Given that Beauty is being asked the question, the probability that heads had come up is 1/3. This doesn't mean the probability of heads itself is 1/3. So I think this is a confusion about what the question is asking. Is the question asking what is the probability of heads, or what is the probability of heads given an awakening?

Bayes theorem:

• x = # of times awakened after heads
• y = # of times awakened after tails
• p(heads/awakened) = n(heads and awakened) / n(awakened) = x / (x+y)
• Yields 1/3 when x=1 and y=2.

Where is the probability of heads? Actually we already assumed in the calculation above that p(heads) = 0.5. For a general biased coin, the calculation is slightly more complex:

• p(H) =probability of heads
• p(T) = probability of tails
• x = # of times awakened after heads
• y = # of times awakened after tails
• p(heads/awakened) = n(heads and awakened) / n(awakened) = p(H)x / (p(H)x + p(T)y)
• Yields 1/3 when x=1 and y=2 and p(H)=p(T)=0.5.

I'm leaving this comment because I think the equations help explain how the probability-of-heads and the probability-of-heads-given-awakening are inter-related but, obviously -- I know you know this already -- not the same thing.