# The Sleeping Beauty paradox **URL:** \ **Category:** Great Debates\ **Created:** [April 12, 2009, 8:50pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882 "2009-04-12T20:50:53Z")\ **Posts on this page:** 20\ **Page:** 9
**Author:** ![Tyrrell\_McAllister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tyrrell_mcallister/32/16772_2.png) [@Tyrrell\_McAllister](https://boards.straightdope.com/u/Tyrrell_McAllister)\ **Post date:** [April 21, 2009, 10:39pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/161 "2009-04-21T22:39:28Z")
> [@begbert2](#): > > Well, not to twist the knife, but my criticism was that you were erroneously assuming that SB knew the truth value of “C appears in the experiment” when in fact that can’t be extrapolated from “I’m seeing C on the calendar”. I might have missed your point here. In fact, SB _does_ know the truth value of “C appears in the experiment”, because C is, by definition, the symbol that appears on the calendar on the day in which she is engaged in this reasoning. So, _for that specific symbol C_, she knows for a fact that it appears during the experiment. I would have agreed with you that “that can’t be extrapolated from ‘I’m seeing C on the calendar’”, but only for the technical reason that the statement “I’m seeing C on the calendar” couldn’t even be given meaning, much less used as a premise for further deductions. ---
**Author:** ![begbert2](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/begbert2/32/3357_2.png) [@begbert2](https://boards.straightdope.com/u/begbert2)\ **Post date:** [April 21, 2009, 10:42pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/162 "2009-04-21T22:42:59Z")
> [@Tyrrell\_McAllister](#): > > I might have missed your point here. In fact, SB _does_ know the truth value of “C appears in the experiment”, because C is, by definition, the symbol that appears on the calendar on the day in which she is engaged in this reasoning. So, _for that specific symbol C_, she does know whether it appears during the experiment. > > I would have agreed with you that “that can’t be extrapolated from ‘I’m seeing C on the calendar’”, but only for the technical reason that the statement “I’m seeing C on the calendar” couldn’t even be given meaning, much less used as a premise for further deductions. My point was that it’s not good enough to “know whether it appears during the experiment.” She has to know whether _ **or not** _ it appears during the experiment, to calculate other odds based on the truth or falsehood of that fact. Because if she can’t accurately detect _all_ cases where the statement is true, she can’t calculate the odds using math that’s based on \*all \*the cases where it’s true. ---
**Author:** ![Tyrrell\_McAllister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tyrrell_mcallister/32/16772_2.png) [@Tyrrell\_McAllister](https://boards.straightdope.com/u/Tyrrell_McAllister)\ **Post date:** [April 21, 2009, 10:45pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/163 "2009-04-21T22:45:32Z")
Just to be clear, I’m not using C to represent a variable. It is the _literal_ symbol that appears on the calendar on the day that she is doing the reasoning that we are discussing. That is, the display literally shows a semi-circular arc that is open on the right-hand side. On another day of the experiment (if there is another day), she’ll reason similarly, but with a different symbol. Of course, the probabilities will work out the same. ---
**Author:** ![Tyrrell\_McAllister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tyrrell_mcallister/32/16772_2.png) [@Tyrrell\_McAllister](https://boards.straightdope.com/u/Tyrrell_McAllister)\ **Post date:** [April 21, 2009, 10:46pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/164 "2009-04-21T22:46:51Z")
> [@begbert2](#): > > My point was that it’s not good enough to “know whether it appears during the experiment.” She has to know whether _ **or not** _ it appears during the experiment, to calculate other odds based on the truth or falsehood of that fact. On the day that she does the reasoning that we’re talking about, she knows _for a fact that_ C appears during the experiment. ETA: I clarified this in the edited version of the post to which you were replying. ---
**Author:** ![Tyrrell\_McAllister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tyrrell_mcallister/32/16772_2.png) [@Tyrrell\_McAllister](https://boards.straightdope.com/u/Tyrrell_McAllister)\ **Post date:** [April 21, 2009, 10:49pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/165 "2009-04-21T22:49:58Z")
accidental post ---
