# 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:** 3
**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 14, 2009, 4:12pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/41 "2009-04-14T16:12:30Z")
> [@](#): > > But because most of her guesses will be after a tails, most of her guesses will be wrong (assuming she always says heads). Here I’m with **LHoD**. When you’re talking about credence, it doesn’t make sense to penalize multiple guesses (which are all the same anyway) about the same event. You’re basically penalizing her for being asked the same question twice. ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [April 14, 2009, 4:23pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/42 "2009-04-14T16:23:10Z")
> [@Tyrrell\_McAllister](#): > > Here I’m with **LHoD**. When you’re talking about credence, it doesn’t make sense to penalize multiple guesses (which are all the same anyway) about the same event. You’re basically penalizing her for being asked the same question twice. “Penalize”? There’s no penalty, just mathematics. The answer to the question “how many of her guesses are incorrect if she always says heads?” is factually 99 out of 100 guesses. ---
**Author:** ![erislover](https://avatars.discourse-cdn.com/v4/letter/e/71e660/32.png) [@erislover](https://boards.straightdope.com/u/erislover)\ **Post date:** [April 14, 2009, 4:30pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/43 "2009-04-14T16:30:28Z")
> [@Indistinguishable](#): > > Wait, what are the first and second question? I only see one question in your post. Poor use of the word “this.” I mean, in the SBP, the first question is… ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [April 14, 2009, 4:32pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/44 "2009-04-14T16:32:01Z")
> [@](#): > > But are you confidant that someone couldn’t point out criteria that you hadn’t thought of, but which, once pointed out, you would agree that you care about? Care about for a specific purpose? Sure, it’s very plausible that this could happen. Care about with no purpose in mind, such that I should always judge things relative to _this_ criteria and not some other? That doesn’t even make sense (except relative to some further criteria for the correctness of criteria…). > [@Tyrrell\_McAllister](#): > > ETA: In other words, it seems that your answer is, “I haven’t figured whether there are criteria that I care about which determine an answer.” Well, my position is “There are lots of different criteria I could care about for different purposes. Some of them, I can see determining the answer of 1/2; some of them, I can see determining the answer of 1/3; yet others, different things. I haven’t yet been shown criteria which I both feel should determine a unique answer, yet which this setup frustrates. And, so far as those two answers go, it is spurious to ask me to decide between the criteria leading to 1/3 and those leading to 1/2; it’s not that one is the right criteria and the other is not (again, no preferred criteria for judging criteria…), they’re just different”. ---
**Author:** ![erislover](https://avatars.discourse-cdn.com/v4/letter/e/71e660/32.png) [@erislover](https://boards.straightdope.com/u/erislover)\ **Post date:** [April 14, 2009, 4:40pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/45 "2009-04-14T16:40:05Z")
> [@Pleonast](#): > > Probability of tails on first flip: 1/2 > Probability of tails on second flip, given a heads a first flip: 1/4 > Probability of heads on second flip, given a heads on first: 1/4 > Odds of tails versus heads: (1/2+1/4):(1/4) = 3:1 LOL You are correct. :o ---
**Author:** ![yoyodyne](https://avatars.discourse-cdn.com/v4/letter/y/a9a28c/32.png) [@yoyodyne](https://boards.straightdope.com/u/yoyodyne)\ **Post date:** [April 14, 2009, 4:52pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/46 "2009-04-14T16:52:18Z")
> [@Pleonast](#): > > “Penalize”? There’s no penalty, just mathematics. The answer to the question “how many of her guesses are incorrect if she always says heads?” is factually 99 out of 100 guesses. And if she always says tails? ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [April 14, 2009, 4:55pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/47 "2009-04-14T16:55:37Z")
> [@yoyodyne](#): > > And if she always says tails? Then 99 out 100 of her guesses will be correct. ---
