I was asked the question “a man states he has two children, and at least one of them are boys, what are the odds that the man has two boys?” I’ve been told the answer is 1/3, but I can’t wrap my head around it. Additionally, there is another version of the problem that states he has at least one boy born on a Tuesday. How does that change the odds? Why?
Edited to add (so people don’t have to sort through replies): the answer is 1/3 because “at least one boy” is accounting for B/G & G/B. The girl can be the first or second child. You can move the odds to 50/50 by rewording the question to “my first of two children is a boy, what are the odds the other child is a boy”
In: Mathematics
TL;DR at the bottom.
So.
I was confident in my answer of 1/3, because this is what is the typical mathematical answer. It lines up with the questions and answers you would get in this field. But I admit, that it is not the sole correct answer in the sense, that the question itself might be ambiguous, if we disregard some reasonable assumptions.
I recommend reading though the [wiki page](https://en.m.wikipedia.org/wiki/Boy_or_girl_paradox). If you don’t get something, try to keep on reading to see if you understand it after getting some more details. I also recommend reading at least some of the sources (and I mean published papers) to get some insight, it helped to clear up some things for me.
The answer depends on assumptions that some make, and some might not.
We assume:
* the father knows the sex of both kids (reasonable)
* the father doesn’t lie about the sex of his kids
* the options are only boy or girl, with a 50-50% probability for each child independently
* if the father has at least a boy, he will say he has at least a boy
Each of these assumptions are required for the answer to stay 1/3.
1) The person that gives the statement only knows the sex of one kid.
In the case of a wording with a father, it mostly makes sense that they know both of their kids’ sex. But if a bunch of two-kid families are randomly selected, then one of their kids are randomly checked if they are boy or girl, the probabilities get tilted back to 1/2. (For details see the wiki page’s Analysis of the Ambiguity section.)
2) The father lies about his kids
This obviously invalidates the “trick” in the question, since if we can’t exclude one of the four options from the BB/BG/GB/GG set, the probability of both kids being boys will depend on the probability of the dad lying about his kids.
3) We don’t live in a “perfect” world
In reality, the probability of a male or female baby being born is not equal, and we can’t necessarily disregard eg. intersex babies either. These all would influence the answer:
eg. 105 boys (50.7%), 100 girls (48.3%), 2 intersex (1.0%) babies would result in a somewhat different probability
4) The father is *not* biased to the extreme
We assume that the dad of GB/BG kids will *always* say that he has at least a boy. If we don’t make this assumption, then we have to factor in, that eg. only half the time does a father say that he has at least a boy, while at other occasions he says he has at least a girl.
Here this probability will again heavily influence what exactly the result is. For example in a 50-50 scenario, we can adjust the probabilities to the following:
BB – 1
BG – 0.5
GB – 0.5
GG – 0
The eliminate the GG option, and we reach 1/(1+0.5+0.5) = 1/2 as the answer.
The above list might not be exhaustive and complete, but it has some examples of how we can divert the answer from the intended 1/3. “Diverting” here is the key, since getting to answers other than 1/3 requires you to go out of your way to “misunderstand” the task. (And that is not what most of the comments saying 1/2 are doing.)
For the Tuesday question, it is also counterintuitive, and can be ambiguous in how the data was acquired.
For the generally correct answers (that a university course’s prof would almost definitely accept), I recommend reading The Guardian’s [article](https://www.theguardian.com/science/2019/nov/18/did-you-solve-it-the-two-child-problem) on it. (For anybody faced with a paywall, here is an [archived version](https://web.archive.org/web/20230821020730/https://www.theguardian.com/science/2019/nov/18/did-you-solve-it-the-two-child-problem) of it. Haven’t read this one, but skimming through the text, it seemed like the contents are the same as on the live page currently.)
TL;DR
Good answers are 1/3 (og question), and 13/27 (if we are given the day of birth of “the” boy), but due to the ambiguity of the wording, we can argue that you could turn it into basically any percentage depending on assumed context.
Sources: [wiki](https://en.m.wikipedia.org/wiki/Boy_or_girl_paradox) (and its sources), [The Guardian](https://web.archive.org/web/20230821020730/https://www.theguardian.com/science/2019/nov/18/did-you-solve-it-the-two-child-problem)
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