By “awakened” here you mean “awakened at all”. I think you’ve shown already that the probability that heads was flipped given that she was awakened at all is 1⁄2, since in both cases she’s awakened at all and the probability of heads is 1⁄2. I think your dispute is with people who don’t think “I was awakened at all” is all that Beauty knows when she wakes up.
Beauty also knows how many times she it likely to have been woken up when the coin lands heads—and the same for tails. She knew that from the start of the experiment.
OK, I see now why you are emphasizing being awoken at all. That is the relevant event, because that is exactly what she experiences and all that she has to base her decision upon.
(But keep in mind that people are just busy answering different questions, they’re not necessarily incorrect for answering a different question.)
This is incorrect.
Given that Beauty is being asked the question, the probability that heads had come up is 1⁄2.
This is bayes’ theorem:
p(H)=1/2
p(awakened|H)=p(awakened|T)=1
P(H|awakened)=p(awakened|H)P(H)/(p(awakened|H)p(H)+p(awakened|T)p(T))
which equals 1⁄2
By “awakened” here you mean “awakened at all”. I think you’ve shown already that the probability that heads was flipped given that she was awakened at all is 1⁄2, since in both cases she’s awakened at all and the probability of heads is 1⁄2. I think your dispute is with people who don’t think “I was awakened at all” is all that Beauty knows when she wakes up.
Beauty also knows how many times she it likely to have been woken up when the coin lands heads—and the same for tails. She knew that from the start of the experiment.
OK, I see now why you are emphasizing being awoken at all. That is the relevant event, because that is exactly what she experiences and all that she has to base her decision upon.
(But keep in mind that people are just busy answering different questions, they’re not necessarily incorrect for answering a different question.)