[justinpombrio]

The question is ill-posed: it does not give you enough information to tell the probability. You know what Mr. Jones has told you, but you don't know under what circumstances he would have told you this.

Suppose that you ask Mr. Jones weather he has a boy and he says yes. Then the probability that he also has a girl is 2/3.

Suppose that you asked Mr. Jones weather he had a boy born on a Tuesday, and he says yes. Then the probability that he has a girl is less than 2/3, because having two boys gives (about) double the chance for one of them to have been born on a Tuesday.

However, suppose that you asked Mr. Jones weather he has a boy, and if so what day his eldest boy was born on, and he says "yes, and on Tuesday". Then the probability that he also has a girl is again exactly 2/3.

Wikipedia has a detailed explanation: https://en.wikipedia.org/wiki/Boy_or_Girl_paradox

[Chinjut]

Everything you say after your first paragraph is correct (presuming people always answer questions with "Yes" or "No" honestly), but…

No one said anything about "Mr. Jones has told you…", here. There was nothing about asking Mr. Jones a question and him providing an answer according to some process.

Rather, the question was simply "Mr. Jones has two children. What is the probability he has a girl if he has a boy born on Tuesday?".

There are implicit conventions involved in reading this, but not particularly problematic ones. This implicitly means "Out of all families with two children, at least one of which is a boy born on Tuesday, what proportion have a girl? [Presuming that out of those families, birth gender and day of the week for the two children are all independently uniformly distributed]". And this is a straightforward counting problem.

So the wording seems fine and the problem well-posed to me.

[ubernostrum]

The problem really is almost one of metagaming.

People who are reading this are likely to have seen, for example, questions which read as if they're asking for a conditional probability ("John is male, 33 years old, and has a degree in English literature; what is the probability he works as a barista?") but are designed to let the questioner turn around and say "Ah-HA! I got you! It was really a question about the base rate (in this case, of baristas)!".

As posed and with knowledge of that issue, this question reads like an attempt to do the opposite: to pose a question which seems like it's asking about the base rate of boys vs. girls, but then the questioner turns around with "Ah-HA! I got you! It was really a question about the conditional probability!"

Once it's phrased in a way that makes explicit that it really is a question about conditional probability, and not an attempt to lure someone into a base-rate trap, there's no paradox.

Complicating things is that analyses usually focus on the day of the week as the crucial factor, when it's easier to get to an intuitive understanding of the probability via dealing with the day-of-week first and then focusing on the small but crucial change that comes from knowing the gender of one of the children. After accounting for day-of-week you are left with 28 equally-probable situations, with at least one girl in 14 of them, for the expected 1/2. Then the fact that you end up at a probability just over 1/2 is due to the elimination of the case in which both children are girls (since we know at least one is a boy), which pushes the final result to 13/27 in favor of the second child being a boy.

[Retric]

There are implicit conventions involved in reading this

Explicitly the question adds no such limits. So, abstractly someone could be asking the question without those limits.

It's like the difference between infinity and how whatever subset of math you work in defines infinity. And yes there are more than one commonly used definition.

[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?

[jwilk]

Following classical probability arguments, we consider a large urn containing two children.

:-O