One key observation is that Al made his observation after being told that he would meet someone who made a particular observation—specifically, the first person to make that specific observation, Bob. This makes Al and Bob special in different ways:
Al is special because he has been selected to meet Bob regardless of what he observes. Therefore his data is genuinely uncorrelated with how he was selected for the meeting.
Bob is special because he has been selected to meet Al because of the specific data he observes. More precisely, because he will be the first to obtain that specific result. Therefore his result has been selected, and he is only at the meeting because he happens to be the first one to get that result.
In the original case, Bob’s result is effectively a lottery ticket—when he finds out from Al the circumstances of the meeting, he can simply follow the Natural Answer himself and conclude that his results were unlikely.
In the modified case, assuming perfect symmetry in all relevant aspects, they can conclude that an astronomically unlikely event has occurred and they have no net information about the contents of the pond.
Al is special because he has been selected to meet Bob regardless of what he observes. Therefore his data is genuinely uncorrelated with how he was selected for the meeting.
Not quite. He was selected to meet someone like Bob, in the sense that whoever the messenger was, he’d have seen 25⁄100 big. He didn’t know he’d meet Bob. But he regards the identity of the messenger as irrelevant.
You can bring out the difference by considering a variant of the case in which both Al and Bob hear about Charlie’s plan in advance. (In this variant, the first to see 25⁄100 big will visit Al.)
What is the relevance of the fact that they observed highly improbable event?
One key observation is that Al made his observation after being told that he would meet someone who made a particular observation—specifically, the first person to make that specific observation, Bob. This makes Al and Bob special in different ways:
Al is special because he has been selected to meet Bob regardless of what he observes. Therefore his data is genuinely uncorrelated with how he was selected for the meeting.
Bob is special because he has been selected to meet Al because of the specific data he observes. More precisely, because he will be the first to obtain that specific result. Therefore his result has been selected, and he is only at the meeting because he happens to be the first one to get that result.
In the original case, Bob’s result is effectively a lottery ticket—when he finds out from Al the circumstances of the meeting, he can simply follow the Natural Answer himself and conclude that his results were unlikely.
In the modified case, assuming perfect symmetry in all relevant aspects, they can conclude that an astronomically unlikely event has occurred and they have no net information about the contents of the pond.
Not quite. He was selected to meet someone like Bob, in the sense that whoever the messenger was, he’d have seen 25⁄100 big. He didn’t know he’d meet Bob. But he regards the identity of the messenger as irrelevant.
You can bring out the difference by considering a variant of the case in which both Al and Bob hear about Charlie’s plan in advance. (In this variant, the first to see 25⁄100 big will visit Al.)
What is the relevance of the fact that they observed highly improbable event?