A third argument tries to add rigor by considering monetary payoffs. If Beauty's bets about the coin get paid out once per experiment, some argue that she will do best by acting as if the probability is one half.half (while others argue that probability one third gives the correct result if decision theory is correctly applied). If instead the bets get paid out once per awakening, one can again argue about whether or not acting as if the probability is one third has the best expected value.
I think if we change the nature of the problem but keep the game principle the same we have an more intuitive answer:
Imagine someone tosses a fair coin and asks you to determine the probability of whether it’s heads. They tell you have not tampered with it and will show you the actual result, and even toss the coin in front of you.
You would naturally say 1/2 . They show you the result (heads or tails) and write it down as the last result on their log. [This is effectively what happens when sleeping beauty wakes up]
Afterwards they show you the log for every flip you made in the past. Secretly they have removed half of the head results, but never the final result, So over time the spread shows 1/3 of heads now. You confront them and they admit it saying it all part of an experiment! [This is the same as being woken up twice for tails and once for heads]
They toss the same coin and ask you to predict heads. As before they will show you the result and put that results as the last result in their log* .
[You now know the rules of the game]
You answer would remain as 1/2. The coin is a fair coin, you can see the toss and the result.
If a betting person is betting on the log results with a bookmaker, then he/she would always bet on tails (as some of the historical heads are removed to keep the balance on 2/3 tails).
Love to have other people's feedback.
Ketan
*Before the coin toss they will remove any excessive historical heads so that the split is always 1/3 heads 2/3 tails but only after they show you the last result.