The Sleeping Beauty Problem: Any halfers here?

[Ken G]

What's more, the number of times she is interviewed on Monday is N, and the number of times she is interviewed on Tuesday is N/2, so clearly she should reckon the chances that it is Monday is 2/3. It seems to me that all the halfers get this calculation wrong.

 

[Stephen Tashi]

This way of putting it, the compound game is certainly not of the form that allows you to compute probability based on how much you would pay for the chance to play. The amount you have to bet and the amount you stand to win are not fixed.

I agree. If Beauty is rational, she realizes that her purchase of a bet on on particular awakening has consequences for any other awakening that happens in the experiment. The definition of credence I quoted assumes a bet where the payoff from the event E is 1 unit of utility.and an buying the bet at price X doesn't affect the payoff.

A web search this morning turns up other people, presumably more experienced in studying credence than me, who point out this complication of the Sleeping Beauty problem. For example:

https://philpapers.org/archive/YAMLSB

 

[stevendaryl]

I agree. If Beauty is rational, she realizes that her purchase of a bet on on particular awakening has consequences for any other awakening that happens in the experiment.

Her bet doesn't have consequences beyond the particular awakening. She's not forced to bet the same way every time she is awakened, it's just that it's in her best interest to do so. I don't see how that's any different from any other type of bet: if there's a unique best way to bet, then you'll bet the same way every time you're in that circumstance. That doesn't mean that one of your bets forces you to make the other bets in any particular way.

The thing that's odd about SB is that the coin toss has consequences beyond being the event that she's betting on.

 

[Stephen Tashi]

Her bet doesn't have consequences beyond the particular awakening. She's not forced to bet the same way every time she is awakened, it's just that it's in her best interest to do so.

Isn't she forced to price the bet the same way every time if the problem stipulates that Sleeping Beauty is rational? Can we beat consistency with some sort of "mixed" pricing strategy where Sleeping Beauty prices the bet by tossing a die or a fair coin of her own?

 

[Ken G]

To all "halfies," I pose this question: what odds should SB take, when awakened, that today is Monday, and what odds should she take that today is Tuesday?

 

[Stephen Tashi]

What's more, the number of times she is interviewed on Monday is N, and the number of times she is interviewed on Tuesday is N/2, so clearly she should reckon the chances that it is Monday is 2/3. It seems to me that all the halfers get this calculation wrong.

In this thread, I think some "thirders" and "halfer" claim calculate credence by first calculating P(coin laded heads | SB is awakened) objectively, using only information in the problem. From that objectively calculated probability, they obtain SB credence since a rational SB does the same calculation.

Other "thirders" and "halfers" arrive at SB's credence by asserting Sleeping Beauty must reason in certain way and add her own subjective probabilities (via the Principle of Indifference) to the given information.

The following remarks only concern the question of whether P(coin landed heads| Sleeping Beauty is awakened) can be objectively calculated. The value denoted by "P(coin landed heads | Sleeping Beauty is awakened:" is not a well defined probability unless there is some stochastic process implicitly or explicitly specified in the given information that tells us how to stochastically pick the situation in the experiment when she is awakened. One natural way to specify such a process is run the experiment a lot of times and then randomly select a situation where she was awakened from the records of the experiments, giving each an equal probability of selected. Using that method, it is correct that there are (probably) about twice as many instances where she was awakened when the coin landed tails as when the coin landed heads. That method supports the "thirder" answer. However, my (perhaps very technical objection) is that the information in the problem does not specify any method for picking what situation occurs when Sleeping Beauty is awakened. One rebuttal to my objection is everybody knows that you are supposed assume that "thirder" sort of method is what is used. This becomes a debate about conventions of interpreting probability word-problems. Another rebuttal to my objection is that a rational Sleeping Beauty would assume that the "thirder" type of method is used. That rebuttal is not relevant to computing the objective value of P(coin landed heads | Sleeping Beauty) is awakened. Instead, it Is relevant to the different question of how P(coin landed heads | Sleeping Beauty is awakened). is calculated subjectively.

