There is no sense in which 1⁄2 is the correct answer
Sure, there is! If the question to the SB is “what heads:tails ratio does the person flipping the coin see?” then the answer is 1:1, not 1:2, and she can provide the correct answer. Or if SB is only asked the original question on Mondays. Each question corresponds to a different observer, and so may result in a different answer.
The standard question is what probability Beauty should assign to Heads after being woken (on Monday or Tuesday), and not being told what day it is, given that she knows all about the experimental setup. Of course if you change the setup so that she’s asked a question on Monday that she isn’t on Tuesday, then she will know what day it is (by whether the question was asked or not) and the answer changes. That isn’t an interesting sense in which the answer 1⁄2 is correct. Neither is it interesting that 1⁄2 is the answer to the question of what probability the person flipping the coin should assigns to Heads, nor to the question of what is seven divided by two minus three...
Sure, there is! If the question to the SB is “what heads:tails ratio does the person flipping the coin see?” then the answer is 1:1, not 1:2, and she can provide the correct answer. Or if SB is only asked the original question on Mondays. Each question corresponds to a different observer, and so may result in a different answer.
The standard question is what probability Beauty should assign to Heads after being woken (on Monday or Tuesday), and not being told what day it is, given that she knows all about the experimental setup. Of course if you change the setup so that she’s asked a question on Monday that she isn’t on Tuesday, then she will know what day it is (by whether the question was asked or not) and the answer changes. That isn’t an interesting sense in which the answer 1⁄2 is correct. Neither is it interesting that 1⁄2 is the answer to the question of what probability the person flipping the coin should assigns to Heads, nor to the question of what is seven divided by two minus three...