**Author:** ![panamajack](https://avatars.discourse-cdn.com/v4/letter/p/47e85d/32.png) [@panamajack](https://boards.straightdope.com/u/panamajack)\ **Post date:** [April 14, 2009, 5:43pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/48 "2009-04-14T17:43:13Z")
To further show the distinction between ‘experiment’ and ‘waking events’, consider an altered version: When she is woken, she is asked what the coin flip result was. If, at any point in the experiment, she guesses correctly, she ‘wins’ that experiment (and a years’s supply of supply of sleeping pills, $500, and an autographed picture of Randy Mantooth). In that case, she ought to be favoring heads slightly, because she’s trying to be correct on a per-experiment basis. She only has one chance to get it right if it’s heads, but more chances if it’s tails. What proportion of the time should she guess heads? phi - 1. In the original statement, she’s making one guess per waking event, and (apparently) trying to be correct for _that event only_, which yields a 1/3-2/3 distribution. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [April 14, 2009, 8:19pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/49 "2009-04-14T20:19:03Z")
Take it to an infinite curse: If it’s tails, the experiment will be carried on forever. If it’s heads, she’ll be put to sleep and asked the question only once. Prior to beginning the experiment, ask her: what are the chances that the next time you wake up, you’ll be free of the curse? Her answer should, logically, be 1/2. When you wake her up, however, if y’all are right, her answer should be: I am certain that the result was tails. I have trouble with this result. The difference between this and Monty Hall is that in MH, you gain information that allows you to recalculate the odds. In this one, you lose information. It appears that the loss of information leads you to calculate the odds very weirdly. ---
**Author:** ![Grumman](https://avatars.discourse-cdn.com/v4/letter/g/43a26b/32.png) [@Grumman](https://boards.straightdope.com/u/Grumman)\ **Post date:** [April 14, 2009, 8:44pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/50 "2009-04-14T20:44:59Z")
> [@Left\_Hand\_of\_Dorkness](#): > > Prior to beginning the experiment, ask her: what are the chances that the next time you wake up, you’ll be free of the curse? Her answer should, logically, be 1/2. Only because you changed the rules. You are only counting the _first_ awakening on each side, not _every_ awakening. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [April 14, 2009, 8:57pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/51 "2009-04-14T20:57:08Z")
> [@Left\_Hand\_of\_Dorkness](#): > > In this one, you lose information. It appears that the loss of information leads you to calculate the odds very weirdly. It’s not that odd that loss of information causes odds to shift, since it’s just the reverse of gaining information, which also causes odds to shift, and can do so arbitrarily drastically. The odd thing is that, between certain points, you seem not to have gained or lost _any_ relevant information. [E.g., in terms of right before the coin is flipped as compared to the first time you wake up, you seem to have all the same facts at hand (at least, once one rephrases them into absolute rather than deictically relative terms): the basic setup, the fact that you will be awake at some point, etc.] ---
**Author:** ![bup](https://avatars.discourse-cdn.com/v4/letter/b/6bbea6/32.png) [@bup](https://boards.straightdope.com/u/bup)\ **Post date:** [April 14, 2009, 8:58pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/52 "2009-04-14T20:58:54Z")
> [@](#): > > Only because you changed the rules. Nevertheless, **LHoD** has demonstrated something useful - the odds that she’s asleep forever are 1/2, but the odds each time she wakes up that she’s been given the permanent curse may be certain, or not. Plus he’s introduced a real creepy factor, which all good paradoxes should have. Certainly, if there were a payout, and she sought to maximize her payout (for her family, since she’s as good as dead), she should guess tails each time and give them an annuity forever. Should she logically say, “I’m certain I’m stuck in the sleep forever package,” though? Now I don’t know. Thank you for stating it that way, **LHoD**. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [April 14, 2009, 9:12pm UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/53 "2009-04-14T21:12:00Z")
Let’s try rephrasing it so that instead of being in terms of time (Monday vs. Tuesday), it’s in terms of space (West vs. East). The inhabitants of the western hemisphere are exact clones of those in the eastern hemisphere; for every Bob\_West, there is a corresponding Bob\_East, and vice versa. A coin is flipped and, depending on the results, either the inhabitants of the western hemisphere are awake while those in the eastern hemisphere are asleep, or both hemispheres are awake. You wake up, and can remember nothing except the basic experimental setup. You have no means of establishing which hemisphere you are in; if your antipodal clone is also awake, then both of you, whichever is which, have the exact same thoughts in your head. What probability should you assign to the coin having been heads? (If your answer is different when the problem is phrased using space rather than time (or using another quantity rather than either of those)… why? What intrinsic relevant connection is there? Is not the theory of probability applicable in equal fashion to any quantity, or is there some special role within it for temporal ones?) ---
