If you answered 1/2 to the original Sleeping Beauty Problem, the answer to this one should be reasonable to calculate. The probability of exactly n flips is (1/2)^n, so the probability of an even number of flips is (1/2)^2+(1/2)^4+(1/2)^6...=1/3.
If you answered 1/3 to the original Sleeping Beauty Problem, I do not think that there is any sensible answer to this one. I do not however consider this strong evidence that the answer of 1/3 is incorrect for the original problem. This could be an example of evidence for infinite set atheism. Analyzing this problem requires taking as given that the experiment can actually be repeated an arbitrarily large number of times, and we have thus far seen mostly evidence that this is not possible in our universe.
If you answered 1/3 to the original Sleeping Beauty Problem, I do not think that there is any sensible answer to this one. I do not however consider this strong evidence that the answer of 1/3 is incorrect for the original problem.
To also expand on this: 1/3 is also the answer to the "which odds should I precommit myself to take" question and uses the same math as SIA to yield that result for the original problem. And so it is also undefined which odds one should take in this problem. Precommitting to odds seems less controversial, so we should transplant our indifference to the apparent paradox there to the problem here.
I got into a heated debate a couple days ago with some of my (math grad student) colleagues about the Sleeping Beauty Problem. Out of this discussion came the following thought experiment:
Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: She will be put to sleep. During the experiment, Beauty will be wakened, interviewed, and put back to sleep with an amnesia-inducing anti-aging drug that makes her forget that awakening. A fair coin will be tossed until it comes up heads to determine which experimental procedure to undertake: if the coin takes n flips to come up heads, Beauty will be wakened and interviewed exactly 3^n times. Any time Sleeping Beauty is wakened and interviewed, she is asked, "What is your subjective probability now that the coin was flipped an even number of times?"
I will defer my analysis to the comments.