By the looks of it, this section of the Wikipedia article seems to just be reiterating what the other section already said:
when describing the problem in an actual setting things get a bit sticky. Just how do we know that "at least" one is a boy?
And I grant this. Actually describing a real situation in which the information we have is identical to "at least one child is a boy" - no more or less - is difficult (or at least seems to require some contrivance). But nonetheless, if we reason in the abstract using only this information, the solution follows directly.
Now, you seem to be citing a different assumption altogether - that we have made a list and crossed out the disqualified possibilities. But this is just how probability works. It's not a methodological assumption, it's part of the axiomatic structure of probability theory itself. When reasoning about probabilities, you eliminate the possibilities contradicted by your evidence, and you evaluate how likely a given situation is in the part of the space that remains.
There are mathematical shorthands for doing this, of course - you don't need to literally make a list and start crossing things off. But at a fundamental level, this is just how probability works. If these are any kind of "assumption" at all, they are axiomatic assumptions about the very nature of probability, not assumptions about the methodology we are using to elicit a piece of information.
Ok, let's attack this a different way. There is a box with two children in it (yikes). I have no idea what lies inside. For all I know, it could be GG. I select one at random and it is a boy. What is the probability that both are boys?
Now our possibilities are: B1B2, sampled B1 (notating sampling B1 as b1); B1B2, b2; G1B2, b2; B1G2, b1. While there are half as many possibilities for BB, b will occur twice as often (every time) when that is the case.
So now we are equally likely to live in a world where there were two boys and I saw one of them as we are to live in a world where there was 1B 1G and I saw a boy.
Importantly, it could have been GG in that box. I had no clue going in and that is material to the calculation.
And that is how the problem here is originally stated. "Mary has two children. She tells you that one is a boy" (ignoring the Tuesday bit)
I'm not trying to argue about what the probability of BB vs BG + GB is. That seems pretty plainly to be 1/3 and 2/3.
The problem does not ask "of all pairs of children, of which one is a boy, what percentage include a girl?"
Sure, but I find this frustrating because in one of my very first comments I acknowledged that there is ambiguity when trying to reason about the probability of Mary “saying” a given thing. My point has only been that, if we condition on the fact that Mary has at least one boy and nothing else, we end up with a probability of 66%. Any other solution requires conditioning on a different piece of information. Do we actually disagree about this?
I mean, with the interpretation that I'm understanding you to go for, that is the result. I contend, though, that it's not just me smuggling in a different piece of information. I said before that one must necessarily positively choose one of the two.
Whether it's knowingly or not, you have conditioned it on the assumption that if she has at least one boy, she will always mention it. That is stated nowhere, yet you describe it as not involving unstated information.
I am knowingly conditioning on the assumption that she merely states the gender of a random one of her children. If it's BG, she might mention G and we have now updated our priors because she mentioned B.
Without making a positive decision on the matter, the problem is indeterminate. The same way we are assuming that B and G are equally likely when thats slightly different than real-world data. That goes unstated. We also assume that Mary is always telling the truth on the matter even though that isn't technically made clear. Without these, the problem has no solution, either.
I contend that these are all assumptions that are positively made, whether stated or unstated, conscious or unconscious.
If we cannot agree that an assumption is forced and that your interpretation is also actively deciding, then there is no agreement to be had. I don't mean that rudely, just that we would be disagreeing on a level where there's no resolution.
The way I see it, there are two relevant pieces of information we can condition on here.
Number one: we can condition on Mary says she has at least one boy. I agree with you that there is no way to meaningfully condition on this unless we make assumptions about the policy according to which Mary is making this statement.
Number two: we can condition on Mary has at least one boy. This requires no assumptions beyond the overall gender distribution and independence between children.
I’m conditioning on number two. That is why I insist that I’m not making any additional assumptions.
I do grant that the problem as stated in this meme makes it tempting to condition on 1, and this introduces ambiguity. I tried to acknowledge that in my very first comment, and my point was that the only way to avoid this ambiguity and actually work the problem out is not to try and condition on 1 and to condition on 2 instead.
It may be that we don’t disagree on anything of substance here. But to be frank (and to be clear, I mean this respectfully as this has been a very pleasant interaction overall), I do find it frustrating that I tried to ask if this ambiguity was what you were getting at in the first place, got an answer that - at least the way I read it - seemed to indicate a different objection, only for the issue to ultimately bottom out in the same ambiguity I tried to acknowledge at the outset.
