I separately think though that if the actual outcome of each coin flip was recorded, there would be a roughly equal distribution between heads and tails.
Importantly, this is counting each coinflip as the “experiment”, whereas the above counts each awakening as the “experiment”. It’s okay that different experiments would see different outcome frequencies.
If you record the moments when the outside observer sees the coin landing, you will get 1⁄2.
If you record the moments when the Sleeping Beauty, right after making her bet, is told the actual outcome, you will get 1⁄3.
So we get 1⁄2 by identifying with the outside observer, but he is not the one who was asked in this experiment.
Unless you change the rules so that the Sleeping Beauty is only rewarded for the correct bet at the end of the week, and will only get one reward even if she made two (presumably identical) bets. In that case, recording the moment when the Sleeping Beauty gets the reward or not, you will again get 1⁄2.
Importantly, this is counting each coinflip as the “experiment”, whereas the above counts each awakening as the “experiment”. It’s okay that different experiments would see different outcome frequencies.
Yes.
If you record the moments when the outside observer sees the coin landing, you will get 1⁄2.
If you record the moments when the Sleeping Beauty, right after making her bet, is told the actual outcome, you will get 1⁄3.
So we get 1⁄2 by identifying with the outside observer, but he is not the one who was asked in this experiment.
Unless you change the rules so that the Sleeping Beauty is only rewarded for the correct bet at the end of the week, and will only get one reward even if she made two (presumably identical) bets. In that case, recording the moment when the Sleeping Beauty gets the reward or not, you will again get 1⁄2.