"Gardner himself tripped up on his simpler Two Children Problem. Initially, he gave the answer as 1/3, but he later realized that the problem is ambiguous in the same way that Peres argues that the Tuesday Birthday Problem is. Suppose that you already knew that Mr. Smith had two children, and then you meet him on the street with a boy he introduces as his son. In that case, the probability the other child is a son would be 1/2, just as intuition suggests."
is the difference between taking "one is a boy" to mean "at least one boy, can't tell you which" and "a specified boy." I'm kind of surprised it ever took Gardner any time to realize that distinction existed.
However, that's not the same ambiguity as the one that determines whether or not the information about the birth day can be informative.
The information about the birth day can only lead to the 13/27 answer if the speaker only would have told you the birth day if it were Tuesday. If the speaker might have told you the birth day was Wednesday if that were the case, the answer is no longer 13/27. To get an answer of 13/27, you have to imagine the speaker answered the question "Was at least one child a boy born on a Tuesday?" or something similar. This much is clear from the article.
In contrast, "at least one is a boy" can be informative without knowing what prompted the speaker to say it. It is still informative even if the speaker might have said under other circumstances "At least one is a girl."
That means that this paragraph is, at best, unclear:
"Everything depends, he points out, on why I decided to tell you about the Tuesday-birthday-boy. If I specifically selected him because he was a boy born on Tuesday (and if I would have kept quiet had neither of my children qualified), then the 13/27 probability is correct. But if I randomly chose one of my two children to describe and then reported the child’s sex and birthday, and he just happened to be a boy born on Tuesday, then intuition prevails: The probability that the other child will be a boy will indeed be 1/2."
Specifying "if I randomly chose one of my two children" does indeed necessitate that the probability the other one is a boy is 1/2, but it misses a possibility. That possibility is that the speaker is reporting not on a specific child, but reporting the gender of at least one child. In that case, the probability is 1/3, unless the the speaker also responding to a prompt like "is at least one of your children a boy born on a Tuesday?" The information about the birth day and the information about gender do not have the same epistemological status in the problem. The informativeness of the day of birth is totally dependent on what prompted the utterance. The information about the gender of at least one child is not. (I have code that demonstrates this in a simulation; I wasn't confident on the point until I had proved it to myself.)
Whether one can accept an answer of 1/3 does depend on an additional assumption necessary for full specification of the problem: that the speaker would be equally likely to report "at least one of my children is an X" in all cases it's true.
Slashdot Mirror
"Gardner himself tripped up on his simpler Two Children Problem. Initially, he gave the answer as 1/3, but he later realized that the problem is ambiguous in the same way that Peres argues that the Tuesday Birthday Problem is. Suppose that you already knew that Mr. Smith had two children, and then you meet him on the street with a boy he introduces as his son. In that case, the probability the other child is a son would be 1/2, just as intuition suggests."
is the difference between taking "one is a boy" to mean "at least one boy, can't tell you which" and "a specified boy." I'm kind of surprised it ever took Gardner any time to realize that distinction existed.
However, that's not the same ambiguity as the one that determines whether or not the information about the birth day can be informative.
The information about the birth day can only lead to the 13/27 answer if the speaker only would have told you the birth day if it were Tuesday. If the speaker might have told you the birth day was Wednesday if that were the case, the answer is no longer 13/27. To get an answer of 13/27, you have to imagine the speaker answered the question "Was at least one child a boy born on a Tuesday?" or something similar. This much is clear from the article.
In contrast, "at least one is a boy" can be informative without knowing what prompted the speaker to say it. It is still informative even if the speaker might have said under other circumstances "At least one is a girl."
That means that this paragraph is, at best, unclear:
"Everything depends, he points out, on why I decided to tell you about the Tuesday-birthday-boy. If I specifically selected him because he was a boy born on Tuesday (and if I would have kept quiet had neither of my children qualified), then the 13/27 probability is correct. But if I randomly chose one of my two children to describe and then reported the child’s sex and birthday, and he just happened to be a boy born on Tuesday, then intuition prevails: The probability that the other child will be a boy will indeed be 1/2."
Specifying "if I randomly chose one of my two children" does indeed necessitate that the probability the other one is a boy is 1/2, but it misses a possibility. That possibility is that the speaker is reporting not on a specific child, but reporting the gender of at least one child. In that case, the probability is 1/3, unless the the speaker also responding to a prompt like "is at least one of your children a boy born on a Tuesday?" The information about the birth day and the information about gender do not have the same epistemological status in the problem. The informativeness of the day of birth is totally dependent on what prompted the utterance. The information about the gender of at least one child is not. (I have code that demonstrates this in a simulation; I wasn't confident on the point until I had proved it to myself.)
Whether one can accept an answer of 1/3 does depend on an additional assumption necessary for full specification of the problem: that the speaker would be equally likely to report "at least one of my children is an X" in all cases it's true.