- Epistemology: Argument and Evidence
- Epistemology: Science, Can It Teach Us Everything?
- Epistemology: The Will to Believe
- Epistemology: Reason and Faith
- Epistemology: Sleeping Beauty
- Epistemology: Rationality
- Epistemology: Paradoxes of Perception #1 (Argument from Illusion)
- Epistemology: Paradoxes of Perception #2 (Argument from Hallucination)
- Epistemology: The Paradox of the Ravens
- Epistemology: The Puzzle of Grue
- Epistemology: The Preface Paradox
- Epistemology: The Value of Knowledge
- Virtue Epistemology
- Epistemology: The Epistemic Regress Problem
In this video, Michael introduces the Sleeping Beauty problem. This is a problem in formal epistemology about how to correctly assign probabilities to an odd scenario in which we flip a coin and, depending on the outcome, wake Sleeping Beauty up according to two different patterns. We’ll discover some very strange consequences for our beliefs as a result.
Speaker: Michael Campbell, Duke University.
Speaker: Michael Campbell, Duke University.
Want to join the conversation?
- I don't think that's right, because irregardless what the consequences of the coin flip are, there is still a 50/50 chance of it being heads or tails. Splitting it into thirds determines the probability that the certain awakening was either a heads or tails awakening, and whether it occurred on monday or tuesday, but doesn't affect the probability of the coin flip itself. So no matter how many times she's woken up, wouldn't it still be a 50/50 chance it's heads or tails?(11 votes)
- I agree: A coin always has a 1/2 chance of being heads. Sleeping Beauty is being asked what the percent likelihood that it was heads based on conditions present AFTER the flip. She could answer, "I am awake, and I know that there is a 50% chance IN THIS MOMENT that it is Tuesday. Given the knowledge I now possess, there is a 1/1,000,002 chance that that one flip WAS heads.(3 votes)
- How could she possibly be woken up one million times on a single day? There isn't enough time available in 24 hours to do this.(0 votes)
- I think that's more of a theoretical point that just makes us rethink our logic leading up into the next point.(4 votes)
- One thing that very much confused me is what is sleeping beauty trying to accomplish with her answer to the question "What is your credence that the coin was heads?" She isn't trying to predict the outcome (With the exception of the Monday when they flip the coin after she goes back to sleep). If she wakes up on a Monday, and she knows it's a Monday, she should answer 50% because she is clearly trying to predict the outcome of the coin toss (Even if the coin was already tossed, it is still 50/50 to her). However if she doesn't know what day it is, 50 seems to be the wrong answer, which means that she isn't trying to make a prediction. So what is she trying to do?(1 vote)
- She is trying to figure out, given that she has been woken up, how confident she should be that the coin was heads.(2 votes)
- I am still a bit confused. What when the variables of Monday and Tuesday were not into the picture. By pure math, it is simply plausible no of outcomes by total no of outcomes.(1 vote)
- I end up feeling so confused. I think the probability of the coin flip is always one-half.(1 vote)
- The probability of Heads given that Sleeping Beauty was woken is 1/3.
Using conditional probabilities: P(Heads) = P(Monday) x P(Heads|Monday) + P(Tuesday) x P(Heads|Tuesday)
Sleeping Beauty knows she was woken, but does not know whether she was woken on Monday or Tuesday. The probability of it being Monday is P(Monday) = 2/3 and the probability of it being Tuesday is P(Tuesday) = 1/3.
If it is Monday, the probability of Heads or Tails is equal, i.e. P(Heads|Monday) = P(Tails|Monday) = 1/2
If it is Tuesday, the probability of Heads is zero, i.e. she would not have been woken, so P(Heads|Tuesday) = 0. The probability of Tails is 1, i.e. if it is Tuesday and she was woken then it must have been Tails, i.e. P(Tails|Tuesday) = 1.
That means that the probability that Heads was flipped -- given that Sleeping Beauty was woken -- is P(Heads) = 2/3 x 1/2 + 1/3 x 0 = 1/3
The probability that Tails was flipped -- given that Sleeping Beauty was woken -- is P(Tails) = 2/3 x 1/2 + 1/3 x 1 = 2/3.
Or have I missed something?(1 vote)
- I don't think there is a problem, sleeping beauty is just observing different experiment. Imagine a similar scenario:
You flip a coin. If it comes up heads, you take one picture of. If it comes up tails, you take 1,000,001 pictures of it. You would repeat this experiment several times, then shuffle all the taken pictures together and pick one randomly. What is the probability, that there is a tail in the picture?(1 vote)
- The problem can be easily transformed into one of trivial, unconditional probability.
The graph that was mentioned had four quadrants in it; one with an “S”, and three with an “A.” The original put the "S" in “Heads/Tuesday,” but there are three other quadrants where it could be. If we change the question to “What is your credence that the coin result is the one whose row has the ‘S’?”, then the answer is the same regardless of which is used. Sleeping Beauty doesn’t even have to know which it is.
Once we accept that, we can perform the same experiment with four volunteers. We write the four possible graphs on four index cards, and deal one to each volunteer (and it won’t matter if we show each volunteer all four, just their own, or none). We then perform the experiment with all four volunteers, based on the same coin flip performed on Sunday Night, and keeping them separated on Monday and Tuesday.
Say you are one of the volunteers, and are awake. You can consider your experiment to be equivalent to the original Sleeping Beauty problem. If you do, there are two seemingly paradoxical solutions.
But you can also look at it as an unconditional problem. You know that there are exactly three volunteers awake at the moment, that each has the same information to base a credence on, and that exactly one of the three has a card where the coin result matches the row with her “S.” Your credence that you are that one can only be 1/3.