**Author:** ![begbert2](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/begbert2/32/3357_2.png) [@begbert2](https://boards.straightdope.com/u/begbert2)\ **Post date:** [April 21, 2009, 11:02pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/166 "2009-04-21T23:02:36Z")
> [@Tyrrell\_McAllister](#): > > On the day that she does the reasoning that we’re talking about, she knows _for a fact that_ C appears during the experiment. > > ETA: I clarified this in the edited version of the post to which you were replying. Whereas the thirder argument you presented assumes that she will know it _regardless_ of whether it appears in the experiment on the day in question or not, in order to arrive at a certainty level of 1/3. Because if you look at the _actual_ cases, across the cases that any given symbol appears on the day in question, enabling her to know about that particular symbol, the odds of heads are 1/2. Only by explicitly including cases where she will not know that that specific symbol is in the experiment will you arrive at 1/3. Claiming that she knows about it becuase she can see it, and then including cases where she won’t know about it because she’s seeing the other symbol and claiming they work too because the other symbol is now in play, is sort of the mathematical equivalent of moving the goalposts - except when you move them to cover symbol B, you uncover the cases covered by symbol C. ---
**Author:** ![Tyrrell\_McAllister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tyrrell_mcallister/32/16772_2.png) [@Tyrrell\_McAllister](https://boards.straightdope.com/u/Tyrrell_McAllister)\ **Post date:** [April 21, 2009, 11:11pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/167 "2009-04-21T23:11:04Z")
> [@](#): > > Whereas the thirder argument you presented assumes that she will know it regardless of whether it appears in the experiment on the day in question or not, in order to arrive at a certainty level of 1/3. Because if you look at the actual cases, across the cases that any given symbol appears on the day in question, enabling her to know about that particular symbol, the odds of heads are 1/2. Only by explicitly including cases where she will not know that that specific symbol symbol is in the experiment will you arrive at 1/3. That’s just not true. Look again at the computation in [post #140](http://boards.straightdope.com/sdmb/showpost.php?p=11045708&postcount=140). Remember that C is literally the symbol that appears on the calendar on the day that she is carrying out that computation. She uses only the fact that C appears during the experiment. There is no assumption that any symbol appears during the experiment other than the one symbol about which she has direct knowledge. ---
**Author:** ![begbert2](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/begbert2/32/3357_2.png) [@begbert2](https://boards.straightdope.com/u/begbert2)\ **Post date:** [April 21, 2009, 11:30pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/168 "2009-04-21T23:30:37Z")
> [@Tyrrell\_McAllister](#): > > That’s just not true. Look again at the computation in [post #140](http://boards.straightdope.com/sdmb/showpost.php?p=11045708&postcount=140). Remember that C is literally the symbol that appears on the calendar on the day that she is carrying out that computation. She uses only the fact that C appears during the experiment. There is no assumption that any symbol appears during the experiment other than the one symbol about which she has direct knowledge. Contrariwise, look at the list of actual possibilities and odds I gave in [post 142](http://boards.straightdope.com/sdmb/showpost.php?p=11045946&postcount=142) as a rebuttal to 140. It is quite clear that if you look at only the cases where she _actually_ sees the literal symbol C, the odds of heads are 1/2 across them. Your post 140 explanation does explicitly map also to cases where C is in the experiment and SB does not know it, and requires their inclusion to reach the thirder conclusion. Put another way - your math is equavalent to the math for “B is _not_ in the experiment” - to the point that it could basically be assumed as being synonymous to “C is in the experiment” as far as the math is concerned. But SB can _never_ know that B is not in the experiment. This should be a clue that your math assumes a more complete knowledge of the situation than is actually available. ---
