This thought-experiment has been on my mind for a couple of days, and no doubt it's a special case of a more general problem identified somewhere by some philosopher that I haven't heard of yet. It goes like this:
You are blindfolded, and then scanned, and ninety-nine atom-for-atom copies of you are made, each blindfolded, meaning a hundred in all. To each one is explained (and for the sake of the thought experiment, you can take this explanation as true (p is approx. 1)) that earlier, a fair-coin was flipped. If it came down heads, ninety-nine out of a hundred small rooms were painted red, and the remaining one was painted blue. If it came down tails, ninety-nine out of a hundred small rooms were painted blue, and the remaining one was painted red. Now, put yourself in the shoes of just one of these copies. When asked what the probability is that the coin came down tails, you of course answer “.5”. It is now explained to you that each of the hundred copies is to be inserted into one of the hundred rooms, and will then be allowed to remove their blindfolds. You feel yourself being moved, and then hear a voice telling you you can take your blindfold off. The room you are in is blue. The voice then asks you for your revised probability estimate that the coin came down tails.
It seems at first (or maybe at second, depending on how your mind works) that the answer ought to be .99 – ninety-nine out of the hundred copies will, if they follow the rule “if red, then heads, if blue then tails”, get the answer right.
However, it also seems like the answer ought to be .5, because you have no new information to update on. You already knew that at least one copy of you would, at this time, remove their blindfold and find themselves in a blue room. What have you discovered that should allow you to revise your probability of .5 to .99?
And the answer, of course, cannot be both .5 and .99. Something has to give.
Is there something basically quite obvious that I'm missing that will resolve this problem, or is it really the mean sonofabitch it appears to be? As it goes, I'm inclined to say the probability is .5 – I'm just not quite sure why. Thoughts?