I don’t think we’re in disagreement here. The reason why I said the “usual answer” is 1⁄13 instead of writing the “answer” is 1⁄13 is that there are disputes about what the question is asking as you’ve pointed out. I also noted the 1⁄7 directly below. But I definitely could have been clearer—the answer can be 1⁄7 or 1⁄13 depending on the interpretation.
As I said, I’m not sure what point you’re trying to make, but if updating from 1⁄7 to 1⁄13 on any of the statements “at least one was born on Tuesday”, “at least one was born on Wednesday”, etc. is part of the point, then I don’t see any model of what you are told for which that is the case.
Maybe this will make it easier. Suppose you meet the first son on Monday and then the second on Tuesday. Your memory is wiped in between. You wake up not knowing the day and the child tells you that they were born on a Tuesday. What are the odds that both were born on Tuesday?
If you pre-committed to only guessing when you met a boy born on a Tuesday, then on average we’d expect you to guess at least once 13⁄49 times and 1⁄49 would have both born on a Tuesday. My point is that this would be an extremely weird way to behave and the proposal of updating on all relevant information is similarly weird.
Suppose you wake up and immediately observe 111, it doesn’t make sense to calculate the odds of heads given:
a) The following events occurred at least once: Wake up, 1, 1, 1
Instead of the odds of heads given:
b) A random wakeup was: Wake up, 1, 1, 1
The only time you’d care about a) is if you pre-committed only guess upon seeing that particular sequence, which would be just as weird as above.
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.
I don’t think we’re in disagreement here. The reason why I said the “usual answer” is 1⁄13 instead of writing the “answer” is 1⁄13 is that there are disputes about what the question is asking as you’ve pointed out. I also noted the 1⁄7 directly below. But I definitely could have been clearer—the answer can be 1⁄7 or 1⁄13 depending on the interpretation.
As I said, I’m not sure what point you’re trying to make, but if updating from 1⁄7 to 1⁄13 on any of the statements “at least one was born on Tuesday”, “at least one was born on Wednesday”, etc. is part of the point, then I don’t see any model of what you are told for which that is the case.
Maybe this will make it easier. Suppose you meet the first son on Monday and then the second on Tuesday. Your memory is wiped in between. You wake up not knowing the day and the child tells you that they were born on a Tuesday. What are the odds that both were born on Tuesday?
If you pre-committed to only guessing when you met a boy born on a Tuesday, then on average we’d expect you to guess at least once 13⁄49 times and 1⁄49 would have both born on a Tuesday. My point is that this would be an extremely weird way to behave and the proposal of updating on all relevant information is similarly weird.
Suppose you wake up and immediately observe 111, it doesn’t make sense to calculate the odds of heads given:
a) The following events occurred at least once: Wake up, 1, 1, 1
Instead of the odds of heads given:
b) A random wakeup was: Wake up, 1, 1, 1
The only time you’d care about a) is if you pre-committed only guess upon seeing that particular sequence, which would be just as weird as above.
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.