**Author:** ![Tyrrell\_McAllister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tyrrell_mcallister/32/16772_2.png) [@Tyrrell\_McAllister](https://boards.straightdope.com/u/Tyrrell_McAllister)\ **Post date:** [April 22, 2009, 12:00am UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/169 "2009-04-22T00:00:10Z")
We’re having a failure to communicate here, but I don’t know what the problem is. Could you point to the first sentence in my post 140 that you think doesn’t follow from what precedes it? I’ll proceed with a line-by-line response to your last post, in the hopes that it will make my position clear. > [@begbert2](#): > > Contrariwise, look at the list of actual possibilities and odds I gave in [post 142](http://boards.straightdope.com/sdmb/showpost.php?p=11045946&postcount=142) as a rebuttal to 140. It is quite clear that if you look at only the cases where she _actually_ sees the literal symbol C, the odds of heads are 1/2 across them. I’m afraid that it’s not “quite clear” to me. You appear to me to be begging the question. > [@](#): > > Your post 140 explanation does explicitly map also to cases where C is in the experiment and SB does not know it, and requires their inclusion to reach the thirder conclusion. No, it doesn’t, because when she doesn’t know that C is in the experiment, she won’t be computing p(coin lands heads | C appears during the experiment) to determine her credence that the coin landed heads. Instead, she’ll computing p(coin lands heads | B appears during the experiment) (supposing that B is the literal symbol that appears on _that_ day). But the probability will again come out to be 1/3. > [@](#): > > Put another way - your math is equavalent to the math for “B is _not_ in the experiment” - to the point that it could basically be assumed as being synonymous to “C is in the experiment” as far as the math is concerned. No, it’s not. If it were, you would be able to derive p(B appears during the experiment | X) = 0 from my equations, where X is some knowledge that she has at the time that she is performing this computation. But you will find that you won’t be able to do this. If you believe that you can, please show the explicit manipulations of the equations that give you this result. > [@](#): > > But SB can _never_ know that B is not in the experiment. This should be a clue that your math assumes a more complete knowledge of the situation than is actually available. It would be a clue if my math made that assumption. But it doesn’t. If anything, my math assumes _less_ knowledge than it should, because I was disallowing the statement “C appears today”, which is intuitively known to SB and stronger than merely "C appears during the experiment. I was disallowing the stronger statement because I believed that it couldn’t be given meaning at all. I now consider that to be an unacceptable dodge, but I don’t yet know how to take the matter head on. ---
**Author:** ![VarlosZ](https://avatars.discourse-cdn.com/v4/letter/v/d78d45/32.png) [@VarlosZ](https://boards.straightdope.com/u/VarlosZ)\ **Post date:** [April 22, 2009, 7:31am UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/170 "2009-04-22T07:31:07Z")
> [@begbert2](#): > > Scenario 2: the two games are on different tapes; you put one of those tapes in at random (perhaps based on a coin flip), and set aside the other. In that case, if the tape is the Game 1 tape, all the touchdowns will be game 1 touchdowns; if the tape is the Game 2 tape, then all the touchdowns will be game 2 touchdowns. It doesn’t matter how many times you awaken in either case. > > The thing to note about scenario 2 is that it doesn’t matter how many touchdowns there are per tape; it still doesn’t change which tape is in the machine, or the odds that that tape was the one you put in the machine. That was determined in advance, and nothing you do afterward will change it. > > SB is in scenario 2. Agree that scenario 2 is a better analogy for the SB situation. Unfortunately, given that, it leads us right back to the same disagreement: the chance that he put in the low-scoring game is 50%, but each particular awakening is probably the result of the high-scoring game, which can be easily shown if he repeats the identical procedure on every Sunday of the season. > [@](#): > > In addition to being intuitive, we can also calculate this. In a given single experiment, the odds that a given awakening is on a given day are as follows: > 7/14: Heads Monday > 1/14: Tails Monday > 1/14: Tails Tuesday > 1/14: Tails Wednesday > 1/14: Tails Thurdsay > 1/14: Tails Friday > 1/14: Tails Saturday > 1/14: Tails Sunday > > These are trivially arrived at by taking the odds of the coin result and multiplying by the odds of a random awakening being on a given day given that coin result. (I normalized the denominators for convenience.) Ok, at least we’re making a \*little \*bit of progress, as I now understand how you get from 8/14 Mondays to 50% heads. > [@](#): > > Nonetheless, the question isn’t how many times monday occurs over a large number of experiments - it’s what the odds are that in the current experiment, the coin is heads. Maybe the problem is that we’re trying to answer different questions. It seems like you’re answering: “How likely is it that the coin landed heads?” Whereas, I’m answering: “How likely is it that this interview was preceded by heads?” The answer to your question must be 1/2, because it’s a fair coin and nothing done after the fact can change that. The answer to the second question, however, must be less than 1/2, because there are more tails-interviews and SB has no way to tell one interview from another. ---
