- 5,918
- 3,110
The probability for Monday is 3/4 and for Tuesday 1/4.PeroK said:Before you vote for 1/2, you may wish to consider this post:
The probability for Monday is 3/4 and for Tuesday 1/4.PeroK said:Before you vote for 1/2, you may wish to consider this post:
Charles Link said:The probability for Monday is 3/4 and for Tuesday 1/4.
Charles Link said:I think the alternative problem I posed answers the question: .99+.01(.001) it is Monday, .01(.001) it is the second day, .01(.001) that it is the 3rd day, etc.
It is somewhat of a puzzle. My logic of post #99 was somewhat of a response to the logic previously posted of adding extra days if it came up tails. I'm trying to demonstrate a possibility using a weighted coin, and extrapolating it to a non-weighted coin. ## \\ ## In some ways, the calculation that Sleeping Beauty does here is similar to what happens in science when we try to compute a probability under the assumption that the state we are considering is completely random. ## \\ ## I think my example in post #99 is worth consideration, because it illustrates the type of assessment that ultimately results: Is Sleeping Beauty likely to be awake on the Monday (with a weighted coin) of a very likely event, or did the unlikely occur, so that she is now part of a long chain of what would occur if the unlikely took place? I can also follow the logic of assigning a 1/3 to the heads condition, but, in some ways, this problem defies logic.PeroK said:That makes no sense to me. The problem is about a coin with 50-50 heads and tails. Please explain your solution for that, especially in light of post #100 and the fact that if it's Monday it's 50-50 heads/tails and if it's Tuesday it's 100% tails.
Probability that the coin is heads: 1/2PeroK said:@Marana If you were the sleeper, please tell me what you would answer to these three questions:
You wake up:
What is the probability that the coin is heads?
If it's Monday, what would be your answer?
If it's Tuesday, what would be your answer?
PeroK said:Or, someone has two children. The probability of two boys is 1/4.
If they have two girls then they come to see you on a Monday; otherwise, they come to see you on a Tuesday. Nothing random. Yet, if they come to see you on a Tuesday, the probability of two boys has increased to 1/3.
Demystifier said:Probability is not only about randomness, but also about absence of knowledge. Suppose that I pick one of the letters A or B, by will. Then I ask you, what is the probability that I picked A? What is your answer?
stevendaryl said:Here's the way that I became a thirder, which I think is convincing (even if it is much more work than the original, one-line argument for 2/3 or 1/2).
Imagine that experimenters are doing this experimenter over and over, with lots of different test subjects (sleeping beauties).
Marana said:Tuesday follows Monday by the laws of the universe.
And I know that, so to deceive you I will be more prone to choose B. But I know that you know that too, so I will deceive you at a higher level by being more prone to choose A. But I know that you know that I know that you know that, so perhaps I should deceive you at an even higher level be being more prone to choose B ... When one takes all this into account, A and B seem about equally likely.Marana said:but I'm not totally indifferent as I'd guess A is more popular
Marana said:It's not just the randomness that concerns me, it is whether it is an experiment. "In probability theory, an experiment or trial is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space."
Demystifier said:And I know that, so to deceive you I will be more prone to choose B. But I know that you know that too, so I will deceive you at a higher level by being more prone to choose A. But I know that you know that I know that you know that, so perhaps I should deceive you at an even higher level be being more prone to choose B ... When one takes all this into account, A and B seem about equally likely.
Charles Link said:The probability for Monday is 3/4 and for Tuesday 1/4.
Charles Link said:@stevendaryl Thank you. Very interesting, but this is one of those that I think I could study for years and not be convinced that there is one answer that is completely correct. It's like the riddle of "Who's on first. (baseball=first base). Is "Who" on first, etc.? It comes up in the Dustin Hoffman, Tom Cruise movie "Rain Man", and Tom Cruise tells his brother that it is a riddle that has no correct answer. :) :)
PeterDonis said:It looks like I'm the only vote in the poll for "it depends on the precise formulation of the problem", which seems strange, since the fact of this thread going on for 6 pages would seem to be evidence in favor of that choice.(A better basis for it, though, IMO is Demystifier's post #67.)
PeroK said:I cannot see any reason that in this experiment we would only count one of the instances in the case of tails.
