is already over, and therefore a 75% chance she is still in the experiment.

P(still in experiment |tails) = 1
P(still in experiment |heads) = 0,5

Therefore, according to Bayes' Theorem
P(tails |still in experiment) = P(still in experiment |tails) * P(tails) / P(still in experiment) = 1 * 0,5 / 0,75 = 2/3
P(heads |still in experiment) = P(still in experiment |heads) * P(heads) / P(still in experiment) = 0,5 * 0,5 / 0,75 = 1/3

So the moment she realizes she is in the experiment she must conclude there's a 2/3 chance the coin was flipped tails.

edit: fixed a little mixup in the middle, the experiment is over on (tuesday & heads)

Is this directed at me? I'm in the 1/3 camp so I agree with your conclusion.

What the conclusion is saying is that if you flip a fair coin (without anything else going on) 1/3 of the time it will come up heads and 2/3 tails.

That's just flipping the coin and looking at it (unconditional)

I am a professional gambler and I understand mathematics fairly reasonably (I would say easily in the top 0.1% of people posting on 2+2), please dont insult me. I am coming around to some parts of the 1/3 view but only because of more interesting reasons (I have read a bunch of papers on it today), not because of most of the reasons in this thread*.

*There are similar problems such as, a couple has two children, atleast one of which is a girl, what is the chance that the younger one is a girl? (2/3). This is not as similar as you would like to believe.

Thanks coffee for clearing up what Justin A was saying.

1 is not unconditionally true. It is only true under these conditions. Therefore, you have to take the word unconditionally out of your conclusion.

Would like the 50/50ers to answer my Memento scenario.

If you think Memento is still 50/50, how about this:


Person w/ 1 minute memory span is told the following:
We will flip a coin. If it's heads, you will spend 10 hours in a green room, and spend 10 minutes in a red room. If it's tails, 10 hours red, 10 minutes green. Due to his short memory this information is written down for him.

He finds himself in a green room. Is it not reasonable for him to assume it's much more likely heads was flipped than tails?

No reason for you to be insulted if you are a successful professional gambler. All that means is that my conclusion is flawed.

The girl problem is well known. Whether it is similar or not would be interesting to discuss but there is no point until you are at 1/3 heads for the coin problem.

How about the change in wording from credence to wager for $1? What is your take on that? Rewording it has the advantage of eliminating any possible confusion around the meaning of "credence".

The problem with the 1/3ers is they mix up probabilities from the experimenter's perspective with SB's perspective. Yes, from outside, MH, MT, and TT each occur 50% of the time. But I don't think you can just assume that the probabilities from the outside perspective carry over to the waking-up perspective. MH, MT, and TT are not 3 separate universes, they are 2 universes: one with one observer, and one with two observers.

The two universes are equally likely, but the 3 observers are not. SB knows when she wakes up that she will be an observer chosen from { MH } 50% of the time and from { MT, TT } 50% of the time. What is wrong with this logic?

The problem here is that the guy doesn't remember how long he's been awake for. Time would be relevant info, if he knew it, but he doesn't. Dividing up each timespan into 5 minute chunks, he is 50% to be any of two observers, and 50% to be any of 120 observers.

Yes but he's given new information (the colour of the room). Kind of like in the SB example if she's told it's Monday.

Well, I'm a 1/2'er so I appreciate what you're saying here. But I'm afraid it just won't fly. I think nearly everyone agrees that it makes no difference whether the coin is flipped Sunday night or Monday night because Beauty will be awakened Monday regardless of the coin's outcome. The only day's awakening that's decided by the coin is Tuesday's. So they might as well wait until Monday night to flip the coin for the Tuesday wakeup decision. For that matter, as jason1990 points out in the 2007 thread, Beauty might be a determinist and figure the outcome of the possibly future Monday night coin flip has already been determined as much as if it had already been flipped.

I think this also puts you in the uncomfortable position of being a 1/2'er if the coin is flipped Sunday night but a 1/3'er if it's flipped Monday night. I'm afraid very few people will buy that. In fact, if you maintain that position long, cognitive dissonance may force you into the 1/3'er camp altogether. You wouldn't want that would you?


PairTheBoard

You are drawing from the space of wake ups not the space of flips. Why isnt that the end of the discussion??? Both an external observer and SB experience identical events during the waking up part. They belong to the same universe. And by the way spare me with the multiple worlds ideas. There is only one world so far in this human adventure and we are all in it. Having or not memory will do nothing to the process of counting how often a wake up is due to tails for all observers involved.

