The difference between feeling something as 0.333 and 0.5 is rather small. Anyway I think that feeling is not very appropriate term here, and betting will be more correct, but for betting bigger difference is needed or repated events.
The problem with betting is that it matters whether it is one case of you making the bet or two cases, and you do not know which it is. This is irrelevant to feeling, so feeling is the way to judge the matter.
I agree that it might be hard to notice the difference when it is so small. This is why the case where you have one wakening vs a million wakening is better. Even in that case, it is obvious that the person will feel uncertain whether the coin landed/​will land heads or tails, so it is obvious that they are halfers.
The difference between feeling something as 0.333 and 0.5 is rather small. Anyway I think that feeling is not very appropriate term here, and betting will be more correct, but for betting bigger difference is needed or repated events.
The problem with betting is that it matters whether it is one case of you making the bet or two cases, and you do not know which it is. This is irrelevant to feeling, so feeling is the way to judge the matter.
I agree that it might be hard to notice the difference when it is so small. This is why the case where you have one wakening vs a million wakening is better. Even in that case, it is obvious that the person will feel uncertain whether the coin landed/​will land heads or tails, so it is obvious that they are halfers.