PeterDonis said:Um, because the experimenter decided to define the experiment that way? The scenario described in post #67 is about betting, and a bet can be whatever the bettors want it to be.
Nobody would be answering 1/3 without the drug. Without the drug the answer is 1/2 on monday and 0 on tuesday (because you keep your memory you always know what day it is).PeroK said:Furthermore, the only thing that makes this problem difficult is the amnesia drug. The 1/2 position can only arise in the case of an amnesia drug. Without the drug, the problem is trivial and the answer is 1/3. And, I suggest, that without the drug, no one would be claiming that the problem is not precise.
Finally, I see the fundamental difference in this thread as the 1/3 position has been backed up by analysis; whereas, it is the 1/2 position that is largely opinion: we can't use relative frequencies; we can't use a Bayesian approach; it's not a random experiment; there is no sample space; conditional probabilities don't apply; etc.
In each case, an analysis of why these objections are invalid has been presented, although I guess now the core analysis has been lost in 6 pages of claim and counterclaim.
I don't think I am being all that radical by saying things like this are not allowed, for the reason I mentioned above. "It is Monday" and "it is Tuesday" are a different kind of information which can't use the usual probability techniques. But I'm not just making that up: it is fact that you may learn both "it is Monday" and "it is Tuesday" for a single flip, and it is a fact that that is absolutely impossible with normal probability techniques.PeroK said:PS And, if the answer is 1/2, then you can deduce with certainty that it is Monday. Proof:
Suppose the probability that it is heads is 1/2 and the probability it is Monday is ##p##. Then the probability it is heads is:
##\frac{p}{2} + 0 = \frac{p}{2} = 1/2##
Hence, ##p = 1##.
Therefore, whatever the answer is, it cannot be 1/2, unless the problem is changed so that Tuesday is excluded.
Marana said:Nobody would be answering 1/3 without the drug. Without the drug the answer is 1/2 on monday and 0 on tuesday (because you keep your memory you always know what day it is).
Marana said:I don't think I am being all that radical by saying things like this are not allowed, for the reason I mentioned above. "It is Monday" and "it is Tuesday" are a different kind of information which can't use the usual probability techniques.
Marana said:It's like saying you "learned" it was 2pm. Then you "learned" it was 2:01. Then you "learned" it was 2:02. Well, no, you're not learning all that in the usual sense. Time is just going forward.
stevendaryl said:You quoted one line of my post, but I don't see that you responded to it. Do you agree that in the setup I described, it makes sense for an observer to assign a 2/3 probability that a randomly selected sleeping beauty has an associated coin flip result of heads? The way I described that thought-experiment seems perfectly amenable to usual probabilistic reasoning. Right?
The next step would be for each sleeping beauty herself to consider herself a random choice. She knows that there are 3N/2 people in the same situation she is in--not knowing whether they have been awakened one time, or two. Of those, she knows that
So it seems completely straight-forward that she would assume that
- N/2 had a coin toss result of heads, and are awakening for the first time.
- N/2 had a coin toss result of heads, and are awakening for the second time.
- N/2 had a coin toss result of tails, and are awakening for the first time.
The final logical leap is to assume that the probabilities that apply in a repeated experiment also apply in a single experiment.
- Her probability of having a result of heads is 2/3
- The probability that she is being awoken the first time is 2/3
- The probability that she is being awoken the second time is 1/3.
I like this idea, but I don't think it is equivalent any more to the original.stevendaryl said:Let me try another variant of the experiment that I think will convince you that you're wrong.
Have you seen the movie "Memento"? The main character has a form of amnesia where he wakes up every morning having no idea what happened the previous day, unless he left notes for himself beside his bed (or pinned to his pajamas, or whatever). So we can redo the Sleeping Beauty problem using such an amnesiac. There is no need to wipe memories, but instead, we just control what notes she has waiting for her on the two mornings, Monday and Tuesday.
We prepare two envelopes. The first envelope says "Read me first" on the outside. Inside is a note explaining the rules of the experiment. Regardless of the coin toss, she gets this note on both Monday and Tuesday. The second envelope says "Read me second" on the outside. Inside is a note saying either "Today is Tuesday" or "Today is either Monday or Tuesday". She only gets the note saying "Today is Tuesday" if it actually is Tuesday, and the coin toss result was tails.