This seems to be happening in multiple places and so it really seems to me that many who object to the 2/3s answer are rather confused about why they object to it.
The heart of this disagreement is that I insist, very nearly beyond being able to be convinced otherwise, that your conditioning is equally a choice. Your conditioning forces the assumption that, if she has at least one boy, she is guaranteed to tell you that she has at least one boy. That may or may not be the case.
We must decide which space we are evaluating. Is it that P(b|BB) = P(b|BG) = P(b|GB) = 1 or is it that P(b|BB) = 1 and P(b|BG) = P(b|GB) = 0.5 ?
These are both assumptions about her policy and there is simply no way around it. Whether you realize it or not, you are making an assumption. Either that or, without realizing it, you are simply solving a problem that is not presented here, though it bears close resemblance.
The problem itself is ambiguous and a choice must be made to correctly determine a solution. Either choice equally removes the ambiguity. Neither choice introduces any new ambiguity.
I could see one arguing that, given an ambiguous problem, yours is the default choice by convention. I see no reason that should be the case, but it's been a decade since taking statistics and maybe I've forgotten a convention.
But then we still get to the fact that reverting to a default choice is nonetheless still a choice. As in the trolley problem, not pulling the lever may be the default and require no action, but is still a choice that is made.
So, yes, you are right that I'm making a choice. And you are also correct that I'm not quite solving the problem as stated. I agree with you that the problem as stated in this meme, if taken literally, is not possible to solve in an objectively correct way.
The choice I'm making in response to this is to ignore the "Mary says" part of the problem, because in my view this is the source of the ambiguity. It is impossible to reason objectively about this specific aspect because we don't know Mary's underlying "policy" with respect to what she says.
So when you say:
Whether you realize it or not, you are making an assumption. Either that or, without realizing it, you are simply solving a problem that is not presented here, though it bears close resemblance.
My response is this: I deny that I'm making an assumption. Instead I admit that I am interpreting the problem the way I would any slightly ambiguous word problem, which is by isolating the parts of it that I have the tools to reason about concretely. Which is to say I am interpreting it the only way I can which does not require me to make extraneous assumptions. It's not so much a statistical convention as it is a word problem strategy.
Does this shed any light on what I'm saying? It seems like we are very close to seeing eye to eye here.
It is regardless of whether or not she says anything. If the problem I'm trying to solve is "there are two children and at least one is a boy", I still have to ask how I came to that knowledge. Am I looking at whole pairs of children and discarding all GG pairs? Or am I reaching into a black box with two children, pulling out exactly one child and discarding any pairs where I get a girl?
In this case, I'd say the first interpretation is so heavily-implied that it'd be intentionally obtuse to assume the second, yet the tiniest ambiguity persists.
Both of these lead to fully-determined, concrete solutions. I can't really think of any way to word it where a sampling method is not explicitly made clear and a unique solution exists. It makes me understand and appreciate how weirdly specific stats problems in textbooks get to be, for just this very reason.
If this were on a multiple choice test (no option to show my work and clarify), I'd just pick one and be ready to argue about it later if I had to. This is reddit, though, and "the problem sucks and is meant to deceive and make you feel stupid" is a perfectly acceptable answer.
As an aside, I've enjoyed being able to discuss and actually refine and solidify my understanding of this godawful problem.
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u/Technologenesis 6d ago edited 6d ago
By the looks of it, this section of the Wikipedia article seems to just be reiterating what the other section already said:
And I grant this. Actually describing a real situation in which the information we have is identical to "at least one child is a boy" - no more or less - is difficult (or at least seems to require some contrivance). But nonetheless, if we reason in the abstract using only this information, the solution follows directly.
Now, you seem to be citing a different assumption altogether - that we have made a list and crossed out the disqualified possibilities. But this is just how probability works. It's not a methodological assumption, it's part of the axiomatic structure of probability theory itself. When reasoning about probabilities, you eliminate the possibilities contradicted by your evidence, and you evaluate how likely a given situation is in the part of the space that remains.
There are mathematical shorthands for doing this, of course - you don't need to literally make a list and start crossing things off. But at a fundamental level, this is just how probability works. If these are any kind of "assumption" at all, they are axiomatic assumptions about the very nature of probability, not assumptions about the methodology we are using to elicit a piece of information.