There is a difference between P(“Heads came up”) and P(“Heads came up” given that “I was just woken up”). Since you will be woken up (memory-less) multiple times if tails came up, the fact that you are just getting woken up gives you information and increases the probability that tails came up.
Let’s consider P(H | JustWoken) = P(H and Monday | JustWoken) + P(H and Tuesday | JustWoken)
Because I have no information about the scientist’s behavior (when he chooses to ask the question), I have to assign equal probabilities (one third) to P(H and Monday | JustWoken), P(T and Monday | JustWoken) and P(T and Tuesday | JustWoken). And it’s impossible to be woken up on Tuesday if Heads came up, so P(H and Tuesday | JustWoken) = 0.
In result, P(H | JustWoken) = 1⁄3.
If anyone doubts that, we could set up a computer simulation (you write the scientist and coin code and I write code for the beauty answering the question) and we bet. But I would require an experimental condition, stating that the scientist will ask the beauty the question every time she wakes up. Under those conditions, a beauty which always bets that “tails came up” any time she gets woken up will win 2⁄3 of the time.
If we could not agree to those conditions (getting interviewed by the scientist on every occasion), the bet would be broken because you know what answer I will give and you have information that I don’t have (strategy for when to interview).
I think the solution to the problem depends on what you want to measure. The probability of being tails per wakening is not the same as the probability of being tails per flip or per day.
There is a difference between P(“Heads came up”) and P(“Heads came up” given that “I was just woken up”). Since you will be woken up (memory-less) multiple times if tails came up, the fact that you are just getting woken up gives you information and increases the probability that tails came up.
Let’s consider P(H | JustWoken) = P(H and Monday | JustWoken) + P(H and Tuesday | JustWoken) Because I have no information about the scientist’s behavior (when he chooses to ask the question), I have to assign equal probabilities (one third) to P(H and Monday | JustWoken), P(T and Monday | JustWoken) and P(T and Tuesday | JustWoken). And it’s impossible to be woken up on Tuesday if Heads came up, so P(H and Tuesday | JustWoken) = 0. In result, P(H | JustWoken) = 1⁄3.
If anyone doubts that, we could set up a computer simulation (you write the scientist and coin code and I write code for the beauty answering the question) and we bet. But I would require an experimental condition, stating that the scientist will ask the beauty the question every time she wakes up. Under those conditions, a beauty which always bets that “tails came up” any time she gets woken up will win 2⁄3 of the time. If we could not agree to those conditions (getting interviewed by the scientist on every occasion), the bet would be broken because you know what answer I will give and you have information that I don’t have (strategy for when to interview).
I think the solution to the problem depends on what you want to measure. The probability of being tails per wakening is not the same as the probability of being tails per flip or per day.