Sleeping Beauty

This semester my “End of the World” class discussed the Doomsday Argument promulgated by Brandon Carter and John Leslie. The argument relies upon certain intuitions about probability, relating directly to what is known as the “Sleeping Beauty” example. If you are interested in the Doomsday Argument, check it out on wikipedia, but I don’t find it convincing because of the conclusion I arrive at after examining Sleeping Beauty.  None of the philosophers we read have my position, which I find fascinating because I think my paper is fairly convincing. Maybe I’ll send it to one of them one day… here is what I wrote.

Sleeping Beauty: A Halfer-Halfer’s Justification

Sleeping Beauty will be put to sleep on Sunday night. She will be woken up either once or twice depending on the toss of a fair coin. Heads (H) – she only wakes up Monday. Tales (T) – she wakes up Monday and Tuesday. A drug is given that makes her forget any previous wake ups. Everyone agrees that on Sunday, the probability of the coin landing H is ½. Upon waking up, however, there are different opinions about whether the probability changes. The Halfer says that probability of H stays the same. The Thirder believes the probability of H is now 1/3. Upon learning that it is Monday, the Halfer thinks the probability of H is now 2/3, while the Thirder thinks it is now ½. I will argue that both of these changes in probability are wrong and that in fact Sleeping Beauty should assume a ½ chance of H at every stage of knowledge (having all the info on Sunday, waking up, and upon learning it is Monday). This position will be called the Halfer-Halfer.

Let’s first take a look at the probability of H upon waking up. I agree with the Halfers in this case, as they state that since no information is learned upon waking up that there should be no probability shift. The Thirders, however, argue that since there are two wake ups with T and only one with H, there is twice the chance that on any given wake up that it occurs in a T[1]. This is clearly mistaken. This theory is related to the Self-Indication Assumption[2], something that is refuted in the Presumptuous Philosopher thought experiment. The fact of one’s own, personal existence provides no evidence as to how many other conscious beings exist. Similarly, because one wakes up does not in any way make it more probable that one has already woken up or will wake up tomorrow. On Sunday, Sleeping Beauty knows that she is going to wake up and not know the day, regardless of H or T. There is no new relevant information upon waking up. Another Thirder explanation points to Sleeping Beauty’s loss of knowledge about her temporal location as relevant. The problem with this theory is that she knows she will be unaware of her temporal location upon awakening. Why, then, should the probability change just because time has passed? Sleeping Beauty is exactly the same and is merely in a different temporal location that she knew was coming. Imagine another kind of fair coin toss. On one side of this coin is an “M” and the other side is an “M – T”. As soon as the coin is flipped, however, the images disappear and are only visible under UV rays. One should still maintain the probability of “M” as ½ despite the fact that it there are no letters visible and time has moved forward (analogous to not knowing the day). Sleeping Beauty is in the exact same situation and thus must maintain that the probability of H is ½.

Assume that the Sleeping Beauty experiment will be repeated 50 times and she must bet as to whether the coin landed H or T every time she wakes up (which is essentially what this problem asks us to do). She will clearly make more money betting on T. But this is misleading. She is only betting on days that she does wakes up. While H and T are roughly going to occur 25 times each with a fair coin, she will wake up 50 times in the T’s and 25 times in the H’s. But she is not making her bet with the intent of winning money. If she were, T is clearly the better choice. She is instead asked what the chances are of H vs. T. Since that probability is 50/50 from an objective position, it is unfair to consider the amount of times she makes her probabilities as a factor in actually making those probabilities themselves. In a way, we must decide the probability for Sleeping Beauty on Tuesday in the case of H, for she will not be conscious. Waking up twice should in no way affect the probability of H. Thus, the Thirders are mistaken.

Upon learning that it is Monday, the Thirders change the probability of H back to 1/2. The Halfers, however, have changed their probability of H to 2/3. Halfers will argue that Sleeping Beauty does indeed gain new information – namely, information about the future and the fact that she is not in it (the future being Tuesday)[3]. Something seems to be going wrong here. Surely gaining information about the future is informative in the sense that Sleeping Beauty knows she is not in it, but it tells her nothing about what the future actually holds. She already knows that she is going to wake up on Monday regardless of the coin landing H or T, so what should cause her to think that it being Monday is any way alters the probabilities she assigned on Sunday? The certainty of her probabilities on Sunday is unquestionable. She is also equally certain that she is going to wake up on Monday. Thus, there is no ground for shifting the possibilities to 2/3 upon learning that it is in fact Monday. What if the experiment was changed so that the coin toss did not occur until Monday night, after Sleeping Beauty had already gone to sleep? It would be absurd for her to change her probabilities of H to 2/3 because the coin hasn’t even been tossed yet! But this is exactly my point. It would make no difference if the coin were tossed Sunday night or Monday night because the results would be the same. If she were informed of the experiment on Sunday, it would not change the fact that she woke up on Monday. And if the experiment were to be repeated multiple times, she would wake up on Tuesday about the same number of times as not waking up on Tuesday. Essentially, what this line of argument shows is that she cannot alter her probabilities until new, relevant information is learned. In this case, the only relevant information is learning it is Tuesday. Thus, on Tuesday she can change her probability of H to 0 and T to 1. If she does not wake up on Tuesday, she is unable to change her probabilities and the observer must then do it for her.