**Author:** ![VarlosZ](https://avatars.discourse-cdn.com/v4/letter/v/d78d45/32.png) [@VarlosZ](https://boards.straightdope.com/u/VarlosZ)\ **Post date:** [April 22, 2009, 7:41am UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/171 "2009-04-22T07:41:49Z")
Note: I reserve the right to retract my declaration that the answer the first question is 1/2. ---
**Author:** ![begbert2](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/begbert2/32/3357_2.png) [@begbert2](https://boards.straightdope.com/u/begbert2)\ **Post date:** [April 22, 2009, 2:57pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/172 "2009-04-22T14:57:53Z")
> [@VarlosZ](#): > > Agree that scenario 2 is a better analogy for the SB situation. Unfortunately, given that, it leads us right back to the same disagreement: the chance that he put in the low-scoring game is 50%, but each particular awakening is probably the result of the high-scoring game, which can be easily shown if he repeats the identical procedure on every Sunday of the season. Not in a given experiment it’s not. In a given experiment it’s certainly the case that if you’re _in_ the case where there’s any possibility at all of seeing the low-scoring game’s touchdown, the odds of seeing a high-scoring game’s touchdown is 0. And vice versa. > [@VarlosZ](#): > > Maybe the problem is that we’re trying to answer different questions. It seems like you’re answering: “How likely is it that the coin landed heads?” Whereas, I’m answering: “How likely is it that this interview was preceded by heads?” The answer to your question must be 1/2, because it’s a fair coin and nothing done after the fact can change that. The answer to the second question, however, must be less than 1/2, because there are more tails-interviews and SB has no way to tell one interview from another. Two points: the question I’m asking _is_ the one dictated by the problem, and secondly, in a single given experiment, your question becomes my question (and assumes the probability results associated with my question). > [@VarlosZ](#): > > Note: I reserve the right to retract my declaration that the answer the first question is 1/2. All rights reserved; got it. 😃 Okay, **Tyrrell McAllister** , I’ve thought of a whole new approach that I think will solve this on its own, preventing me from even having to explain the logic behind my prior post. 🙂 When doing probabilities, you have to take into account _all_ the information you have available, right? For example if you did the Monty Hall problem while taking into account that you know Monty opened a door (with a goat), and fail to take into account that you _also_ know that he isn’t picking randomly, you’d get probabilities for the doors at 1/2 and 1/2, not 1/3 and 2/3 with is the final correct answer. SB knows that if the experiment landed on tails, that there is another day besides the one she is answering any given question on, correct? And she also knows that no symbol appears on both days. So, when she looks at the calendar and sees C, she not only learns “C appears in the experiment”, she _also_ learns that it is _not_ the case that “C appears on another day in this experiment”. You have to take that into account when calculating your probabilities to get the _correct_ result. I’m too rusty on my probabalistic math to do this (subtracting probabilities conditioned on facts that have a nonzero correlation with one another? Umm…), but I bet if you do it, you get a halfer result. ---
**Author:** ![Tyrrell\_McAllister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tyrrell_mcallister/32/16772_2.png) [@Tyrrell\_McAllister](https://boards.straightdope.com/u/Tyrrell_McAllister)\ **Post date:** [April 22, 2009, 5:31pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/173 "2009-04-22T17:31:07Z")
> [@begbert2](#): > > SB knows that if the experiment landed on tails, that there is another day besides the one she is answering any given question on, correct? And she also knows that no symbol appears on both days. > > So, when she looks at the calendar and sees C, she not only learns “C appears in the experiment”, she _also_ learns that it is _not_ the case that “C appears on another day in this experiment”. > > You have to take that into account when calculating your probabilities to get the _correct_ result. I’m too rusty on my probabalistic math to do this (subtracting probabilities conditioned on facts that have a nonzero correlation with one another? Umm…), but I bet if you do it, you get a halfer result. If I understand you, you’re saying exactly what I said in the last paragraph of my last post, except that I hesitate to condition on “C appears today (rather than some other day of the experiment)” before