**Author:** ![bup](https://avatars.discourse-cdn.com/v4/letter/b/6bbea6/32.png) [@bup](https://boards.straightdope.com/u/bup)\ **Post date:** [April 15, 2009, 12:33am UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/54 "2009-04-15T00:33:05Z")
**Indistinguishable** , I don’t know anymore. To make the new situation you describe fair, you have to assume there’s an equal likelihood you were in the east or the west before you went to sleep. 50/50. Also, _and I think this is important_, in your experiment, you’re only woken up once. To make the experiment fair, those in the west have to be woken up twice. There’s a 50/50 chance you’re in the west, but if you are, you’ll be asked and be right twice or be wrong twice. Here’s where I’m starting to go with this ‘paradox’: the amnesia bit has essentially made two events out of what it’s calling one event. And now I bother to read the wikipedia page, which agrees, as I understand it. The first event is heads/tails (or whether you went to sleep in the east or the west). The second event, _which is a function of the first event_, is the event of being asked about the experience. Let me change the experiment this way, which I think is fair: Instead of guessing what the probability is, Sleeping Beauty is told what the outcome is. The first event, heads/tails, is 50/50. The second event, which is\* her learning about the outcome\*, is 1/3 heads, 2/3 tails. ---
**Author:** ![Chessic\_Sense](https://avatars.discourse-cdn.com/v4/letter/c/7c8e57/32.png) [@Chessic\_Sense](https://boards.straightdope.com/u/Chessic_Sense)\ **Post date:** [April 15, 2009, 1:09am UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/55 "2009-04-15T01:09:09Z")
To me, it’s just ambiguous wording, and not that impressive. Consider the following conversation between you and me: “I just flipped a fair coin. What is it?” “Umm…heads” ::it was tails, so I ask you to repeat it:: “What did you say?” “I said heads” Now the problem arises when we debate whether or not that’s two guesses or one. If it’s one guess, then SB should say 1/2. If it’s two guesses, then she should say 1/3. This is because she’s asked twice about one flip. If I had a deck of cards and only repeatedly asked you (a hundred times) if I drew the Ace of Spades, then you should certainly guess that. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [April 15, 2009, 1:29am UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/56 "2009-04-15T01:29:34Z")
> [@bup](#): > > Nevertheless, **LHoD** has demonstrated something useful - the odds that she’s asleep forever are 1/2, but the odds each time she wakes up that she’s been given the permanent curse may be certain, or not. > > Plus he’s introduced a real creepy factor, which all good paradoxes should have. Oh, I’ve just gotten started :). I’m about to give Sleeping Beauty the ability to manipulate time itself. She hears of the witch’s plan. She knows that there’s a 50% chance of not being cursed, but that when she wakes up, there’s a 100% chance of her being cursed. She despairs: this is a fate worse than death. So just before the last words of the curse are uttered, she slips a golden knife into her bodice. And then she sleeps. And then she awakes. As discussed before, the chance of being cursed is 100% at this point, right? Watch carefully. Desperate to escape this curse, she plunges the golden knife into her breast, killing herself. She is steadfast; she always sticks to her plan. (Note this isn’t just “likely sticks to her plan”: she’s like those nutty islanders who always tell the truth, only she always sticks to a plan). Because of this, as she lies dying, she changes the odds of being cursed from 100% to 50%: she knows that this must be the first time she was woken, since she knows that the first time she awoke, she would follow through on her plan to kill herself and never be woken again. As I understand it, her act gave her no new information, since she knew what she was going to do; but the commission of the act changed the odds of an event that happened in the past. An event that happens now CANNOT affect an event in the past. Can it? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [April 15, 2009, 1:56am UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/57 "2009-04-15T01:56:41Z")