Since most philosophers fall on either the Halfer or Thirder side, I am somewhat surprised that none disagree with arguments from both positions. Ultimately, my position is that no new information is ascertained after Sunday and therefore any shift in probabilities is unjustified. It is also unfair to consider how many times the probability is being made by Sleeping Beauty as support for the Thirder’s shift. Therefore, Sleeping Beauty should maintain a probability of H in all three cases, also known as being a Halfer-Halfer.


[1] Elga argues for this in “Self Locating belief and the Sleeping Beauty problem” (Analysis, 2000). He believes that P(probability upon waking up) is equal for H, T1, and T2, each representing what is possible before being told what day it is. Since P(H)=P(T1)=P(T2), and, he argues, these sum to one, it must follow that P(H)=1/3. I explain why this is misleading in the next paragraph.

[2] SIA– “Given the fact that you exist, you should (other things equal) favor hypotheses according to which many observers exist over hypotheses on which few observers exist.” (Bostrom, 2000) Though SIA can be used to argue for the Thirder position, I believe this can be easily refuted and thus concentrate on refuting Elga’s argument.

[3] Lewis makes this claim in “Sleeping Beauty: reply to Elga” (Analysis, 2001).

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2 thoughts on “Sleeping Beauty

  1. Good argument, Joel. Philosophy sort of twists your mind into a pretzel, huh? Your photos and blog are so interesting and exciting. I’m really enjoying reading it and keeping up with your trip. I hope this is a wonderful experience for you and that you share what you learn… 🙂 Safe travels.

  2. “Another Thirder explanation points to Sleeping Beauty’s loss of knowledge about her temporal location as relevant. The problem with this theory is that she knows she will be unaware of her temporal location upon awakening. Why, then, should the probability change just because time has passed?”

    Um, it doesn’t change because time has passed. You are misrepresenting the thirder claim, which is that she knows (before going to sleep) that she will awaken with no knowledge of what day it is. Therefore she goes to sleep knowing that the prior probability of heads is 50%, and also knowing that on any given future awakening, there will be a 1/3 probability of Monday-heads, a 1/3 probability of Monday-tails, and a 1/3 probability of Tuesday-tails. These probabilities represent different things (different states of knowledge, if you will); they do not represent the same thing at two different times.

    I think you are confusing the prior probability of a coin toss coming up heads with the modified probability (from SB’s point of view) that a coin toss came up heads given that “I am awake and don’t know what day it is”. The 1/3 probability is just what Bayes’ Theorem would predict, from the prior 50% probability (which isn’t disputed) and the strength of the evidence that “I am awake and don’t know what day it is” (an occurrence which is twice as likely given tails as it is given heads).

    The 1/2 probability that SB arrives at on learning that it is Monday is just what Bayes’ Theorem would predict given the 1/3 probability on waking, coupled with the strength of the evidence that “I now know that it is Monday” (If heads, it is certain that today is Monday, whereas if tails, it could be Monday or Tuesday with 50% likelihood of each). (Note that this observation refutes the Doomsday Argument, if we substitute “This is a short-lived civilization” for heads, and “I have a low birth rank” for “This is Monday”: Once the Self-Indication Assumption and the temporal information are both incorporated, the probability reverts to the prior. Neither the SIA nor the effect of birth rank is invalid per se, but neither provides a complete picture by itself. And the whole picture is that the only thing we know about the likelihood we are in a short-lived vs. long-lived civilization is the prior.)

    The zero probability that SB arrives at on learning that it is Tuesday is just what Bayes’ Theorem would predict given the 1/3 probability on waking, coupled with the strength of the evidence that “I now know that it is Tuesday” (if heads, there is no chance today is Tuesday; if tails, that is 50% likely).

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