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# Epistemology: Sleeping Beauty

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.

## 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?
• 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.
• 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.
• I think that's more of a theoretical point that just makes us rethink our logic leading up into the next point.
• 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.
• 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.

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)