> [@bup](#): > > Also, _and I think this is important_, in your experiment, you’re only woken up once. To make the experiment fair, those in the west have to be woken up twice. In the rephrasing, everything which used to be temporal is now spatial. Thus, instead of being woken up at two times if the coin is tails, you are now woken up at two places (i.e., both you and your antipodal clone are woken up). ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [April 15, 2009, 2:04am UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/58 "2009-04-15T02:04:56Z")
If we make everything spatial instead of temporal, couldn’t we phrase it this way? Get ten volunteers. Put them all in separate rooms. Tell them that you’ll flip a coin. If it comes up heads, you’ll ask nine of them, “how do you think the coin came up?” If it comes up tails, you’ll ask one of them, “How do you think the coin came up?” Anyone who guesses correctly gets a cookie. In this case, it seems perfectly obvious that, whenever you’re asked how the coin came up, you should say, “heads.” ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [April 15, 2009, 2:55am UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/59 "2009-04-15T02:55:28Z")
I would say that is an acceptable spatial translation, yes. For the time being, I shall refrain from staking a position on the intuition you draw from that rephrasing. Instead, I’d like to also offer up two other reformulations for discussion: A) Sleeping Beauty is put to sleep and then a coin is flipped. If the result is heads, she is woken up and interrogated. If the result is tails, she is cloned into many copies, each with the exact same memories and none of which is any more “the original” than another (if for some misguided reason you cannot countenance this last part, imagine killing off whoever you would otherwise consider the original). Each clone, of course, is placed in a separate room to be woken and interrogated. Supposing you, as a Sleeping Beauty, wake up and remember this whole setup; what probability should you assign to the coin flip having been heads? B) God is debating the details of the creation of the universe. He flips a coin and either, in the heads case, creates a universe with just one person in it, or, in the tails case, creates a universe with many people in it (though, as always, with each kept unaware of the existence of the rest). Given this setup, as you ponder the implications of your lonely existence, what probability should you assign to the coin flip having been heads? (Whatever your answers to these, how would they change if the coin was weighted to have a 99% probability of coming up heads, but with a million people being created in the tails case?) ---
**Author:** ![Alan\_Smithee](https://avatars.discourse-cdn.com/v4/letter/a/2acd7d/32.png) [@Alan\_Smithee](https://boards.straightdope.com/u/Alan_Smithee)\ **Post date:** [April 15, 2009, 3:03am UTC](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882/60 "2009-04-15T03:03:39Z")
> [@Left\_Hand\_of\_Dorkness](#): > > Oh, I’ve just gotten started :). I’m about to give Sleeping Beauty the ability to manipulate time itself. > > She hears of the witch’s plan. She knows that there’s a 50% chance of not being cursed, but that when she wakes up, there’s a 100% chance of her being cursed. She despairs: this is a fate worse than death. So just before the last words of the curse are uttered, she slips a golden knife into her bodice. > > And then she sleeps. And then she awakes. > > As discussed before, the chance of being cursed is 100% at this point, right? > > Watch carefully. > > Desperate to escape this curse, she plunges the golden knife into her breast, killing herself. She is steadfast; she always sticks to her plan. (Note this isn’t just “likely sticks to her plan”: she’s like those nutty islanders who always tell the truth, only she always sticks to a plan). > > Because of this, as she lies dying, she changes the odds of being cursed from 100% to 50%: she knows that this must be the first time she was woken, since she knows that the first time she awoke, she would follow through on her plan to kill herself and never be woken again. > > As I understand it, her act gave her no new information, since she knew what she was going to do; but the commission of the act changed the odds of an event that happened in the past. > > An event that happens now CANNOT affect an event in the past. Can it? Dude, you just blew my mind! [Previous page](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882.md?page=2) [Next page](https://boards.straightdope.com/t/the-sleeping-beauty-paradox/492882.md?page=4)