In regard to the interpretation of word problems, the information does tell us that we must consider "whenever" Sleeping Beauty is awakened. So a stochastic process that selects the situation that applies when "Sleeping Beauty is awakened" cannot leave out any of the situations. However, we aren't required by the given information to pick the situation that applies from any particular distribution. From a Bayesian point of view, it is natural to assume a prior distribution where each situation has an equal probability of being chosen.

 

[Ken G]

However, my (perhaps very technical objection) is that the information in the problem does not specify any method for picking what situation occurs when Sleeping Beauty is awakened.

Thanks for explaining further, but I don't understand what you mean by a "method for picking". SB is in a real situation, and she is really offered odds to make various bets. I think the issue of the coin toss is causing some confusion, so that's why I asked a different question-- what odds should she take that it is Monday? Regardless of what reasoning you use, this is a perfectly straightforward question, since she could really be offered various odds, and she could actually make or lose money in the long run by accepting or rejecting those odds. So what would you say? What odds should she take that it is Monday? You are allowed to assume any "method for picking" that you like, as long as the experimenters actually flip a coin, actually waken SB, actually apply the amnesia elixir, and actually offer the odds in question that it is Monday. I can't see any method you could imagine where those odds are not simply 2/3, in the sense that SB will certainly make money in the long run by accepting any odds higher than 3-to-2 in favor, and receiving payoff if it is indeed Monday. So we need to start from a place where we can agree on this, before we even consider the coin toss.

You see, as soon as one raises the issue of the odds of what day it is, the halfer response is immediately refuted, because if it is Monday, the coin toss has a 50-50 chance, and if it is Tuesday, the coin toss was tails. Ergo, the only way you can be a "halfer" is if you think you know it can't be Tuesday, or else your reasoning is inconsistent.

 

[Buzz Bloom]

I am a halfer.

I now realize I made an error, and am now a thirder. My error was a misreading of the problem statement to be saying if tails occurs, then a choice is made between a Monday or a Tuesday interview. For N flips of the coin this leads to the expected event distribution:

Heads & Awake Monday - Expected to occur N/2 times
Heads & Awake Tuesday - Expected to occur zero times
Tails & Awake Monday - Expected to occur N/4 times
Tails & Awake Tuesday - Expected to occur N/4 times​

I wonder if all halfers make this mistake.

The correct interpretation is that tails leads to both a Monday and Tuesday Interview. For N flips of the coin this leads to the expected event distribution:

Heads & Awake Monday - Expected to occur N/2 times
Heads & Awake Tuesday - Expected to occur zero times
Tails & Awake Monday - Expected to occur N/2 times
Tails & Awake Tuesday - Expected to occur N/2 times​

Thus interviews with heads occurs only 1/3 of the time.

 

[PeterDonis]

This is getting frustrating.

I can understand that reaction, and I don't want to rehash everything that's already been said, but I think I observed before that IMO the Sleeping Beauty problem is not a good illustration of the concept of "credence", and @Stephen Tashi is referring to one reason why: the amount that Beauty will end up wagering on the coin flip depends on the result of the coin flip (because she ends up wagering once if the coin is heads, but twice if the coin is tails, and each wager is identical because she has the same information each time). So the odds that she is willing to accept on heads are affected, not just by the odds that the coin will turn up heads, but by the weird structure of the experiment that makes the payoffs skewed. In your formulation, the skew is the ratio P(A|H) / P(A|T) = 1/2.

I understand that, in the references you have given, the concept of "credence" is defined in such a way that you can indeed argue that this weird skewed structure still leads to a valid answer of 1/3 to the question "what is your credence now that the coin turned up heads?" But I am also sympathetic to reactions like that of @Stephen Tashi (since I have a similar reaction), which are basically along the lines of: you're setting up a weird, skewed experiment and then trying to claim that the answer 1/3 is just "the credence that the coin came up heads" instead of "the credence that the coin came up heads given that you've skewed the payoffs". Yes, if you actually put me into this experiment, and you explained to me exactly what you meant by "credence that the coin came up heads", I would have to insist on 2:1 odds on heads (i.e., P(Heads) = 1/3). But I would still feel like you were abusing language by phrasing the question that way.

