I am gonna argue that the outcome of the coin flip and when Beauty is awakened are two completely different events.

The coin flip with pull a subset of one from the sample space.

Our sample space contains only two possible outcomes:

H - Monday/Heads
T - Monday/Tails and Tuesday/Tails

So our event will have a 1 to 1 odds of H or T. That is a 50% chance each. That part we should all be able to agree on.

Now the second event is when Beauty determines her credence, and I am arguing that this is actually independent of the coin flip, because Beauty does not know the results of the coin flip. She can only make her determination based on what she knows about the experiment, which would mean the event from the coin flip is out of play here. It does not matter if it landed on H or T, as Beauty does not have that information.

The question is what is her credence, or what is her belief, that is independent of the actual coin flip and completely in the hands of how Beauty decides to interpret the experiment.

That is not at all how it works.

A simple random sample is a subset from the population where n members of the population have a chance of being selected by random chance, such as a fair coin flip. The actual distribution of the population doesn't matter. This subset is also called an event, and probability is the ratio of the event over the total number of possible outcomes from the population.

You are not a very honest person.

Also since when did a Bayesian approach become the gold standard, what about a Classical approach?

There is a chance mechanism, the coin flip, it will give us one event from the population, then the ratio of the event over the total possible outcomes will give us the probability.

Possible outcomes

M1 Heads and Monday
T2 Tails and Monday
T2 Tails and Tuesday

One could stop there or, one could argue that since Beauty is asked the credence that there is a second chance mechanism in Beauty's answer. This is not updating a prior with new information, it would be a new event and the population would be dependent on Beauty's selection.

The fact that we can come up with the 1/3 and justify it without the experiment even being ran just proves that it is not a posterior, it is a prior.

That is were both Lewis and Elga error, in assuming that is the linchpin between which one is right.

In the Monty Hall problem, the host gives you information that changes the probabilities that you assign to each door. That information is new to you.

Similarly, in the Sleeping Beauty problem, awakening provides information that enables you to rule out one of the four states. However since you have no information distinguishing the remaining states, you should be indifferent about which state you are currently in. — Andrew M

Beauty doesn't gain relevant new information when awakening, she knew all this before hand. If we do the experiment on you, then you have a prior belief that it is 1/3, what new information would then update that? Priors need relevant new information which would allow us to update it, not just any old information that you think happened.

This was my prior response, that I already posted before a few pages back.

I am not convinced it contradicts the 1/3, but instead just leads back to the initial problem of the inherited subjectivity in the notion that you should start with a prior belief and update that prior into a posterior belief when you get new information. This carries that subjectivity into any posterior, and as we see here those priors can look very different based on the viewer. Not really objective science. — Jeremiah

I already posted both of their responses.

I understand a no spoilers approach, after you exhaust it on your own you should give them a read.

I am not convinced it contradicts the 1/3, but instead just leads back to the initial problem of the inherited subjectivity in the notion that you should start with a prior belief and update that prior into a posterior belief when you get new information. This carries that subjectivity into any posterior, and as we see here those priors can look very different based on the viewer. Not really objective science.

Don't know what you mean. — Srap Tasmaner

That was Lewis' argument in that paper I linked.

Elga rejects my premiss: his (E7) contradicts my (L1). I reject Elga’s
premiss: my (L6) contradicts his (E1).

Starting on page 3 he lays both arguments out.

http://fitelson.org/probability/lewis_sb.pdf

She knows it must be either Monday or (Tuesday and Tails) — Andrew M

Sure, she knew that would be the case before even starting, but when she is awaken she has no new information on if it is Monday or Tuesday. You are not exactly describing a condition that is new to her knowledge.

Do you agree that P(Heads|Awake) = 1/3? — Andrew M

No, I have said many times this problem has more than one rational and valid answer.

Would you consider the 2/3 debacle an argument against the 1/3 argument and for the 1/2 argument?

The OP version is clearly confusing you, hence the need of the above variation. — Michael

Your "above variation" is confusing you and honestly the colors of the balls would not make any difference. Monday and heads is still only one out of the three possible outcomes.

