There's my devil's advocate view. I can see what you're saying though.
Edit: Additionally, even though the awakenings are not in the same 'stream of consciousness' if one tails waking happen, they both happen. They are linked even if you're trying to view them in isolation.
Let's assume there are 100 people undergoing this experiment. Half of them will flip heads, half tails. The half that flip heads will be woken up a total of 50 times. The half that flip tails will be woken up a total of 100 times.
So, we have a total of 150 wake-ups. 100 of those came from tails, and 50 from heads. So, if you're woken up and have no prior knowledge, you have a 100/150 = 2/3 chance that you flipped tails, and a 50/150 = 1/3 chance that you flipped heads.
Put another way, when you are woken up, it could be one of three cases:
-- Awakened on the only time for heads -- Awakened on the first time for tails -- Awakened on the second time for tails
Two of those cases correspond to tails, one to heads. So, it's twice as likely that you are being awakened due to a tails flip.
Of course, do you really think otherwise? If I promised to put a pound in your moneybox every time you correctly guessed the coin state on each awakening, would you really fall to sleep confident in a 50/50 guess strategy in your 1000 awakenings to 1 scenario?
to a halver, the likelihood of heads and tails are still each 1/2, but the expected winnings of betting tails would of course be larger.
it would be like saying i'll flip a coin, and if you guess correctly that it's tails i'll give you a million dollars. obviously i will guess that it's tails, even though i still believe each outcome is equally likely.
Let us assert that the probability of the coin flip being heads is 1/2.
Now, you have awoken. Regardless of if you've awoken to the Heads flip, or the first time to the Tails flip, or the second time to the Tails flip, the original flip's chance was still 1/2.
The possible misunderstanding comes from the fact that you will awake TWICE to Tails, which is more than to Heads! But the thing is this is irrelevant to the question, because you don't know whether this is the first or the second time, and you only need to wonder whether the original flip (recall it is probability 1/2) was heads or not. Potentially you could think that you'd be "wrong" more often, but we are only looking at one specific instance of you waking up in isolation, for which you have no additional information.
For example, consider waking up 1000 times if the flip is tails. Upon waking up, do you think the probability heads becomes 1/1001? I think regardless of waking up the first time or the 1000th time, you have no more information so it might as well have been the first time, and hence the probability is 1/2.