 

[PeterDonis]

The value denoted by "P(coin landed heads | Sleeping Beauty is awakened:" is not a well defined probability unless there is some stochastic process implicitly or explicitly specified in the given information that tells us how to stochastically pick the situation in the experiment when she is awakened.

As I just posted in response to @Dale , I understand this reaction and where it is coming from. However, I would point out, in defense of his position, that there is no "stochastic" process involved at all in the experiment, except for the initial coin flip itself. Once the coin flip is made, everything is perfectly deterministic. The thing that makes the situation weird, and gives rise to intuitive objections to accepting 1/3 as "the credence that the coin came up heads", is, as I said in my post just now, that the structure of the experiment skews the total payoffs by making the total amount wagered dependent on the results of the coin flip--if the coin comes up heads, Beauty only ends up wagering once, while if it comes up tails, she ends up wagering twice, and each wager must be identical since she has the same information each time (because of the amnesia drug). This is not a "stochastic" result; it's just an odd but deterministic feature of the experimental setup.

So, as I've said before and repeated in my last post, if I were actually put into this experiment, and it were explained to me exactly what the experimenter meant by "credence that the coin came up heads" (which is the odds I will accept on a wager that the coin came up heads, given that the number of times I will end up making that wager depends on the result of the coin flip as just described), I would have to insist on 2:1 odds on heads, to make up for the skewed payoff structure. I might still object to using the language "credence that the coin came up heads" to describe this result, but that's a question of words, not results: once I know what the experimenter means by that (even if I would not use the same ordinary language to convey that meaning), I know how I have to bet.

So my question to you would be, if you were put into the experiment under the same conditions, with the same knowledge of what the experimenter meant by "credence that the coin came up heads" (even if you didn't agree with his choice of words), how would you bet?

 

[Buzz Bloom]

she ends up wagering once if the coin is heads, but twice if the coin is tails

Hi Peter:

I have not read all 500 posts in this thread, and I have missed any discussion of wagering. I do not understand how wagering is part of the problem. I do not see the connection between wagering and credence. Can you explain this?

Regards,
Buzz

 

[Stephen Tashi]

Thanks for explaining further, but I don't understand what you mean by a "method for picking". SB is in a real situation, and she is really offered odds to make various bets.

I agree that the question posed n the Sleeping Beauty Problem concerns "credence" and this leads to questions about a betting strategy.

The question of whether P(heads | SB awakened) can be objectively calculated is a side issue, but it's one that comes up again and again. I'm saying that calculating P(heads| SB) awakened from the information in the problem objectively requires knowing (or deducing) a particular probabiiity distribution F on the situations { (heads, Monday, awakened) , (tails, Monday, awakened), (tails, Tuesday, awakened)}. The distribution F defines the "method of picking" the situation when SB is awakened. The distribution F implies a particular value for P(heads | SB is awakened) by using objective calculations for conditional probabilities.

A "thirder" choice for F is to assign each situation an equal probability. That is information not given in the problem, although it is a plausible Bayesian assumption.

Suppose the process for picking the situation when SB is awakened is to run the experiment a large number of times and select a situation when SB is awakened at random from the situations that happened in the experiments, giving each such situation an equal probability of being selected. This method does imply F is a uniform distribution on the situations. However, the information in the problem does not specify this particular method for picking the situation.

When we get into betting strategies, it is relevant to run the experiment a large number of times and consider how each situation that arises in the experiment affects the strategy. We have to do this because the problem says that the betting occurs each time Sleeping Beauty is awakened. So calculating the result of a strategy by using probability theory must use probabilities that are consistent with the thought that each situation that arose in a large number of experiment is considered exactly once .

 

[PeterDonis]

I do not see the connection between wagering and credence.