She wakes up and reads her first envelope, and is asked her subjective probabilities for various things. I believe everyone would agree that her answers would be:
All four situations are equally likely.
- There is a 1/4 chance that it's Monday and the coin toss result was Heads.
- There is a 1/4 chance that it's Monday and the coin toss result was Tails.
- There is a 1/4 chance that it's Tuesday and the coin toss result was Heads.
- There is a 1/4 chance that it's Tuesday and the coin toss result was Tails
Now, she reads the second note and finds out that she is not in situation 4. The other three situations are still equally likely, though.
Marana said:If so, I agree with you from the outside observer perspective, but not from sleeping beauty's perspective.
The outside observer can think of her as a random choice and assign 2/3 probability, but as I see it sleeping beauty can't consider herself a random choice.
I like this idea, but I don't think it is equivalent any more to the original.
One of the weaknesses with the thirder arguments I've seen is that they don't seem to model all of sleeping beauty's information. Sleeping beauty is well aware of the rules, of the way monday tails and tuesday tails are inextricably linked by the passage of time, and of the week she will next wake up in.
That last part is important in your example. If sleeping beauty never has any memory, then she can't be aware of what week she is going to wake up in. This is added uncertainty. She is lost not only within the week, but between weeks.
So it could be argued that she is now more likely to be in a tails week (since they are longer).
stevendaryl said:I think that if we transform the problem into one where Sleeping Beauty's answer has consequences (by having her bet on the coin flip, for example), then both halfers and thirders will agree on the answer, provided that the consequences are spelled out in sufficient concrete detail. So to me, that means that there isn't actually a disagreement about mathematics. It's a disagreement about the meaning of probability, when it is abstracted away from consequences ...
PeroK said:why would one calculate a probability on that basis?
PeterDonis said:Um, because that's how one interpreted the words "what is your subjective credence that the coin came up heads?"
The issue isn't that the math is unclear once we've decided which math we're using. The issue is that the problem is not stated in math, it's stated in ordinary language, and ordinary language is vague. Once you remove the vagueness by specifying exactly what mathematical calculation corresponds to "subjective credence", of course there's a unique right answer. But you can't just declare by fiat that your preferred mathematical calculation is the only possible one corresponding to the vague ordinary language used in the problem statement. You don't get to decide how other people interpret vague ordinary language.
PeroK said:and doesn't have to be self-consistent or obey the rules that "real" probabilities do.
PeroK said:The problem with the alternative calculations is that they are not consistent with what is meant by a probabilty.
PeterDonis said:I don't see this at all. The two answers, 1/3 and 1/2, are both well-defined conditional probabilities, just different ones:
1/3 is the conditional probability that the coin came up heads, given that Beauty has just been awakened and that we are randomly choosing between the three possible conditions under which Beauty can be awakened (Monday heads, Monday tails, Tuesday tails).
1/2 is the conditional probability that the coin came up heads, given that it was flipped.
PeroK said:Read the Wikipedia page to see the real issue
PeterDonis said:Here is how the Wikipedia page sums up the thirder argument:
"Since these three outcomes are exhaustive and exclusive for one trial, the probability of each is one-third"
In other words, Beauty's "subjective credence" should correspond to the first conditional probability I gave.
Here is how the Wikipedia page sums up the halfer argument:
"Sleeping Beauty receives no new non-self-locating information throughout the experiment because she is told the details of the experiment. Since her credence before the experiment is P(Heads) = 1/2, she ought to continue to have a credence of P(Heads) = 1/2 since she gains no new relevant evidence when she wakes up during the experiment."
In other words, her "subjective credence" should correspond to the second conditional probability I gave.
So I don't see how I've failed to describe the issue correctly. I've obviously left out a lot of details in each argument, but so what? I've correctly identified the two conditional probabilities, and that was all I was trying to do. The rest of the details are just attempts to justify why one particular conditional probability is the "right" one, the one that the words "subjective credence" should be interpreted to refer to.
PeroK said:the halfer argument denies that that information is valid for the purposes of calculating probabilities.
