[Chinjut]

Sure, and if the quibble was along the lines of "You never explicitly said boys and girls are 50-50 distributed! You never explicitly said elder and younger childrens' birth genders are independent! You never explicitly said birth-days-of-the-week are uniformly…", then that would be fair, if pedantic.

But this "You know what Mr. Jones has told you, but you don't know under what circumstances he would have told you this" objection is objecting to some other problem than the one posed; the problem posed had nothing to do with Mr. Jones saying anything.

I understand the reason for worrying about this, because many probability riddles ARE poorly worded or presented in such a way as that this becomes an issue, but it wasn't the case here. (Note: I haven't watched rest of the video and have no comment on it; I'm just considering the wording of this individual question within it)

There was never any claim that Mr. Jones said anything, and no one was called to infer anything from any actions taken by Mr. Jones. He could be a lifelong mute. Rather, the fact that Mr. Jones has two children was presented, by an omniscient narrator, and then a counting question was asked.

(Indeed, Mr. Jones himself is completely irrelevant to the problem asked, except as a way of framing the counting question to be about two-children families. The question asked might as well have been "What proportion of two-children families with a boy born on Tuesday have girls?". It was very slightly differently worded, but not in such a way as makes "We don't know what Mr. Jones was asked!" a relevant objection.)

[adverbly]

I also fell into the same ambiguity trap, and I think that the objection about explicit wording is a fair one to make.

"What proportion of two-children families with a boy born on Tuesday have girls?" seems completely clear to me. I would have answered that question relatively quickly.

But the original question had me very confused. I felt a strong desire to ask more about the situation. A great deal of my intuition wanted to say that "well there is nothing special about Tuesday... Any boy that he has is going to be born on some day of the week, and if whatever day of the week that son is born on is included as this line item in the question, then that line item is irrelevant."

I wouldn't have fallen into that same trap in the case of the "What proportion of two-children families..." version because the "Any boy that he has is going to be born on some day of the week" logic doesn't apply.

Tuesday seemed like it might have been arbitrary in the original question, where it seems explicit in your rephrased version.

[Chinjut]

I mean, it's just as arbitrary in my rephrased version. I could just as well ask "What proportion of two-children families with a boy born on Monday have girls?". But, very well, the different wording prompted differing intuitions for you; so it goes.

[justinpombrio]

> 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.

[x0x0]

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.

[justinpombrio]

> 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.

Do other people agree?

[Chinjut]

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."

[justinpombrio]

It's the fact that it differs from 2/3. If the day of the week was not mentioned the (conventional) answer is exactly 2/3. Not 1/2.

[Chinjut]

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.