> Rather, the fact that he had two children was presented, by an omniscient narrator.
That the narrator is omniscient doesn't change anything. The question still remains: under what circumstances would the narrator have told you, e.g., that "he has a boy born on Tuesday" vs. "he has a girl born on Tuesday". Perhaps this omniscient narrator really likes girls, in which case they would tell you about a girl if Mr. Jones had any girls. Then since they told you "Mr. Jones has a boy born on Tuesday", you know definitely that Mr. Jones has no girls.
Ignoring the source of your knowledge doesn't make that source any less important. And the standard convention you're talking about corresponds to a source of knowledge where you ask a yes/no question and get a yes, which is frequently unrealistic. This is why it disagrees with people's intuition, and this problem is called a paradox.
As a probability problem with the standard assumptions, it's a well defined question. If you saw this in Bertsekas or Sheldon Ross, the sampling would be clear.
And I also think you're incorrect about why it's a paradox. People are just bad at understanding and estimating things in conditional probabilities. Further, the answer changes based on the sampling regime, which (as mentioned) was not explicitly stated but is clear to almost any student that's taken a discrete probability class.
> And I also think you're incorrect about why it's a paradox. People are just bad at understanding and estimating things in conditional probabilities.
This is a testable prediction. I predict that making the source of your knowledge explicit eliminates the paradox.
To me, it feels strange that "the probability that Mr. Jones has a girl given that he has a boy born on Tuesday" is ~1/2. However, it feels normal that "You ask Mr. Jones weather he has a boy born on Tuesday, and he says yes. What is the probability that he has a girl?" is ~1/2.
It's not that the probability is close to 1/2 that makes it paradoxical for most people. It's that the probability differs from 1/2 at all. As in the OP of this very thread saying "Somehow knowing the day of the week the boy was born changes the result. It's completely bizarre."
Yes, that's also "paradoxical", though probably not the paradox that would trip people up first unless they'd seen the other problem first. But, you're right that I may have misread which departure from expected answer was bugging the OP. Nonetheless, everything else I stated still holds.
That the narrator is omniscient doesn't change anything. The question still remains: under what circumstances would the narrator have told you, e.g., that "he has a boy born on Tuesday" vs. "he has a girl born on Tuesday". Perhaps this omniscient narrator really likes girls, in which case they would tell you about a girl if Mr. Jones had any girls. Then since they told you "Mr. Jones has a boy born on Tuesday", you know definitely that Mr. Jones has no girls.
Ignoring the source of your knowledge doesn't make that source any less important. And the standard convention you're talking about corresponds to a source of knowledge where you ask a yes/no question and get a yes, which is frequently unrealistic. This is why it disagrees with people's intuition, and this problem is called a paradox.