Let me try to explain why this is the wrong answer.
Let's assume there are 100 people undergoing this experiment. Half of them will flip heads, half tails. The half that flip heads will be woken up a total of 50 times. The half that flip tails will be woken up a total of 100 times.
So, we have a total of 150 wake-ups. 100 of those came from tails, and 50 from heads. So, if you're woken up and have no prior knowledge, you have a 100/150 = 2/3 chance that you flipped tails, and a 50/150 = 1/3 chance that you flipped heads.
Put another way, when you are woken up, it could be one of three cases:
-- Awakened on the only time for heads
-- Awakened on the first time for tails
-- Awakened on the second time for tails
Two of those cases correspond to tails, one to heads. So, it's twice as likely that you are being awakened due to a tails flip.
Let's assume there are 100 people undergoing this experiment. Half of them will flip heads, half tails. The half that flip heads will be woken up a total of 50 times. The half that flip tails will be woken up a total of 100 times.
So, we have a total of 150 wake-ups. 100 of those came from tails, and 50 from heads. So, if you're woken up and have no prior knowledge, you have a 100/150 = 2/3 chance that you flipped tails, and a 50/150 = 1/3 chance that you flipped heads.
Put another way, when you are woken up, it could be one of three cases:
-- Awakened on the only time for heads -- Awakened on the first time for tails -- Awakened on the second time for tails
Two of those cases correspond to tails, one to heads. So, it's twice as likely that you are being awakened due to a tails flip.