It needs two blue balls and one red ball.

Listen I am not doing the example gambit; you people come up with all these examples and some can be representative of the problem, while others are not and at that point they become a straw-man, whether it was intentional or not. And if I argue a straw-man then I give false creditiably to it. The best response I can have here, is to suggest we stick to the main problem of interest.

I know many of you think I am just being a troll when I call straw-man, but I know where these numerous examples lead. They are fine in sort order, but people have a tendency to take them off the central path and if you follow them then you end up in La La Land and the discussion has steered way off track.

Let's just stick with the OP, then we all know we are debating the same problem and not something else.

It's self-locating information that she can update on. — Andrew M

However, it is not. When awakened Beauty does not know if it is Monday or Tuesday.

One should update one's probabilities when given new information. — Andrew M

The problem with a Bayesian approach is that in order to update you need a prior to update, which in Beauty's case could be based on either the coin flip or the possible times she will be awakened. That's the issue with Bayesian probability and what makes it so controversial, it is subjective.

And that does not change the fact that P(T) = P(T & M) = 50%

I agree the 1/2 argument has its merits as a valid argument, but not for the reasons you are listing. Probability is not reallocated evenly between Tuesday and Wednesday since she is awaken both of those days from the same chance mechanism. From the coin flip she has a 50% chance of being awakened on Tails and Tuesday, and a 50% chance of being awakened on Tails and Monday. Not 25% each as you suggested. This means they are equally likely as Heads and Monday.

Tails grantees Monday as well. She is awakened on Monday regardless of if it is tails or heads.

I am, you are reallocating the 50% for a tails flip across two days; however, it is not reallocated across two days, it is the two days. When Beauty is awakened there are three possible events in which she is awakened. Two of them are determined by the same outcome of the chance event, which is the coin flip, they both have a 50% chance of happening, so they are equally likely. The other possible outcome is on heads, and has a 50% chance of happening so it is equally likely as well. Since all three are equally likely, Beauty's credence that it is Heads and Monday is 33%.

Right I agree with that, and have from the start.

I never disagreed with that, I don't really feel you are understanding what I am saying.

Consider instead that if it was heads then she's woken on Monday and if it was tails then she's woken on Tuesday and Wednesday. What's the chance that she's woken on Monday? 50%. Therefore what's the chance that it was heads? 50%. — Michael

And a 50% chance of that she is awakened on Tuesday and Wednesday.

Therefore a 33% of Monday and Heads. What you are not considering is that from the chance mechanism 50% is allocated to two of the possible awakening as the same outcome. Not split between them, but instead, in terms of the coin flip, they are not separate events, they are the same outcome.

Here you might find these papers interesting.

https://www.princeton.edu/~adame/papers/sleeping/sleeping.html

http://fitelson.org/probability/lewis_sb.pdf

It is a fair coin flip, so the chance of Monday and Heads is 50% and the chance for Monday and Tails is 50%. Since on one tails flip she is awakened on two days then the chance of Tuesday and Tails is also 50%. Therefore, Beauty has a 33% chance of being awake on a heads

That is not new information, she knew she'd be awakened beforehand. New relevant and significance information to reallocating creditably would be if she was told what day it was on Monday.

There is an interesting discussion to be had about it. — andrewk

And yet the other one was generating more discussion.

And this is consistent with the fact that we know that, given a fair coin toss, there's a 50% chance that it landed heads. We shouldn't change our view of that just because we might be woken up twice rather than once. — Michael

Yes, Beauty is aware the coin is fair, but she has also been told the details of the experiment and knows there are three possible events in which she is awakened. If awaken on Tuesday she would not know it is Tuesday; she only knows there are three possible outcomes in which she will be awakened. To Beauty, who does not know if it is Tuesday or Monday when she is awaken, Tails and Tuesdays and Tails and Monday are both valid outcomes. Beauty has to consider three possibilities and only one of them is the desired outcome.