PeterDonis said:No, it doesn't. It just denies that the probability you calculate using that information is the "subjective credence that the coin came up heads". Which is vague ordinary language. If Beauty was asked the question "what is the conditional probability that the coin came up heads, given that this particular time at which you are awakened is randomly chosen from three equally likely possibilities?" then there would be no debate. But that isn't the way the question is asked in the standard version of the problem.
PeroK said:Read the Wikipedia page
PeroK said:Why is my subjective credence not the probability that I can calculate?
PeroK said:Why is the subjective credence that a coin is heads 1/2 in the first place?
PeroK said:if you have to give a numerical value to subjective credence (and that is the only possible value), then it must be 1/3. In other words, 1/3 cannot be wrong.
PeroK said:the halfers are using probability theory selectively (to get 1/2 in the first place), then denying it to get their subjective credence
PeterDonis said:Since you referenced this page, I'll go ahead and critique another aspect of the thirder argument as it's presented there:
"Suppose Sleeping Beauty is told and she comes to fully believe that the coin landed tails. By even a highly restricted principle of indifference, her credence that it is Monday should equal her credence that it is Tuesday since being in one situation would be subjectively indistinguishable from the other. In other words, P(Monday | Tails) = P(Tuesday | Tails), and thus
P(Tails and Tuesday) = P(Tails and Monday).
Consider now that Sleeping Beauty is told upon awakening and comes to fully believe that it is Monday. She knows the experimental procedure doesn't require the coin to actually be tossed until Tuesday morning, as the result only affects what happens after the Monday interview. Guided by the objective chance of heads landing being equal to the chance of tails landing, it should therefore hold that P(Tails | Monday) = P(Heads | Monday), and thus
P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday)."
This argument is based on supposing that Beauty has information that she doesn't actually have, and the information is different in the two cases. In other words, first it is argued that the following two conditional probabilities are equal: P(Monday|Tails) = P(Tuesday|Tails). In other words, if Beauty knows the coin came up tails, it is equally probable that she was awakened on either day.
Second, it is argued that the following two conditional probabilities are equal: P(Tails|Monday) = P(Heads|Monday). In other words, if Beauty knows that it is Monday, it is equally probable that the coin came up heads or tails.
Now that I've correctly stated the actual conditional probabilities, it should be obvious that the third stage of the argument is invalid, since it is arguing, in essence, that P(Monday|Tails) = P(Tails|Monday). But that is only true if we fill in the numerical values in the two conditional probabilities above (using the constraint that they must add up to 1), and hence observe that P(Monday|Tails) = P(Tails|Monday) = 1/2. But the argument purports to show that P(Tails and Monday), which is how it describes both of the conditional probabilities P(Monday|Tails) and P(Tails|Monday), is 1/3. This is obviously false.
Here's another argument based on conditional probabilities. First we need to find an expression for a conditional probability of Heads that is conditioned on the information Beauty actually has. This is P(Heads|Awakened), where "Awakened" means Beauty has just been awakened but doesn't know which day, Monday or Tuesday, she has been awakened in. Then the obvious way to proceed is to write:
P(Heads|Awakened) = P(Heads|Monday) P(Monday|Awakened) + P(Heads|Tuesday) P(Tuesday|Awakened)
Since the conditions of the experiment tell us that P(Heads|Tuesday) = 0, and we know from the above that P(Heads|Monday) = 1/2, we now have only to evaluate P(Monday|Awakened). I would expect a thirder to claim that P(Monday|Awakened) = 2/3, by arguing that there are three possible "awakenings", Monday & Heads, Monday & Tails, and Tuesday & Tails, and that these are all equally probable. But a halfer could argue that when Beauty is awakened, it isn't a random choice between those three alternatives. She gets awakened on Monday, and then the coin flip result is checked to see if she is awakened again on Tuesday. And then we get into all the arguments about relative frequencies and Bayesian priors and so on. But notice that, if we are arguing about that stuff, that means we have reinterpreted Beauty's "subjective credence that the coin came up heads" as the conditional probability P(Heads|Awakened), which, as far as I can tell from the Wikipedia page, nobody has even brought up.