See this Wikipedia article (which is linked to from the one on the Sleeping Beauty problem that is linked to in the OP of this thread):

https://en.wikipedia.org/wiki/Credence_(statistics)

 

[Buzz Bloom]

See this Wikipedia article (which is linked to from the one on the Sleeping Beauty problem that is linked to in the OP of this thread):

Hi Peter:

I get the relationship between "credence" and "probability", but I do not get any relationship to wagering from the puzzle statement. I do nopt see from the puzzle statement that Sleeping Beauty is to make a wager.

Regards,
Buzz

 

[PeterDonis]

I get the relationship between "credence" and "probability",

Did you read the Wikipedia article? It describes the relationship between credence and wagering.

I do nopt see from the puzzle statement that Sleeping Beauty is to make a wager.

The puzzle statement does use the term "credence", which, according to at least one common definition of that term, implies a wager, as described in the Wikipedia article on credence that I linked to. This has been discussed at great length in this thread, so I'm afraid you'll have to read the 500 posts if what's in the Wikipedia articles isn't enough.

 

[Buzz Bloom]

Did you read the Wikipedia article? It describes the relationship between credence and wagering.

The puzzle statement does use the term "credence", which, according to at least one common definition of that term, implies a wager

Hi Peter:

Yes I did, and the discussion there is about ways to think about solving the puzzle, not about anything inherent in the problem statement itself. As I interpret the discussion, it was that SB's credence could be determined by the limits she used to decide what bets she would decide are profitable. Thus (if she were a thirder) she would accept a bet paying her odds of 2:1 + a small amount more, but not 2:1 + a small amount less. The determination of a boundary between accept and not could be determined by the interview without any bets actually being made. Thus making a wager is not implied, but wagers might be discussed.

Regards,
Buzz

 

[PeterDonis]

SB's credence could be determined by the limits she used to decide what bets she would decide are profitable

Yes, and that is how the term "wager" is being used in this discussion. You can think of it as actual wagers, or as hypothetical ones; it doesn't matter. The question of whether 1/3 or 1/2 is the correct credence is the same either way.

 

[bahamagreen]

Do both halfers and thirders agree that she would answer...?

What is the chance that the coin landed heads?
"One half"

What is the chance that today is Monday?
"Two thirds"

What is the chance that the coin landed heads and that today is Monday?
"One third"

If betting is included as part of the experiment and SB's rationale, what is her credence Wednesday when she is debriefed and informed, "Oh, you didn't win any money; of course you can't recall anything about it but you were incorrect".

 

[Stephen Tashi]

I'm coming into this late, but it seems very clear to me the correct answer is 1/3, on the simple grounds that Sleeping Beauty knows that if she guesses "the coin landed heads" every time she is awakened, and if the experiment is repeated every week for a year, then she will clearly have been correct 1/3 of the time, and none of those events will seem any different to her. So that's 1/3 credence, and I can see no other meaningful way to define the concept of "credence."

1/3 is the correct answer to some questions - but which questions?

The answer 1/3 is a correct answer to the question: "In large number of repetitions of Sleeping Beauty experiment, what is the expected ratio of ( The number of situations that arose when Sleeping Beauty was awakened when the coin landed heads) to the (Total number of situations that arose in the experiment)?

The answer (1/3)$ is also correct answer to the question: When awakened, what is a fair price for Sleeping Beauty to pay for an agreement that she must always pay that price when awakened and only gets $1 on those awakenings where the coin landed heads.

The controversy (in some minds) is whether Sleeping Beauty should interpret the question "What is your credence that the coin landed heads" literally. If she takes a literal interpretation, she attempts to apply the definition of credence given by https://plato.stanford.edu/entries/probability-interpret/#SubPro

This boils down to the following analysis:

Your degree of belief in E is p iff p units of utility is the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E.

She must define a price for the bet "you get $1 if the coin landed heads" , not for the agreement "you pay the price every time you are awakened and you get $1 if the coin landed heads.".

The assumption that Sleeping Beauty is rational compels her to pay the same price for a bet each time she awakens because she can't tell one awakening from another. Thus if she pays X for a bet on Monday and the coin landed tails, she will (i.e must) pay X for the same bet when it is offered on Tuesday. Sleeping Beauty knows the way the experiment is conducted so she knows of such a possibility.

