I might be misunderstanding your conditions, but I would say no, waking up only on tails would not count, unless of course you got to make your choice after the researcher indicates they're about to put you back to sleep. Then you obviously know that the coin flip was tails! But in any case where you have just awoken after having had your memory wiped, and you have not been told anything by the researcher, you have been given no new info.
I think, if you are told it's your first awakening, then you still must assume 50% heads or tails: in both cases, you will always have a first awakening, so there is no new info there. However, since you will not be told this in your second awakening, if that occurs, you know immediately the coin must have come up tails.
>I think, if you are told it's your first awakening, then you still must assume 50% heads or tails: in both cases, you will always have a first awakening, so there is no new info there.
How is this not a direct violation of Bayes theorem? The probability of learning that it is the first awakening differs based on what the coin flip was, so it requires an update.
When you wake up, you don't know which awakening it is. You claim the probability of heads is 50%.
Now, you are told it is the first awakening. The probability of learning that if heads is twice as much as the probability of learning that if tails, so your Bayes factor is 2:1. You therefore cannot still believe that the probability of heads is 50%.
I think, if you are told it's your first awakening, then you still must assume 50% heads or tails: in both cases, you will always have a first awakening, so there is no new info there. However, since you will not be told this in your second awakening, if that occurs, you know immediately the coin must have come up tails.