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Recently there have been surprising discussions and disputes amongst philosophers and physicists about an elementary probability problem called the Sleeping Beauty problem. The following remarks are, as usual in this blog, the result of discussions with Nicoletta Sabadini.
The problem goes as follows: A beauty is told that the following procedure will be carried out. On Sunday a fair coin will be tossed without her knowing the result. She will go to sleep. Then on Monday one of two possibilities will occur.
In the case that the toss of the coin resulted in tails she will be wakened and asked her opinion of the probability that the result of coin was heads. She will then have her memory of what happened on Monday erased and will be put to sleep. On Tuesday (again in the case of tails, without a further toss of the coin) she will be wakened and asked her estimate of the result of the coin toss being heads.
In the case of heads, on Monday she will be asked her estimate of the probability that the result of the coin toss was heads. In that case she will not be asked again.
It seems clear intuitively that, when this procedure is carried out, in all three responses she has learnt nothing about the result of the coin toss, and that she should answer in each case $1/2$.
Strangely a considerable number of philosophers and physicists make an elementary error in the calculation of probabilities and believe that she should answer $1/3$.
I see also that a recipient of the COPSS Presidents' Award (the Nobel Prize of Statistics, according to Wikipedia), Jeffrey S. Rosenthal, believes the $1/3$ answer.
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