Why are you using the notation P(W) when you mean E(W)? And if you can get an expectation for it, you must know the probability of it.

Sorry, but if we're randomly selecting a waking then it is not true that you're on the heads path 50% of the time. In a pair of runs, one head, one tail, you are woken 3 times, twice on the tails path.

Randomly selecting a waking does not imply a uniform distribution. On the contrary, we know the distribution is not uniform.