SB is not experiencing a flip, she is sleeping when it happens. SB is experiencing the event of waking up which can be right after the flip or a day later. Go ahead and hope its 50% and ignore my posts. I maintain the 2/3 tails camp or 1/3 heads is confident and the 50% camp is searching for ways to convince themselves. Show me a pathological result that is obvious by applying the 2/3 logic. But do it inside this problem, do not create unrelated convoluted examples.


We now go to the 1 time trial? Well in the one time trial what i can do is take a billion times her clones and create a multiple times 1 trial and of course the probability assigned in the 1 time ought to be used in the probability of the many resulting in a big problem once we use 50% for that. The resulting compounding of 1 trial clones yields the 1 billion times repeated trial of a single person since they are clones. Hence a problem for the billion that now doesnt yield 2/3 anymore when it should.

Again why isnt the 50% camp answering the question what is the probability upon waking up that its Monday?

The only way it could be flipped Monday night is if on heads she is awakened on Tuesday and told that the experiment ended Monday and there is no question.

Why is everyone avoiding the monetary expression of credence? Is it that "credence" means something fundamentally different to you than "What would you wager to win a $1?" or does the monetary question make 1/3 heads clearly the correct answer?

I believe this is begging the question. You can really get either answer depending on how you do the sampling. If you take one sample of her credence in each run of the experiment you will see her 50% credence is correct. If you check her credence once for each run by asking her to guess based on her credence and based on 50% she guesses Heads every time then you will confirm her credence of 50% is correct by seeing her guess of Heads is correct in half your samples.

To my mind it makes no sense to check the validity of a person's credence in an event (the coin is Heads) by checking it more times for one outcome than the other.

Is she right 1/2 the "time" or 1/3 the "time". Both really. Guessing heads, She's right 1/2 the time "per experiment run" and 1/3 the time "per awakening". Which is the correct way to verify the validity of her credence? To my mind, to verify the validity of her credence in the outcome of a coin flip you should check it "per coin flip". To me, this is "obvious".


PairTheBoard

Probably the one that's defined in the question.

This is so backwards. You can't define in what setting she's waken up by "how many observers she is". This just isn't a logically acceptable factor. However, time is. Time is simple and straightforward and relevant to the problem at hand.

MT and MH occur with equal probability. This much is undeniable.

Just because he doesn't know the current time doesn't mean he can distinguish between any time based events. In this example what I mean is, he can tell he's awake at a point in time. Out of all the possible points in time he can be awake, it's much more likely he's in the 10 hour camp than the 10 minute.

Not sure what you mean here.

They occur with equal probability from the experiment's perspective, but that doesn't prove she should have equal credence in them when she wakes up.

Yes I'm aware that the experiment is no different if the coin is flipped Sunday or Monday evening. But that's irrelevant since SB doesn't know what day it is when waking up.

You're just asserting your answer. Each point in time is equally likely from an outside perspective, that doesn't prove it is from the perspective of the guy during the experiment.

I don't think you understand. She is awakened Monday regardless of the coin. She is put back to sleep after her Monday awakening regardless of the coin. The only time the coin comes into play is for what they do with her Tuesday. If the coin is heads they let her keep sleeping until the experiment is over. If tails they wake her again Tuesday then put her back to sleep again until the experiment is over - Wednesday or the next Sunday if they like. That Tuesday choice can be made just as well based on the outcome of a coin flipped Monday night as on the outcome of a coin flipped Sunday night. It makes no difference for how the experiment is conducted. This is not hard.


PairTheBoard

OK, fine. It doesn't matter when the coin is flipped as long as Monday's question is asked regardless and Tuesday's is dependent on the coin being tails. Also, she cannot know under which condition she is answering. The 2/3 bias then exists towards tails.

Again, reword the question. She is asked "What would you pay to earn $1 if the coin is heads?" If you answer that question then you know your credence for the proposition of the coin being heads.

Can anyone refute my post #107? If not, I don't understand how you could still argue the 50/50 side...

Just to be clear, she is given new info when she wakes up. Before she wakes up it could be monday or tuesday, and heads or tails. The fact that she wakes up within the experiment means its not tuesday AND heads.

To the 50/50 people: what should her credence be towards the day being monday when she awakes within the experiment?