Is there ambiguous wording in vos Savant's statement of the boy or girl paradox, and if so in what way?

Question

Is there ambiguous wording in vos Savant's statement of the boy or girl paradox?

Say that a woman and a man (who are unrelated) each have two children. We know that at least one of the woman's children is a boy and that the man's oldest child is a boy

She is asked what are the chances of the other child being a boy? I am fairly sure the standard two ways of interpreting the boy or girl paradox both still work.

Specifically, that two different procedures for determining that "at least one is a boy" could lead to the exact same wording of the problem. But they lead to different correct answers:

From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of ⁠ 1 / 3 ⁠.

From all families with two children, one child is selected at random, and the sex of that child is specified to be a boy. This would yield an answer of ⁠ 1 / 2 ⁠.[3][4]

Am I wrong: is only one answer right?

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As I understand it, from having thought for a while, that's all, in one solution "at least one boy" specifies one of the children. For what it's worth, I feel that if the language of the question is ambiguous, then it resides in the determiner 'the' before 'woman', whether it refers to a specific woman or not, as if it does then you specified one of her children to be a boy (rather than selected from unspecified two child families with at least one boy).

Answer 1

You are correct. You can get 1/3 or 1/2 depending on exactly how the question is specified, or indeed other values. To get the answer 1/3 to the boy-or-girl paradox you have to word the question very carefully, and most popular accounts of the problem fail to do so.

To get a single definite answer, we have to put a value to the probability that we come to know that one of the children is a boy given that at least one of them is a boy. Suppose we ask the woman, "Is it the case that at least one of your children is a boy?" and she answers, "Yes". This rules out two girls, but it leaves the other three possibilities equiprobable, so we have the 1/3 answer.

Suppose we ask the man, "Tell us the gender of your elder child," and he answers, "Boy". This rules out girl>girl and girl>boy so it leaves boy>girl or boy>boy and so we have 1/2.

Now suppose instead we ask the man, "Tell us the gender of one of your children," and he answers, "Boy". Or suppose that a propos of nothing the man just volunteers the information that one of his children is a boy. If he has one boy and one girl and he did a mental coin flip to decide which answer to give, then we get the 1/2 answer. But we may have no idea how he came to give his answer, so the probability of two boys could be anything between 1/3 and 1.

The problem with saying, "At least one of the woman's children is a boy," is that this underspecifies the problem. A frequentist might say that it does not properly specify the experiment. A Bayesian might say that it fails to specify how we acquired the information that one of the children is a boy, and without this we have no value for P(E|H) which we need to perform a Bayesian update.

The entry in Wikipedia has quite a good account of the ambiguity of the problem.

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Updated to do the math properly from a Bayesian perspective. Let BB, GB, BG and GG represent the four possible states of the genders of the children. These partition the set of all possible states and are assumed to be equiprobable. Let E be the evidence that we learn that at least one child is a boy.

                                   P(BB).P(E|BB) 
P(BB|E) =      -------------------------------------------------------------
               P(BB).P(E|BB) + P(BG).P(E|BG) + P(GB).P(E|GB) + P(GG).P(E|GG) 

P(BB) = P(BG) = P(GB) = P(GG) = 0.25.

P(E|BB) = 1. P(E|GG) = 0.

P(E|GB) and P(E|BG) are unspecified.

If we assume P(E|BG) = P(E|GB) = 1, then P(BB|E) = 1/3.

If we assume P(E|BG) = P(E|GB) = 0.5, then P(BB|E) = 1/2.

If we assume other values for P(E|BG) and P(E|GB) we can get anything between 1/3 and 1.

Answer 2

Vos Savant's question makes the distinction between the woman, who has at least one boy, and the man whose oldest child is a boy.

In case of the man the chance is 1/2. This is because in his case the options are:

1. Two boys
2. Boy and girl, boy is oldest.
3. Boy and girl, boy is youngest.
4. Two girls.

It is clear that only options 1 and 2 are under consideration. 2/4 = 1/2

Now, the import part in the wording of the explanation is 'one child is selected'. That is, the child is determined, either 'selected', 'the oldest', 'the tallest'.

This produces the following table:

1. Two boys
2. Boy and girl, boy is selected
3. Boy and girl, girl is selected
4. Two girls

The reasoning is as with the 'oldest boy': 1/2 rather than 1/3. The weird thing though, is that it still works with the addition that the boy is selected randomly...

That is why it is important to use the phrase has at least one boy if you want the answer to be 1/3...

Maybe Eliezer Yudkowski's englightenment can shed some light too. He thinks it has to do with Bayes' Rule.