Sleeping Beauty


Sleeping Beauty (SB) was a beautiful girl and well-trained in statistics, but she needed
some money and so she decided to participate in a medical trial. This was actually a
strange experiment with a sleeping pill, which had a severe side-effect, affecting and
essentially erasing the short-term memory. The experiment went as follows:

On Monday the experimenters would flip a coin (with a priori probability 1/2 for head
or tail respectively) and they would put her to sleep using the drug. If the coin showed
head, SB would awake on Tuesday, the experimenters would ask her one question,
which she had to answer correctly (her paycheck depended on the answer being truthful)
and this was the end of the experiment.
However, if the coin showed tail on Monday, they would put her to sleep, ask her the
question on Tuesday and then put her to sleep again. Then she would wake up on
Wednesday and the same question would be asked, which she had to answer once again.

Of course, since the drug erases the short-term memory, SB would not know whether it
was Tuesday or Wednesday when she would wake up, but she would remember all her
statistics knowledge and also the details of the experiment.
And so she participated in the experiment and she woke up and she was asked the one
question: "What is the probability that the coin showed head ?"
SB thought that this was a strange question, but also one she could answer easily:

1) "The a priori probability for head was p(H) = 1/2, on Monday. I have no new
information, therefore the probability is unchanged p(H) = 1/2." She was about to give
this answer, when she decided to double check her result, after all she did not want to risk
her paycheck.

2) "If I would know what day it is, my answers would differ. If I would know that
today is Wednesday, then the probability for head would be zero; If I would know that
today is Tuesday the probability for head would be 1/2. Of course, I do not know which
day it is, but I can calculate the probability that today is Tuesday or Wednesday to check
my previous result. If the outcome was head, today can only be Tuesday and if it was tail,
today is either Tuesday or Wednesday with probability 1/2 for both, since I have no further
information. Thus the probability that today is Tuesday is p(Tue) = (1/2) + (1/2)(1/2) = 3/4
and the probability that today is Wednesday is p(Wed) = 1/4. Now,  I can use this to calculate
p(H) as follows: p(H) = (1/2)*p(Tue) + 0*p(Wed) = (1/2)*(3/4) = 3/8.
So the probability for head is p(H) = 3/8 ?  Ooops!"
At this point SB understood that her reasoning was inconsistent and she did the calculation
more carefully as follows:

3) "p(Tue) = p(H) + (1/2)( 1 - p(H) ) , p(H) = (1/2) p(Tue) + 0 and thus
p(H) = (1/2)[ p(H) + (1/2)( 1 - p(H) )] with the final solution p(H) = 1/3."
This solution made sense to her, since she could not distinguish the three cases (awaking on
Tue only, awaking on Tue before being put to sleep again, awaking on Wed) in any way and
thus they seemed to have the same probability. Thus p(H) = 1/3.

But then, her first answer, p(H) = 1/2, seemed so much better.
And at this point SB understood what a weird experiment this really was ...

update: You may have noticed that SB wakes up Tuesday and/or Wednesday in my story,
while most webpages (e.g. if you followed the link above) use Monday and/or Tuesday.

update: Barry Clarke has a nice webpage about this problem and Lev Vaidman discusses SB
in his paper on Probability and the Many-Worlds Interpretation of Quantum Theory.

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