What Sleeping Beauty needs is a bet on "The coin landed heads" that has a simple payoff of $1 instead of additional consequences.

 

[.Scott]

But there hasn't been a satisfactory account of why the probability of heads changes.

The numbers work out the same as in Sleeping Beauty. But in this case, the fact that I am picked is additional information that changes the conditional probability of heads. In the Sleeping Beauty case, the fact that she is asked the probability upon waking is no additional information, since it was a certainty that that would happen.

The problem is not that she has additional information upon being awakened, it is that she has less. Or perhaps it is better to explain it as having different information about the coin toss.

Before being put to sleep the first time, she knows it's 50/50. And she knows that she will be awakened once or twice. But when she is awakened, she doesn't know which day it is and she doesn't know if this is one of one or one of two.

Clearly, if she makes a wager each Monday or Tuesday when she is awoken, she should use the 33.3:66.6 odds, not the 50:50.

In general, there is a 50:50 shot that the flip of a coin will be heads or tails. But once you add information about the outcome of the coin toss, the odds change. Credence is determined by what you know and don't know. So as that changes, so does the credence.

It's the same for those conducting the experiment. At the start of the experiment, they would say there is a 50:50 shot that it was heads. But once they flip the coin, it changes to 100:0 one way or the other.

 

[Dale]

I am a halfer.

Then every time you wake I will sell you the following bet for $0.40: a bet that pays $1 if the coin landed heads, $0 if the coin did not land heads. How much money do you expect to make?

 

[Ken G]

I'm saying that calculating P(heads| SB) awakened from the information in the problem objectively requires knowing (or deducing) a particular probabiiity distribution F on the situations { (heads, Monday, awakened) , (tails, Monday, awakened), (tails, Tuesday, awakened)}. The distribution F defines the "method of picking" the situation when SB is awakened. The distribution F implies a particular value for P(heads | SB is awakened) by using objective calculations for conditional probabilities.

Well that is certainly true, but surely the question asserts a simple procedure that determines F. The experimenters flip a coin, and if they get heads, they wake up SB only on Monday. If tails, they wake her Monday and Tuesday. Any time they wake her, they allow her to bet at some given odds that the day is Monday. They also apply amnesia elixir so she cannot remember if she has been wakened before. That's it, that's all you need-- with that situation, is it not perfectly clear she will make money, after many identically repeated experiments, if she accepts any odds more favorable to her than a 2/3 chance it is Monday, and will lose for any odds less favorable than that?

Now of course, the experimenters can cheat in any way they like. They can simply lie to her. But if the setup is what is given here, and the experimenters follow it, then there is no question how she makes money. We can do the same thing with repeated bets on "heads" or "tails," but I'm trying to make sure the two problems are correctly coupled. In other words, no one can self-consistently claim that the odds of "heads" are 50-50, without also claiming it is definitely Monday, because it should be clear that only on Mondays are those the correct odds of a "heads," given nothing but the setup I just described.

 

[Buzz Bloom]

How much money do you expect to make?

Where are you getting that from?

Hi Dale and steven:

I had a change of heart and am now a thirder. Please see my post # 498.

Regards,
Buzz

 

[Stephen Tashi]

Well that is certainly true, but surely the question asserts a simple procedure that determines F. The experimenters flip a coin, and if they get heads, they wake up SB only on Monday. If tails, they wake her Monday and Tuesday.

That correctly describes the situations can possibly be selected. However, it doesn't describe a probability distribution F telling how to determining which situation applies "when Sleeping Beauty is awakened" - unless you assume that each possible situation has an equal probability of being the one that applies. I agree that a typical Bayesian approach is to make such an assumption.

A "halfer" method of picking which situation applies when Sleeping Beauty is awakened is: Flip the coin and run the experiment. From those situations that arise the experiment after the coin flip, pick one of the situations at random, giving each situation an equal probability of being selected.

"Thirders" have various objections to the "halfer" method, such as "You aren't forcing each situation in an experiment to be selected when coin lands Tails. You might pick Monday and omit Tuesday". However, any random method of picking a single situation might omit one of the situations in a series of 3 independent selections.