The “million wakings” variant is no longer a paradox. To make it easier, consider the original experiment to be one waking (on Monday) after Heads, and wakings on the N consecutive days starting with Monday after Tails. This is changed to use 2*N graphs, for all possible combinations of N “A”s in one row, and one “A” and N-1 “S”s in the other. N+1 volunteers are awake on any given day, and only one has the “S”s in the coin’s row.(1 vote)
(intro music) Hi, there! I'm Michael Campbell, a graduate student in philosophy at Duke University, and I'm going to try to explain to you something called the "Sleeping Beauty problem." If you like interesting puzzles, you'll love this one. So, let's just dive right in. Image, one day, that some philosophers get together, and decide to run an experiment on poor old Sleeping Beauty. Right now, it's Sunday, and they tell her that the following will take place. First of all, they will, of course, put Sleeping Beauty to sleep. Then, they will flip a fair coin, which just means that there's a fifty-fifty chance that the coin will land heads or tails. Depending on the result of that coin flip, which they won't tell Sleeping Beauty, one of two scenarios will occur. If the coin comes up heads, Sleeping Beauty will be woken up on Monday, and then given a sedative which puts her to sleep and erases her memory of ever having been woken up. On Tuesday, they will let her sleep all day, before waking her up on Wednesday, when the experiment ends. But if the coin comes up tails, then Sleeping beauty will be woken up on a Monday, given the memory-wiping sedative, and woken up on Tuesday, and then given the sedative again. Like before, she will be woken up on Wednesday, and the experiment ends. We can make all this a little bit clearer by drawing a graph. On one side of the graph, we can write "H" for "Heads," and "T" for "Tails." On the other side, let's write "M" for "Monday," and "T" for "Tuesday." Now, in the Heads-Monday box, we can write "A" for "Awake." But in the Heads-Tuesday box, we'll write "S" for "Sleep." In both the Tails-Monday and the Tails-Tuesday boxes, we can write "A" for "Awake." Now, the first time we wake Sleeping Beauty up, we ask her the following question: "What is your credence that the coin came up heads?" "Credence" just means something like "probability," so the question is basically "What do you think the chance is that the coin came up heads?" Now you might think that the answer is clearly "one-half." After all, the coin was fair. It's a rule of probability theory that probabilities have to add up to one. So we can write "one-half" in the Monday-Heads box. And given that the coin a one-half chance of coming up tails too, we can divide that one-half by two, splitting the probability evenly between the two scenarios. Problem solved! If only it were that easy. If you look closely at the graph, you'll see that there are three awakenings in total. Two of those awakenings take place if the coin comes up tails, but just one takes place if the coin comes up heads. So given that there are three possible awakenings, and only one of them happens when the coin comes up heads, it seems like there is a one-in-three chance that on any given awakening, it's a heads awakening. Remember, Sleeping Beauty doesn't know the result of the coin toss, or whether she's been woken up beforehand. So, the answer to the question "What probability should Sleeping Beauty assign "to the coin having come up heads when she's woken up?" is "one-third." In that case, we can write "one-third" in each of these "Awake" boxes. But that sounds crazy, doesn't it? Before Sleeping Beauty was put to sleep, but after she was told all about the experiment, she knew that the coin was fair. If someone had asked her then, "What are the chances that the coin will come up heads?", she'd clearly say "one-half." And it seems like she doesn't learn anything new when she wakes up for the first time, because she knew all about the experiment beforehand. Given all this, why should we think that her answer changes to one-third? Confused? Well, let me try and make you even more confused. Notice that Sleeping Beauty wakes up on a Monday, regardless of whether the coin comes up heads or tails. So it doesn't seem to matter if we toss the coin on Sunday night or if we toss it on Monday evening, after we've woken Sleeping Beauty up and put her back to sleep again. Imagine that when we wake Sleeping Beauty up on Monday, we tell her that it's Monday. Now she knows that she is not in the Tuesday-Tail scenario, but that she's either in a Monday-Heads or a Monday-Tails scenario. Given that the coin is fair, and the coin will be tossed after Sleeping Beauty goes to sleep again on the Monday, shouldn't she think that the probability that the coin will come up heads is one-half? Well, it seems so. But if you think that Sleeping Beauty should say "one-half" when she doesn't know what day it is, I can prove to you, using some very simple assumptions about how probabilities work, that Sleeping Beauty ought to say, when she's told it's Monday, that the coin has a two-thirds chance of coming up heads. You'll have to take my word for this now, but check out this article by David Lewis if you want to be sure. So, we can say that the answer is "one-half," but then we have to accept that Sleeping Beauty ought to say that the probability that the coin comes up heads is two-thirds when she's told it's a Monday. That seems really strange. So why not go with the one-third view? Well, there are problems with that answer too. Remember that we thought the answer should be one-third, because there are three total potential awakenings, and only one of them happens when the coin lands heads. Let's change the experiment a little bit, but apply the same logic. Everything is as it was before, except that instead of being woken up once on Tuesday, we wake Sleeping Beauty up one million times on Tuesday. Now there are one million and two total potential awakenings. When she's woken up, from Sleeping Beauty's perspective, she could be in any one of those million and two awakenings. But given that so many of them will happen on a Tuesday if the coin comes up tails, it seems that Sleeping Beauty should say that the chances that the coin came up heads should only be one-in-a-million-and-two, because only one of those million and two awakenings happen when the coin comes up heads. But that just seems crazy too. The coin was fair. So how can it be that we ought to think that there was just a one-in-a-million-and-two chance that the coin came up heads? So whether we go for the one-third view or the one-half view, it seems like we're going to run into problems. Which view is best? I'm not sure. What do you think? Thanks for watching! Subtitles by the Amara.org community