I have a satisfactory way (to me) to understand the meaning of the word “today”. At the time that I wrote the OP, I was denying that the word “today” had any meaning except for the meaning provided by the random calendar, according to which “today” means “the day that C appears on the calendar.” (This is precisely why I introduced the random calendar: to give the word “today” meaning.) I know realize that “today” should be meaningful apart from the random calendar, but I don’t yet know how exactly to do that. And until I understand how to give “today” meaning (apart from the random calendar), I don’t know what happens when you condition on statements that contain that word. The upshot is that I will probably settle on being either a thirder or a halfer (instead of the halfer-in-some-cases position that I had before). But I don’t yet know which. ---
**Author:** ![begbert2](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/begbert2/32/3357_2.png) [@begbert2](https://boards.straightdope.com/u/begbert2)\ **Post date:** [April 22, 2009, 5:48pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/174 "2009-04-22T17:48:57Z")
> [@Tyrrell\_McAllister](#): > > If I understand you, you’re saying exactly what I said in the last paragraph of my last post, except that I hesitate to condition on “C appears today (rather than some other day of the experiment)” before I have a satisfactory way (to me) to understand the meaning of the word “today”. > > At the time that I wrote the OP, I was denying that the word “today” had any meaning except for the meaning provided by the random calendar, according to which “today” means “the day that C appears on the calendar.” (This is precisely why I introduced the random calendar: to give the word “today” meaning.) > > I know realize that “today” should be meaningful apart from the random calendar, but I don’t yet know how exactly to do that. And until I understand how to give “today” meaning (apart from the random calendar), I don’t know what happens when you condition on statements that contain that word. > > The upshot is that I will probably settle on being either a thirder or a halfer (instead of the halfer-in-some-cases position that I had before). But I don’t yet know which. “C appears on another day in this experiment” doesn’t include the word “today”. However, I’ll try to spread my lack of confusion. Here’s how to define “another day” without referring to “today”. SB knows that if tails is flipped, there will be two interviews. She can, reasonably, label them as Interview One and Interview Two in an arbitrary manner, so as to be able to mentally distinguish between them. SB also knows that when she is interviewed, that interview will _either_ be Interview One, or Interview Two (assuming the coin landed on tails). She can, based _only_ on the above information, define “another day” as “The set of days including the single day on which interview 2 occurs, if the current interview is interview 1 and the coin was tails. Or the set of days including the single day on which interview 1 occurs, if the current interview is interview 2 and the coin was tails. Or the empty set, if the coin was heads”. That can be defined based _only_ on information explicitly given in the experiment setup and the ability to recognize that the interview that is happening to her is one of the interviews in the set of experiment interviews. If she can’t figure this much out, then she certainly doesn’t have the ability to calculate _any_ probabilities, in part because she’ll be unable to connect the symbol on the calendar with the interview she’s experiencing or either of those with the experiment itself. ---
**Author:** ![Tyrrell\_McAllister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tyrrell_mcallister/32/16772_2.png) [@Tyrrell\_McAllister](https://boards.straightdope.com/u/Tyrrell_McAllister)\ **Post date:** [April 22, 2009, 5:52pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/175 "2009-04-22T17:52:38Z")
> [@begbert2](#): > > She can, based _only_ on the above information, define “another day” as “The set of days including the single day on which interview 2 occurs, if the current interview is interview 1 and the coin was tails. Or the set of days including the single day on which interview 1 occurs, if the current interview is interview 2 and the coin was tails. Or the empty set, if the coin was heads”. How is she defining “the current interview” if she can’t already define “today” (and hence “another day”)? ETA: Give me some credit ;). If it were that easy, I’d have figured it out myself :). ---