 

[Stephen Tashi]

Hi Dale and steven:

I had a change of heart and am now a thirder. Please see my post # 498.

The "thirders" have a valid position if certain plausible assumptions are made.

 

[Ken G]

That correctly describes the situations can possibly be selected. However, it doesn't describe a probability distribution F telling how to determining which situation applies "when Sleeping Beauty is awakened" - unless you assume that each possible situation has an equal probability of being the one that applies.

The only probability distribution involved in waking her up is described by the coin flip, there is no ambiguity, no assumptions-- it's all spelled out in the instructions.

A "halfer" method of picking which situation applies when Sleeping Beauty is awakened is: Flip the coin and run the experiment. From those situations that arise the experiment after the coin flip, pick one of the situations at random, giving each situation an equal probability of being selected.

Each situation does have equal probability, that's what gives the thirder result. There are three situations, correct?

 

[Dale]

I observed before that IMO the Sleeping Beauty problem is not a good illustration of the concept of "credence",

I agree. It is not a good illustration. You have to apply the definition of credence in this problem, but I don't think many people get the "now I get it" feeling from the problem.

 

[Dale]

The definition of credence I quoted assumes a bet where the payoff from the event E is 1 unit of utility.and an buying the bet at price X doesn't affect the payoff.

That is correct. That is the definition of credence and that bet is the one that the halfer solution gets wrong. So do you now understand that 1/2 is not a valid solution to the problem?

 

[andrewkirk]

On my analysis, the Thirder argument has the flaw that it assumes each of the two wakings in the Tails situation have the same probability as the waking in the Heads situation. That is not the case.

The probability of being woken on Monday in the Heads situation is just the probability of Heads, which is 0.5.

The probability of being woken on Monday in the Tails situation is 1/2 the probability of Tails, which is 0.5 times 0.5 = 0.25. The same goes for Tuesday-Tails.

So we have three potential wakings, with probabilities 0.5, 0.25, 0.25. For the latter two, the coin is Tails, for the first one, it is Heads. When SB is woken, she just uses that information to work out the probability that the coin was Heads, which is 0.5.

The error in the Thirder position is to assume that the three probabilities are the same. There is no mathematical rule to justify that.

I can see why Nick Bostrom argues for the Thirder position, because he uses the same logic to argue that, if simulation sophisticated enough to produce consciousness in the simulants is possible, then we are almost certainly in a simulation now, because the simulators will create very many simulations, all equally likely to each other and to the possibility that we are not in a simulation. He omits from that calculation the probability of such technology being developed, just as the Thirder solution omits the probability of Tails.

We can use the same Bostrom argument against the Thirder position by noting that there are at least a thousand different sub-scenarios for the Heads-Monday option, being differentiated by which temperature band of the thousand bands of width one-millionth of a degree, centred around standard body temperature, SB's body temperature on waking in the Heads case falls into. We now have one thousand equally likely scenarios for the Heads case and only two for the Tails case (we don't do temperature binning for that case), so the probability of Heads is 1000/1002, a virtual certainty.

PS: I've only read the first page of this thread.

PPS: Strictly speaking, a probability is simply a measure on a set of possible outcomes where the measure of the entire set is 1. Subject to that restriction, we are free to define the measure however we wish, which means assigning any probabilities we wish, as long as they don't contradict the criteria for a valid measure.

So there's no such thing as a 'correct' probability.

 

[Ken G]

On my analysis, the Thirder argument has the flaw that it assumes each of the two wakings in the Tails situation have the same probability as the waking in the Heads situation. That is not the case.

It suffices to count the events after the experiment is repeated N>>1 times. There will be N wakings on Monday and N/2 wakings on Tuesday. SB knows only, in each waking, that she samples equally from both those sets, so she samples in each instance from 3N/2 wakenings, and on N of them, it is Monday. This suffices to tell her that there is a 2/3 chance it is Monday, or if you prefer, she will break even in the long run by taking 3 to 2 odds. Do you say that is not the case?

So we have three potential wakings, with probabilities 0.5, 0.25, 0.25.

