If the memory wipe is the only fantastic aspect of this situation, then when the child you see when you wake says they were born on Tuesday (and I assume you know that both children will always say what day they were born on after you wake up), you should consider the probability that the other was also born on Tuesday to be 1⁄7. The existence of another wakening, which will of course be different in many respects from this one (e.g., the location of dust specs on the mirror in the room), is irrelevant, since you can’t remember it (or it hasn’t occurred yet).
I’ve no idea what you mean by “guessing only when you met a boy born on Tuesday”. Guessing what? Or do you mean you are precommitted to not thinking about what the probability of both being born on the same day is if the boy doesn’t say Tuesday? (Could you even do that? I assume you’ve heard the joke about the mother who promises a child a cookie if he doesn’t think of elephants in the next ten minutes...) I think you may be following some strange version of probability or decision theory that I’ve never heard of....
“Or do you mean you are precommitted to not thinking about what the probability of both being born on the same day is if the boy doesn’t say Tuesday?”—exactly. In these kinds of scenarios we need to define our reference class and then we calculate the probability for someone in this class. For example, in anthropic problems there’s often debate about whether our reference class should include all sentient beings or all humans or all humans with a certain level of intellectual ability. Similarly, the question here is whether our reference class is all agents who encounter a boy born on a Tuesday on at least one day or all agents who encounter a boy. I see the second as much more useful, unless you’ll only be offered an option if at least one boy was born on a Tuesday.
“You should consider the probability that the other was also born on Tuesday to be 1/7”—exactly!
My point is that you only get the 1⁄13 answer when you pre-commit to guessing when you wake up and the boy tells you that you were born on Tuesday. Further this involves collapsing guessing twice as though you’d only guessed once and abstaining in the majority of cases when you wake up the second time (or the abstaining on Monday if you guessed on Tuesday). The number of scenarios where you care about this is vanishingly small. Similarly, we shouldn’t be conditioning on the sequences you observe when you wake up.
If the memory wipe is the only fantastic aspect of this situation, then when the child you see when you wake says they were born on Tuesday (and I assume you know that both children will always say what day they were born on after you wake up), you should consider the probability that the other was also born on Tuesday to be 1⁄7. The existence of another wakening, which will of course be different in many respects from this one (e.g., the location of dust specs on the mirror in the room), is irrelevant, since you can’t remember it (or it hasn’t occurred yet).
I’ve no idea what you mean by “guessing only when you met a boy born on Tuesday”. Guessing what? Or do you mean you are precommitted to not thinking about what the probability of both being born on the same day is if the boy doesn’t say Tuesday? (Could you even do that? I assume you’ve heard the joke about the mother who promises a child a cookie if he doesn’t think of elephants in the next ten minutes...) I think you may be following some strange version of probability or decision theory that I’ve never heard of....
“Or do you mean you are precommitted to not thinking about what the probability of both being born on the same day is if the boy doesn’t say Tuesday?”—exactly. In these kinds of scenarios we need to define our reference class and then we calculate the probability for someone in this class. For example, in anthropic problems there’s often debate about whether our reference class should include all sentient beings or all humans or all humans with a certain level of intellectual ability. Similarly, the question here is whether our reference class is all agents who encounter a boy born on a Tuesday on at least one day or all agents who encounter a boy. I see the second as much more useful, unless you’ll only be offered an option if at least one boy was born on a Tuesday.
“You should consider the probability that the other was also born on Tuesday to be 1/7”—exactly!
My point is that you only get the 1⁄13 answer when you pre-commit to guessing when you wake up and the boy tells you that you were born on Tuesday. Further this involves collapsing guessing twice as though you’d only guessed once and abstaining in the majority of cases when you wake up the second time (or the abstaining on Monday if you guessed on Tuesday). The number of scenarios where you care about this is vanishingly small. Similarly, we shouldn’t be conditioning on the sequences you observe when you wake up.