**Author:** ![begbert2](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/begbert2/32/3357_2.png) [@begbert2](https://boards.straightdope.com/u/begbert2)\ **Post date:** [April 22, 2009, 6:06pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/176 "2009-04-22T18:06:04Z")
> [@Tyrrell\_McAllister](#): > > How is she defining “the current interview” if she can’t already define “today” (and hence “another day”)? > > ETA: Give me some credit ;). If it were that easy, I’d have figured it out myself :). (I think I’ll label the interview if the coin was heads as Interview Zero.) She’d be accomplishing this by noticing that she _is_ being interviewed, and applying the knowledge that the interview is one of the interviews of the experiment, and thus _must_ be a single member of the set [Interview Zero, Interview One, Interview Two]. The ‘current’ label is defined as an arbitrary label on the interview she is experiencing (as opposed to the other interviews, which she is clearly not currently experiencing). As noted, this association must be made to deduce that her current experience -which notably includes the calender with it’s C symbol- has anything at all to do with the experiment. Thus, in order to be _capable_ of knowing that “C appears in the experiment” (which is necessary for the thirder conclusion), the only way to reach that knowledge is by first identifying the interview in question as a single specific (if unknown) member of the set of interviews in the experiment. And after that identification is made, “the current interview” is as valid an arbitrary label for it as any. So if she has the _ability_ to learn “C appears in the experiment”, then she does indeed also learn that it is not the case that “C appears on another day in this experiment”. ---
**Author:** ![VarlosZ](https://avatars.discourse-cdn.com/v4/letter/v/d78d45/32.png) [@VarlosZ](https://boards.straightdope.com/u/VarlosZ)\ **Post date:** [April 22, 2009, 6:07pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/177 "2009-04-22T18:07:20Z")
> [@begbert2](#): > > Two points: the question I’m asking _is_ the one dictated by the problem. . . I don’t think that’s a given; the question can be parsed different ways. “What is your credence now for the proposition that our coin landed heads?” For one thing, the inclusion of the word “now” suggests the relevance of SB’s temporal location vis-à-vis the coin flip and the experiment. For another, the phrase “_your_ credence” is somewhat ambiguous, suggesting, like the word “now,” that what is being asked about is the probability relative to SB’s current predicament, not the probability as calculated by an objective observer. Presumably, there’s a reason why the question in the paradox is never phrased as simply as: “What are the odds that the coin we flipped came up heads?”, which would be the the most obvious and least confusing way to phrase it if that was the piece of information being sought. For an example of someone demonstrating that SB’s “credence” is 1/3, see Elga, [here](http://www.princeton.edu/~adame/papers/sleeping/sleeping.html), arguing the thirder position. Now, nominally he’s answering the question: “When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?”, but if you read the article it becomes pretty clear that he would consider that question synonymous with the one in the OP. > [@Adam Elga](#): > > . . . H1 = HEADS and it is Monday. > T1 = TAILS and it is Monday. > T2 = TAILS and it is Tuesday. > > . . . Let P be the credence function you ought to have upon first awakening. > > . . . If (upon first awakening) you were to learn that the toss outcome is Tails, that would amount to your learning that you are in either T1 or T2. Since being in T1 is subjectively just like being in T2, and since exactly the same propositions are true whether you are in T1 or T2, even a highly restricted principle of indifference yields that you ought then to have equal credence in each. But your credence that you are in T1, after learning that the toss outcome is Tails, ought to be the same as the conditional credence P(T1| T1 or T2), and likewise for T2. So P(T1| T1 or T2) = P(T2 | T1 or T2 ), and hence P(T1) = P(T2). > > The researchers have the task of using a fair coin to determine whether to awaken you once or twice. They might accomplish their task by either > > 1. first tossing the coin and then waking you up either once or twice depending on the outcome; or > 2. first waking you up once, and then tossing the coin to determine whether to wake you up a second time. > > Your credence (upon awakening) in the coin’s landing Heads ought to be the same regardless of whether the researchers use method 1 or 2. So without loss of generality suppose that they use - and you know that they use - method 2. > > Now: if (upon awakening) you were to learn that it is Monday, that would amount to your learning that you are in either H1 or T1. Your credence that you are in H1 would then be your credence that a fair coin, soon to be tossed, will land Heads. It is irrelevant that you will be awakened on the following