So you are now claiming that there are the same number of wakings on Monday as on Tuesday? You appear to be arguing that SB should think there is a 50% chance it is Monday. Of course that's not true, as can be seen if we do it for 99 days, not just 2. On heads, we awake her only on day 1, on tails, we awake her every day for 99 days. Do you say she should assess a 50% chance that it is Monday, based on the argument you just gave?

 

[Stephen Tashi]

An answer can always be changed by doing something different.

You assume that there is something there that we can be different from.

You could change the odds of poker by assuming every hand has equal probability, but that's no way to win poker.

I agree that many books pose poker problems and omit to explicitly state that all deals are to be regarded as equally probable. A traditional interpretation is for the reader to assume the author of the problem means that to be the case. However, it's not traditional (except perhaps among ultra-Bayesians) that the existence of N possibilities in an arbitrary probability word problem can be taken to imply that each possibility has probability 1/N.

As I said before a person can take the position that the author of the Sleeping Beauty problem intended to say (or for us to assume) that the situation "When Sleeping Beauty is awakened" has an equal probability of being any of the 3 possible situations. Or a person can take the position that we are permitted to use a Bayesian approach and explicitly assume each situation has as equal probability.

To design an empirical test to decide whether the "halfer" or "thirder" or some other answer is correct, the problem gives enough information to simulate many runs of the experiment. In those runs, the events C =(tails, Monday, awake) and D = (tails, Tuesday, awake) are not mutually exclusive events. In fact P(C|D) = 1. Both events occur in the experiment when the coin lands tails.

However, when we contemplate how to implement a distribution F to simulate the event "Sleeping Beauty is awakened", we must have a procedure that stochastically selects a single situation when she is awakened. In implementing that procedure A and B are mutually exclusive events. This makes it clear that simulating the situation when Sleeping Beauty is awakened does not treat events in the same way as the probability space for the experiment, where A an B are not mutually exclusive events.

I think you take it for granted that the way to to implement F would be to pick a situation at random from those that occurred in the runs of the trials, giving each the same probability. Or we could step through each situation that did occur and compute some frequencies, which , I think, amounts to the same assumption. I agree these are reasonable approaches. However, they aren't specified in the statement of the problem.

 

[Ken G]

You assume that there is something there that we can be different from.

All halfers should answer this question. If the experiment is done for 99 days, and on heads, there is only an awakening on Monday, and on tails, there are 99 days of awakening, what should SB assess as her expectation that it is Monday? Can you possibly think there is any reasonable interpretation of that scenario where she does not expect it is vastly more likely that the day is not Monday? Yet if the day is not Monday, then the toss was a tails.

 

[andrewkirk]

SB knows only, in each waking, that she samples equally from both those sets, This suffices to tell her that there is a 2/3 chance it is Monday. Do you say that is not the case?

I would say there is a 0.75=0.5+0.25 chance it is Monday, being the sum of the probs for Heads-Monday and Tails-Monday.

If Tails leads to 99 extra awakenings rather than one, then the prob of Heads-Monday is 0.5 and that of Tails & <day =n>, where 1<=n<=100, is 0.5/100=0.005. So in that case the probability that it is Monday is 0.5 + 0.005= 0.505.

 

[Stephen Tashi]

So you are now claiming that there are the same number of wakings on Monday as on Tuesday?

I think you are claiming that "numbers of things" has a specific relation to probabilities, which it does when each thing is equally probable. If the 3 possible situations A = (heads, Monday, awake), B = (tails, Monday, awake), C = (tails, Tuesday, awake) have the same probability then you have a point. But must they have the same probability?

 

[Stephen Tashi]

All halfers should answer this question. If the experiment is done for 99 days, and on heads, there is only an awakening on Monday, and on tails, there are 99 days of awakening, what should SB assess as her expectation that it is Monday? .

I think the answer is ( (1/2)(1) + (1/2)(1/99) , which may offend people's intuition, but it doesn't contradict any information given in the problem.

Let me make it clear that I'm not a "halfer". My vote in the poll was for:
It depends on the precise formulation of the problem