Update This has been moved up from its original 29 Feb 2012 date on 23 March 2012 to allow additional comments.
Update This has been moved up from its original 29 Feb 2012 date again on 21 May 2012 to allow additional comments.
Are you awake? I copied and pasted this direct from Wikipedia (I know, I know):
Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details. On Sunday she is put to sleep. A fair coin is then tossed to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and then the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday and Tuesday. But when she is put to sleep again on Monday, she is given a dose of an amnesia-inducing drug that ensures she cannot remember her previous awakening. In this case, the experiment ends after she is interviewed on Tuesday.
Any time Sleeping beauty is awakened and interviewed, she is asked, “What is your credence now for the proposition that the coin landed heads?”
I interpret credence as probability. Think about the answer before reading further.
Suppose she is awakened on Monday. She knows it is Monday, she knows that if the coin landed heads she would have been awakened. She also knows that if the coin landed tails she would have been awakened. In other words, regardless of the way the coin landed, she will be awakened on Monday.
Formally, her evidence is C = “A coin with two sides, labeled head and tail, will be flipped and only one side can show”, M = “It is Monday”, and also E = “The details of the experimental protocol.” She is asked to compute (or form):
Pr( head | C & E & M) = 1/2.
Since knowledge that it is Monday and of the experimental protocol tells her she will be awakened on Monday no matter what happens with the coin, the probability is numerically (but not logically) equivalent to Pr( H | C ), which everybody agrees is 1/2.
Now suppose T = “it is Tuesday.” Here the problem becomes ambiguous (at least, as Wikipedia has it). If the coin originally came up heads she will have been awakened Monday. There are then no words, i.e. no evidence, which tells us what she does Monday night into Tuesday. Perhaps she stays awake all Monday night celebrating Barack Obama’s defeat (we’re imagining this experiment taking place in November). Therefore, when it is Tuesday she will know that she was awakened Monday, because she will have recalled all the events which took place since her awakening; in particular, she will know she was not awakened on Tuesday.
So if queried on Tuesday about the coin she will have a different response than when she was queried on Monday. That is,
Pr( head | C & E & T & Up all night with memories) = 1,
and this will be the same if she goes to sleep on Monday but remembers being awakened on Monday, because of course she was not administered any drugs. So really all we need is her memory of what happened on Monday; she needn’t stay awake. That is,
Pr( head | C & E & T & Memories of Monday) = 1.
But if it is Tuesday and she is awakened and she hasn’t any memory of what happened on Monday, then
Pr( head | C & E & T & No memories of M ) = 0,
because she knows she was awakened on Tuesday and she knows she has no memory of Monday, whereas she would have those memories if the coin was heads. So she knows the coin was tails.
Since all that was too easy, I suspect the problem has been stated poorly. More evidence for this is that the Wikipedia discussion of other authors’ solutions appear to suggest that Sleeping Beauty has no idea what day it is. If that is so, her only knowledge is that she was awakened. Actually, not quite. She still knows that the day(s) she is awakened will be one of two, M or T.
We have already worked out the solutions conditional on her knowing the day. So we need to fit in the uncertainty in the day. She knows (and remember, we’re going on her information)
Pr( M | C & E ) = Pr( T | C & E ) = 1/2.
Here we don’t need memories, because if it’s M there aren’t any to be had, and if it’s T they’ve been wiped away (this is implicit in our new understanding of E). So what she wants is
Pr( head | C & E & M ) x Pr( M | C & E) + Pr( head | C & E & T ) x Pr( T | C & E) = 1/4.
Now Pr( head | C & E & M ) = 1/2 since if she assumes it is Monday, she knows by C & E that she will be awakened no matter what, just as above. But, Pr( head | C & E & T ) is different, since if she assumes it is Tuesday she knows that the coin must have been tails, so Pr( head | C & E & T ) = 0.
This is different than the other solutions, which were 1/3 and 1/2, so it’s possible that I have misinterpreted the experiment or that I have made a bone-headed mistake. What do you think?