day if and only if the coin lands Tails - in this circumstance, your credence that the coin will land Heads ought to be 1/2. But your credence that the coin will land Heads (after learning that it is Monday) ought to be the same as the conditional credence P(H1 | H1 or T1). So P(H1| H1 or T1) = 1/2, and hence P(H1) = P(T1). > > Combining results, we have that P(H1) = P(T1) = P(T2). Since these credences sum to 1, P(H1) = 1/3. It’s worth reading the following section (it’s brief), but I won’t quote it here as doing so would mean reproducing the larger part of the article. > [@begbert2](#): > > . . . and secondly, in a single given experiment, your question becomes my question (and assumes the probability results associated with my question). I’m not sure exactly what you mean here. Are you differentiating between an experiment and an interview? If so, why, since my question specifically addresses an interview? ---
**Author:** ![Tyrrell\_McAllister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tyrrell_mcallister/32/16772_2.png) [@Tyrrell\_McAllister](https://boards.straightdope.com/u/Tyrrell_McAllister)\ **Post date:** [April 22, 2009, 6:19pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/178 "2009-04-22T18:19:52Z")
I think that you’re just not getting why I’m unhappy with just considering the meaning of “today” to be already self-evident. It seems to be one of those things where one person’s self-evident fact is another person’s paradoxical swamp of confusion. > [@begbert2](#): > > (I think I’ll label the interview if the coin was heads as Interview Zero.) > > She’d be accomplishing this by noticing that she _is_ being interviewed, and applying the knowledge that the interview is one of the interviews of the experiment, and thus _must_ be a single member of the set [Interview Zero, Interview One, Interview Two]. The ‘current’ label is defined as an arbitrary label on the interview she is experiencing (as opposed to the other interviews, which she is clearly not currently experiencing). My problem is with your use of the word “current”. If I didn’t have that problem, she could have just defined “today” to mean “the current day” Now, you get close to my thinking when you propose considering the word “current” to be an “arbitrary label”. That, indeed, is exactly what I was trying to make explicit with my random calendar. The display on the calendar provided the “arbitrary label” for “today” (or, equivalently, “the current day” or “the current interview”). But, as you saw, that approach did not lead to the halfer conclusion. > [@](#): > > As noted, this association must be made to deduce that her current experience -which notably includes the calender with it’s C symbol- has anything at all to do with the experiment. Thus, in order to be _capable_ of knowing that “C appears in the experiment” (which is necessary for the thirder conclusion), the only way to reach that knowledge is by first identifying the interview in question as a single specific (if unknown) member of the set of interviews in the experiment. And after that identification is made, “the current interview” is as valid an arbitrary label for it as any. That would be one way to do it, except that, from my perspective, you’re deducing a non-problematic statement (“C appears during the experiment”) from a problematic one (“C appears during today/the current day/the current interview”) > [@](#): > > So if she has the _ability_ to learn “C appears in the experiment”, then she does indeed also learn that it is not the case that “C appears on another day in this experiment”. If I restrict myself only to statements I’m comfortable with, I can only say this: If she learns “C appears in the experiment”, then she does indeed also learn that it is not the case that “C appears on another day in this experiment _than the day on which it appears_”. ---
**Author:** ![begbert2](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/begbert2/32/3357_2.png) [@begbert2](https://boards.straightdope.com/u/begbert2)\ **Post date:** [April 22, 2009, 6:52pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/179 "2009-04-22T18:52:34Z")
> [@VarlosZ](#): > > I don’t think that’s a given; the question can be parsed different ways. “What is your credence now for the proposition that our coin landed heads?” For one thing, the inclusion of the word “now” suggests the relevance of SB’s temporal location vis-à-vis the coin flip and the experiment. For another, the phrase “_your_ credence” is somewhat ambiguous, suggesting, like the word “now,” that what is being asked about is the probability relative to SB’s current predicament, not the probability as calculated by an objective observer. Presumably, there’s a reason why the question in the paradox is never phrased as simply as: “What are the odds that the coin we flipped came up