I responded—note that this was completely spontaneous—”What on Earth do you mean? You can’t avoid assigning a probability to the mathematician making one statement or another. You’re just assuming the probability is 1, and that’s unjustified.”
Is it? We have observed the mathematician making the statement. Assuming observation matches reality, and the statement is true, the probability of the mathematician having made the statement should be 1 or close to it because it has already happened. In every world, as long as the mathematician already makes this statement, the statement being something other is not possible. This eliminates the possibility of it being girl-girl and through orthodox statistics brings us to 1⁄3 yada yada. I’ve even run a small program to test it out, and it is very close to 1⁄3.
If the mathematician has one boy and one girl, then my prior probability for her saying ‘at least one of them is a boy’ is 1⁄2 and my prior probability for her saying ‘at least one of them is a girl’ is 1⁄2. There’s no reason to believe, a priori, that the mathematician will only mention a girl if there is no possible alternative.
I have been pondering about this statement for hours on end. Assume I accept that the prior probabilities still need to be substantially considered despite the evidence, I am still confused about how there is a prior probability of 1⁄2 each for the mathematician saying that “at least one of them is a girl” and “at least one of them is a boy” (if she has a boy and a girl). Does this not assume that she can only make two statements about her state and no other? Aren’t there many other ways she could have stated this such as “I have a boy and a girl” or simply “I have two girls” and “I have two boys”? Despite our prior probabilities for statements, the last two statements make the probability of both being boys 0 or 1. This is of course assuming the mathematician does not lie.
Finally, I’d like to understand how adding the possibility of the mathematician stating at least one girl increases the possibility of both being boys, rather than decrease it.
Wait.
The reason the probability increases is that since the mathematician chooses this statement despite having two options, it is now more likely there are two boys. I see. I got the intuition but I’d like this in mathematical notation. This still does not seem to fix the problem of already having many statements to choose from, making the assumption that the prior probability of her choosing to say at least one boy 1⁄2 dubious.
But I seem to now understand the reasoning behind it in the event that if the prior for making the statement is 1⁄2, the answer is indeed 1⁄2. Though this now seems to bring to forth how far back one needs to go to reach optimal probability and how there may be so many little subtle observations in real life which substantially impact the probability of events. Very exciting!
I’d very much like to see your work for the question! This is my first comment, I apologize for its length and any folly involved which is purely my own.
Yes, I’m baffled as well. Eliezer says that the prior P(“at least one of them is a boy”|1 boy 1 girl) + P(“at least one of them is a girl”|1 boy 1 girl) = 1, which is nonsensical given that, in fact, the mathematician could have said many other things (given 1 boy 1 girl). But even if this were true, it still doesn’t tell us the probability P(“at least one of them is a boy”|two boys). Regardless of whether she has one boy or two boys, “at least one of them is a boy” is a very unusual thing to say, and it leads me to suppose that she had two children born as boys, one of whom is transgender. But how do I assign a probability to this? No idea.
If the mathematician herself had said “what is the probability that they are both boys?” it becomes more likely that she’s just posing a math problem, because she’s a mathematician… but that’s not how the question was posed, so hmm.
I am a bit confused here.
Is it? We have observed the mathematician making the statement. Assuming observation matches reality, and the statement is true, the probability of the mathematician having made the statement should be 1 or close to it because it has already happened. In every world, as long as the mathematician already makes this statement, the statement being something other is not possible. This eliminates the possibility of it being girl-girl and through orthodox statistics brings us to 1⁄3 yada yada. I’ve even run a small program to test it out, and it is very close to 1⁄3.
I have been pondering about this statement for hours on end. Assume I accept that the prior probabilities still need to be substantially considered despite the evidence, I am still confused about how there is a prior probability of 1⁄2 each for the mathematician saying that “at least one of them is a girl” and “at least one of them is a boy” (if she has a boy and a girl). Does this not assume that she can only make two statements about her state and no other? Aren’t there many other ways she could have stated this such as “I have a boy and a girl” or simply “I have two girls” and “I have two boys”? Despite our prior probabilities for statements, the last two statements make the probability of both being boys 0 or 1. This is of course assuming the mathematician does not lie.
Finally, I’d like to understand how adding the possibility of the mathematician stating at least one girl increases the possibility of both being boys, rather than decrease it.
Wait.
The reason the probability increases is that since the mathematician chooses this statement despite having two options, it is now more likely there are two boys. I see. I got the intuition but I’d like this in mathematical notation. This still does not seem to fix the problem of already having many statements to choose from, making the assumption that the prior probability of her choosing to say at least one boy 1⁄2 dubious.
But I seem to now understand the reasoning behind it in the event that if the prior for making the statement is 1⁄2, the answer is indeed 1⁄2. Though this now seems to bring to forth how far back one needs to go to reach optimal probability and how there may be so many little subtle observations in real life which substantially impact the probability of events. Very exciting!
I’d very much like to see your work for the question! This is my first comment, I apologize for its length and any folly involved which is purely my own.
Cheers.
Yes, I’m baffled as well. Eliezer says that the prior P(“at least one of them is a boy”|1 boy 1 girl) + P(“at least one of them is a girl”|1 boy 1 girl) = 1, which is nonsensical given that, in fact, the mathematician could have said many other things (given 1 boy 1 girl). But even if this were true, it still doesn’t tell us the probability P(“at least one of them is a boy”|two boys). Regardless of whether she has one boy or two boys, “at least one of them is a boy” is a very unusual thing to say, and it leads me to suppose that she had two children born as boys, one of whom is transgender. But how do I assign a probability to this? No idea.
If the mathematician herself had said “what is the probability that they are both boys?” it becomes more likely that she’s just posing a math problem, because she’s a mathematician… but that’s not how the question was posed, so hmm.