PeterDonis said:I still don't see this at all. The conditional probability P(Heads|Flipped) is 1/2--everyone appears to agree on that. The halfers are simply saying that this conditional probability is the one that corresponds to the vague ordinary language "subjective credence that the coin landed heads".
stevendaryl said:I think the disagreement is about whether (and how) to update the a-priori probability of heads in light of new information. Certainly, if I told you that I flipped a second coin, and at least one of the two result was "heads", then your subjective credence that the first coin was heads would change. (I think it would change from 1/2 to 2/3). So usually, "subjective credence" includes whatever information is available.
stevendaryl said:The controversy is over whether Sleeping Beauty has any information that would allow (force?) her to update her estimate, from 1/2 before the experiment to 2/3 upon being awakened.
PeroK said:There is no reason not to use probability theory in this case.
I don't even understand the problem.Demystifier said:
Marana said:And I think we can agree that there are situations where you can't condition on things like "it is monday". For example, suppose you flip a coin while the clock says 2:00. "It is 2:00". While you compute the probability of heads you notice that the clock now says 2:01. "It is 2:01". Using conditioning, you notice that this is impossible (in conditioning something you know can't become false, nor can something you know is false become true) and give up. Or more likely, you don't give up because you intuitively know that you can't simply condition as normal with "it is 2:00" and "it is 2:01".
Boing3000 said:I don't even understand the problem.
The fair coin if flipped once. So 1/2 is the only answer possible.
A reasonable person don't wager.PeroK said:Credence is a statistical term that expresses how much a person believes that a proposition is true.[1] As an example, a reasonable person will believe with 50% credence that a fair coin will land on heads the next time it is flipped. If the prize for correctly predicting the coin flip is $100, then a reasonable person will wager $49 on heads, but they will not wager $51 on heads.
I agree, but then it is irrelevant to the question "What is your credence now for the proposition that the coin landed heads?".PeroK said:The probability that it is Monday is 2/3 and Tuesday 1/3.
I have no idea how Beauty or anyone else could do that calculation. "Waking up" have never changed anything to how you process the world, except maybe on the February 2 (arghh I lied, I am not serious anymore)PeroK said:With this information she can calculate that
The probability that the coin was Heads is 1/3 and that it was Tails is 2/3.
If the question was "how many time do you wake up", I would agree Beauty should answer 3/2.PeroK said:She could choose to ignore these calculations. But, again, by the definition of "credence" it is unreasonable to ignore information you have or to refuse to perform calculations on that data - for fear of losing a bet.
stevendaryl said:Well, that's the step where probability changes from frequency (which is just a matter of counting) to likelihood (which is a measure of confidence in the truth of something). To apply probability in our own lives, we are always in unique situations. But if you imagine that your situation is one instance of an ensemble of similar situations, then you can reason about likelihood. If you don't take such a step, then I don't see how probability is either useful or meaningful.
It is at issue in the most popular argument for 1/3. The argument goes: after conditioning on "it is monday", we must get 1/2. That is, P(monday and heads)/P(monday) = P(monday and tails)/P(monday) = 1/2. Therefore P(monday and heads) = P(monday and tails).stevendaryl said:That's an interesting problem--how to make sense of time-dependent statements such as "It is now 2:00". Later finding out that it is 2:01 doesn't contradict the truth of the earlier statement. However, I think you're barking up the wrong tree in relating it to the Sleeping Beauty problem--that's not really what's at issue.
Boing3000 said:A reasonable person don't wager.
Look, for once, I'll try to be dead serious. This problem is in the "positive trolling" category, exactly like the chicken and eggs. Those problems have perfectly fine and simple answer, until you add some "hidden" confusion based on tricky language where everyone feels untitled to plug their own meaning.
In sleeping beauty, the trolling start when you speak about days name. This is irrelevant because of the drug and the closed room.
So if the coin is fair, and the experiment is fair, Beauty does not even have to be put to sleep to answer the question. Actually the coins does not even have to be flipped before Tuesday. So asking the question on Monday is definitely some sort of lying if the experimenter choose to do so.I agree, but then it is irrelevant to the question "What is your credence now for the proposition that the coin landed heads?".I have no idea how Beauty or anyone else could do that calculation. "Waking up" have never changed anything to how you process the world, except maybe on the February 2 (arghh I lied, I am not serious anymore)If the question was "how many time do you wake up", I would agree Beauty should answer 3/2.
But that is not the question, isn't it ?