heads?”, which would be the the most obvious and least confusing way to phrase it if that was the piece of information being sought. > > For an example of someone demonstrating that SB’s “credence” is 1/3, see Elga, [here](http://www.princeton.edu/~adame/papers/sleeping/sleeping.html), arguing the thirder position. Now, nominally he’s answering the question: “When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?”, but if you read the article it becomes pretty clear that he would consider that question synonymous with the one in the OP. > > It’s worth reading the following section (it’s brief), but I won’t quote it here as doing so would mean reproducing the larger part of the article. Let’s look at this. He points out, correctly, that if it’s tails that monday and tuesday are equally likely to occur. Then he points out, correctly, that if it’s monday heads and tails are equally likely to occur. And then from this, he deduces that, if it’s monday, heads and tails are _not_equally likely to occur. Hmm, deducing X = 1/3 from X = 1/2 doesn’t sound right. What could the problem be? He’s taking the odds of it being tuesday and the odds of it being heads, and then equating apples to oranges. > [@VarlosZ](#): > > I’m not sure exactly what you mean here. Are you differentiating between an experiment and an interview? If so, why, since my question specifically addresses an interview? I mean that, in a single experiment, my answer is true for _each and every_ interview. The odds are based on the fact that the number of times you are asked the question doesn’t influence the answer. ---
**Author:** ![begbert2](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/begbert2/32/3357_2.png) [@begbert2](https://boards.straightdope.com/u/begbert2)\ **Post date:** [April 22, 2009, 7:09pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/180 "2009-04-22T19:09:06Z")
> [@Tyrrell\_McAllister](#): > > I think that you’re just not getting why I’m unhappy with just considering the meaning of “today” to be already self-evident. It seems to be one of those things where one person’s self-evident fact is another person’s paradoxical swamp of confusion. > > My problem is with your use of the word “current”. If I didn’t have that problem, she could have just defined “today” to mean “the current day” > > Now, you get close to my thinking when you propose considering the word “current” to be an “arbitrary label”. That, indeed, is exactly what I was trying to make explicit with my random calendar. The display on the calendar provided the “arbitrary label” for “today” (or, equivalently, “the current day” or “the current interview”). But, as you saw, that approach did not lead to the halfer conclusion. I maintain that if she can’t correlate C with a _specific_ day in the experiment, she can’t correlate it to the experiment at all. Because the experiment is composed of nothing but specific days - and your math specifically correlates C with specific days of the experiment. As for your calendar’s C being an eplicit arbitrary label for “today”, that only works if you correctly account for the fact that your calendar _also_ conveys the information “C doesn’t appear on another day in the experiment.” To review: # Options Odds (\* is the day of interrogation) 1 H B\* .25 2 H C\* .25 3 T B\* C .125 4 T B C\* .125 5 T C\* B .125 6 T C B\* .125 Cases where C is in the experiment: 2 3 4 5 6 Cases where C is “today”: 2 4 5 Cases where C is another day in the experiment: 3 6 Notice how nicely the cases work out when you include all the available information. > [@Tyrrell\_McAllister](#): > > That would be one way to do it, except that, from my perspective, you’re deducing a non-problematic statement (“C appears during the experiment”) from a problematic one (“C appears during today/the current day/the current interview”) That’s the _only_ way to get to the non-problematic experiement. Your non-problematic statement has a foundation of sand (according to you). > [@Tyrrell\_McAllister](#): > > If I restrict myself only to statements I’m comfortable with, I can only say this: > > If she learns “C appears in the experiment”, then she does indeed also learn that it is not the case that “C appears on another day in this experiment _than the day on which it appears_”. How does she recognize that C appears in the experiment if she can’t connect that she is experiencing _one specific single interview_ in the experiment? If she can’t do that, she can’t connect the symbol C on the calender to the experiment at all, because she won’t be able to connect her current experience with any of the specific separate cases that she could be in if she was in the experiment. [Previous page](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882.md?page=8) [Next page](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882.md?page=10)