But now I don't know if that's misdirection too. — Srap Tasmaner

None of it is misdirection, this problem has several possible rational answers. We are not talking about just different points of view, or different readings of the semantics, it can rationally be answered in multiple ways from the same point of view and the same interpretation of the semantics. This is because the problem is probing the relationship between knowledge and probability, in this case Beauty's knowledge.

And actually, Pattern-chaser's first answer is still a valid take. Beauty can't be sure of what day she is awakened, as she is given no new information when she is awakened, which means a 1/2 chance for Monday is a rational response. This would follow Bayesian philosophy on probability which suggest we should update our probability models when we get new information.

I didn't count it twice. I am saying P (Tails and Monday) and P (Tails and Tuesday) have the same likelihood of occurring because they are determined by the same chance event, but for Beauty the coin flip generates three possible answers. You have to distinguish between the coin flip and Beauty determining the probability it landed on heads based on the possible times she could be awakened.

P (Tails and Monday) and P (Tails and Tuesday) are the same flip. Are you really suggesting they have a different chance of occurring when they both occur on the same chance event?

It is a fair coin, fair means 50% chance the coin come up heads and and 50% chance it comes up tails. When Beauty is awakened she has to provide the probability it is heads. I am arguing at this time 33%, as there are three possible outcomes where she is awakened and only one of those outcomes is heads.

The coin is only flipped once.

P (Tails and Monday) and P (Heads and Monday) are mutually exclusive — T Clark

Clearly . . . .

Also, P (Tails and Monday) and P (Tails and Tuesday) are not independent. — T Clark

They are the same flip. It is pointless to argue independence. That is like saying I rolled a die and got 4 and it is not independent because I got 4. If the coin is tails she will be awakened on Monday and awakened on Tuesday, therefore they have the same probability.

Also, you left out P (Heads and Not Tuesday). — T Clark

She is not awakened on Tuesday if the coin is heads.

The possible outcomes:

- - M T
H: A, S
T: A, A

If she is awakened before Wednesday, there is one awake on heads and two awake on tails. Of the possible awakens there is a 1/3 chance it is Monday and heads.

Consider it this way:

P(Tails|Monday) = P(Tails|Tuesday) Right? Because they are on the same flip they have the same probability.

Now if the coin is fair then P(Heads and Monday) = P(Tails and Tuesday)

Therefore P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday).

It must add up to one, thus the 1/3.

Btw, there are other ways to look at this, Pattern-Chaser was not necessarily wrong. I only presented this side because they led with the 1/2 argument; however, they are correct in pointing out Beauty has gained no additional information. Really all she knows is what she was told before the experiment.

Well let's break it down.

If the coin flip is heads then Beauty is awakened on Monday but sleeps though Tuesday.

If the coin flip is tails Beauty is awakened on Monday and awakened on Tuesday.

So we have:

H: A, S
T: A, A

So since there are three possible awakenings and only one is when the coin comes up heads, then won't that mean she has a 33% chance of it being heads?

What is truly childish is pretending these paradoxes belong only in select conversation.

I never read any Nietzsche, and I am likely much older than you think.

Its fine to be unproductive and timewasting, — Akanthinos

Your entire argument boils down to a child throwing a fit because the neighbor kid is playing with his/her toys.

devalues the discipline. — Akanthinos

:brow:

Anyone with a Analytical Philosophy course completed, upon being asked "who shaves Russell's Barber" out of context, will simply look the interlocutor with amused pity. — Akanthinos

And you claim you are not an elitist. . . .

What is oddly backwards about your entire argument, is that I am using these paradoxes to generate discussion, which is their central reason for engagement. While you are pretending they belong to your fantasy club of those you consider your fellows. You seem to think there is a specific context, under certain terms and with select people these matters should be discussed. I, on the other hand, discuss these topic and much more complicated notions with everyone and anyone. I have had conversations about multi-variable calculus with people who have no greater than high school algebra, because I assume that if I can figure it out and understand it then they can as well.

Yes, I can be a troll, I am well aware of that and fully admit it, but clearly you are also not without your egoistical hang ups as well.