I have two children, at least one of whom is a boy born on a day that I’ll tell you in 5 minutes.
“[A] boy born on a day that I’ll tell you in 5 minutes” is ambiguous. There are two possible meanings, yielding different answers.
If “a boy born on a day that I’ll tell you in 5 minutes” means “a boy, and I’ll tell you the name of a boy I have in 5 minutes” then the answer is 1⁄3 as Liron says.
However, if “a boy born on a day that I’ll tell you in 5 minutes” means “a boy born on a particular singular day that I just wrote down on this piece of paper and will show you in 5 minutes”, then this is equivalent to saying “a boy born on a Tuesday” and the answer is 13⁄27.
The reason why the second meaning is equivalent to “a boy born on a Tuesday” is because it’s a statement that at least one of the children is a particular kind of boy that only 1/7th of boys are, just like how “a boy born on a Tuesday” is a statement that at least one of the children is a particular kind of boy that only 1/7th of boys are. (Conversely, for the first interpretation: “a boy born on a day that I’ll tell you in 5 minutes” is a statement that at least one of the children is a a boy, period.)
Another way to notice the difference if it’s still not clear:
When told “I have two children, at least one of whom is [a boy born on a particular singular day that I just wrote down on this piece of paper and will show you in 5 minutes]”, you assign a 1/7th credence to the paper showing Sunday, 1/7th to Monday, 1/7th to Tuesday, etc.
Then, conditional on the paper showing Tuesday, you know that the parent just told you “I have two children, at least one of whom is [a boy born on [Tuesday and I will show you the paper showing Tuesday in 5 minutes]]”, which is equivalent to the parent saying “I have two children, at least one of whom is a boy born on Tuesday”.
So you then have a 1/7th credence that the paper shows Tuesday, and if it’s Tuesday, your credence that both children are boys is 13⁄27. So your overall credence, reflecting your uncertainty about what day the paper shows is (13/27)*(1/7)+(13/27)*(1/7)+(13/27)*(1/7)+(13/27)*(1/7)+(13/27)*(1/7)+(13/27)*(1/7)+(13/27)*(1/7)=13/27.
“[A] boy born on a day that I’ll tell you in 5 minutes” is ambiguous. There are two possible meanings, yielding different answers.
If “a boy born on a day that I’ll tell you in 5 minutes” means “a boy, and I’ll tell you the name of a boy I have in 5 minutes” then the answer is 1⁄3 as Liron says.
However, if “a boy born on a day that I’ll tell you in 5 minutes” means “a boy born on a particular singular day that I just wrote down on this piece of paper and will show you in 5 minutes”, then this is equivalent to saying “a boy born on a Tuesday” and the answer is 13⁄27.
The reason why the second meaning is equivalent to “a boy born on a Tuesday” is because it’s a statement that at least one of the children is a particular kind of boy that only 1/7th of boys are, just like how “a boy born on a Tuesday” is a statement that at least one of the children is a particular kind of boy that only 1/7th of boys are. (Conversely, for the first interpretation: “a boy born on a day that I’ll tell you in 5 minutes” is a statement that at least one of the children is a a boy, period.)
Another way to notice the difference if it’s still not clear:
When told “I have two children, at least one of whom is [a boy born on a particular singular day that I just wrote down on this piece of paper and will show you in 5 minutes]”, you assign a 1/7th credence to the paper showing Sunday, 1/7th to Monday, 1/7th to Tuesday, etc.
Then, conditional on the paper showing Tuesday, you know that the parent just told you “I have two children, at least one of whom is [a boy born on [Tuesday and I will show you the paper showing Tuesday in 5 minutes]]”, which is equivalent to the parent saying “I have two children, at least one of whom is a boy born on Tuesday”.
So you then have a 1/7th credence that the paper shows Tuesday, and if it’s Tuesday, your credence that both children are boys is 13⁄27. So your overall credence, reflecting your uncertainty about what day the paper shows is (13/27)*(1/7)+(13/27)*(1/7)+(13/27)*(1/7)+(13/27)*(1/7)+(13/27)*(1/7)+(13/27)*(1/7)+(13/27)*(1/7)=13/27.