It seems obvious that the answer is 1/2. When Beauty wakes up she does not have any more or any less information than before the experiment. All she knows is that she has woken up and that this would have happened whether the coin landed heads or tails. She has no reason to believe that heads is more or less likely to have happened than tails. So the answer is 1/2. Or is it?
Let's examine it more carefully. Suppose this experiment were repeated 1,000 times. We would expect to get 500 heads and 500 tails. So Beauty would be awoken 500 times after heads on Monday, 500 times after tails on Monday, and 500 times after tails on Tuesday. In other words, only in a third of the cases would heads precede her awakening. So the right answer for her to give (seeing she is clever as well as beautiful) is 1/3.
This is the correct answer from Beauty's perspective. Yet to the experimenter the correct probability is 1/2. How do we reconcile the disparity between these two probability calculations? The reason there are two solutions is that they are solutions to two different questions. One is based on the percentage of runs of the experiment where the coin comes up heads, which is 1/2. The other is based on the percentage of interrogations where the coin comes up heads, which is 1/3. The way the original question is phrased determines the answer. In this case Sleeping Beauty is asked for her own credence, which is the percentage of interrogations where the coin comes up heads, so the answer is 1/3.
This still does not explain the disparity. The probability of an event may be defined as the theoretical knowledge that an observer has of the various paths leading to the present and their relative frequencies. The experimenter sees 1,000 paths leading to the present, consisting of 500 heads and 500 tails. Beauty sees not 1,000 but 1,500, of which 500 originate in heads and 1,000 in tails.
It may appear that the different probabilities, as determined by Beauty and the experimenter, are due to their different levels of knowledge. This is not so. Beauty's amnesia and ignorance of the day of the week are irrelevant. Even her waking up is irrelevant. The only factor that is relevant to the different probability calculations is sampling. If Beauty knew whether it was Monday or Tuesday then she would give the following odds. If it were Monday then the odds would be 1/2. If Tuesday then the odds would be 100% tails. Putting the two cases together we still get 1/3 chance of heads.
The deciding factor is not Beauty's lack of knowledge but how often each of the branches is sampled. By sampling the tails branch more times than the heads branch we guarantee that the probability of tails is higher than that of heads. To make it even clearer, suppose the setup was that Beauty would be awakened only after tails, not at all after heads. She would reply on being wakened that it had to be tails. This is the extreme case but it illustrates what is going on.
This also applies to the experimenter, provided they are asked the question at the same time as Beauty. If the experimenter were asked during the experiment they too would give the probability as 1/3. The reason why the experimenter's answer is given as 1/2 above is because it is tacitly assumed that the experimenter is being asked the question before the experiment or after it is over, but not during. This is the key difference that decides whether the probability of heads is 1/2 or 1/3. It is purely a matter of sampling. The answer of 1/3 arises simply because we sample twice as much on the tails branch.