Before being told anything, you should estimate a 2/3 chance that it's Monday (not a 75% chance). There are three possibilities: heads/Monday, tails/Monday, and tails/Tuesday, all of which are equally likely.
I disagree that these situations are equally likely. We can understand it better by taking the extreme example. I will be much more surprised to hear that the coin was tails and that we are now at day #500,000, then that the coin was heads and that it is the first day. So obviously these two situations do not seem equally likely to me. And in particular, it seems equally likely to me that the coin was or will be heads, and that it was or will be tails. Going back to the non-extreme form, this directly implies that it seems half as likely to me that it is Monday and that the coin will be tails, as it is that it is Monday and that the coin will be heads. This results in my estimate of a 75% chance that it is Monday.
Because tails results in two awakenings, and you are calculating probability per awakening, that boosts the probability of tails, so it would be incorrect to put 50% on heads/Monday and 25% on tails/Monday. Tails/Monday is not half as likely as heads/Monday; it is equally likely.
I am not calculating "probability per awakening", but calculating in the way indicated above, which does indeed make Tails/Monday half as likely as heads/Monday.
Only in the scenario where you were woken up either on Monday or Tuesday, but not both, would the probability of tails/Monday be 25%.
I am not asking about the probability that the situation as a whole will somewhere or other contain tails/Monday; this has a probability of 50%, just like the corresponding claim about heads/Monday. I am being asked in a concrete situation, "do you think it is Monday?" And I am less sure it is Monday if the coin is going to be tails, because in that situation I will not be able to distinguish my situation from Tuesday. And this is surely the case even when I am woken up both on Monday and Tuesday. It will just happen twice that I am less sure it is Monday.
And based on the above reasoning, being told that it is Monday does indeed lead me to expect that the coin will land heads, with a probability of 2/3.
We can understand it better by taking the extreme example. I will be much more surprised to hear that the coin was tails and that we are now at day #500,000, then that the coin was heads and that it is the first day.
You should not be more surprised in that situation. The more days there are, the more that the extra tails awakenings push down the probability of heads. With 500000 awakenings, the probability gets pushed down by a lot. Now heads is 1/500001 per-awakening probability, same as tails-day-1 and tails-day-500000
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