The daughter paradox

[tolman]

The 'Jane' is clearly a red herring.

Either we potentially could have been asked about children of any mother in the population, and just happen to have been asked about Jane (equivalent to having a potential search space of a whole population, and a selection procedure leading to us being given information which was neutral regarding mother's names)

Or maybe we only ever could have been asked about Janes, but that would still only have an effect equivalent to shrinking the potential size of the population by an appropriate factor, without affecting the relative proportions of pairs of children of any particular gender, or with any particular name.

So the 'Jane' really does seem to be irrelevant, and 'A mother' would have been just as good.

If you decide, as you effectively have, that in the case of the second question, you could only *ever* have been given information about Emma-Louises who are daughters of Janes, and be (effectively) asked about the gender of the unnamed sibling, that is, that you could never have been given any *other* information, you are choosing to make a fairly strong assumption about the intentions of the person giving you the information.

If, on the other hand, you allowed that the person asking the question could potentially have said something different, and that you just happened to have been given the information you were given, that would mean that the particular girl's name you happened to have been given is just as meaningless as the mother's name you happen to have been given.

Personally, I fell into making different semi-conscious assumptions about the questions in initially arriving at my answers of 1 in 3 for both questions- the main assumption being that the questions were actually about 'pairs of children which contained a girl', rather than about 'pairs of children which in one particular case had happened to contain a girl'.
Making the 'other' assumptions would have led me to answers of 1 in 2 for both questions, which I think in hindsight are maybe more defensible from the point of having the most general interpretation.

You have chosen to make different assumptions, and effectively interpreted the second question as narrowly as you possibly could, but haven't interpreted the first question in as narrow a way as you could have done.
To that extent, you have chosen to create a paradox.

[Pebble]

Neither the name of the mother nor the name of the daughter changes the question(s).

Replace Jane with Mildred or Clare it makes no difference.

Replace Emma-Louise with Anne or Evelyn it makes no difference.

However, changing from a named woman to an un-named woman has no effect - simply enlarges the population of mothers being considered. Changing from a named to an unidenifiable daughter does change the qualification placed on how the population of mothers is chosen.

Every woman has a 1/4 chance of two boys, 1/4 chance of two girls and a 1/2 chance of mixed gender children when conceiving as required to produce a two children family. Yet you accept that if we add a qualifier of 'at least one daughter' then the mothers chances are reduced to 1/3 and 2/3 respectively. Why is this? There has been no actual change in the mothers potential offspring!

All one has done is changed the selection procedure for choosing the mothers that one enters into the calculation. All I am doing is following the same logic with a daughter who is identified in such a way as to impact on the selection of mothers to be included.

[tolman]

Changing from a named to an unidenifiable daughter does change the qualification placed on how the population of mothers is chosen.

But it only changes things meaningfully if you effectively decide that you could only ever have been told about that specifically-named daughter, that is, that having been told about Emma-Louise, you conclude that it would not have been possible for you to have been given any other girl's name.

If you could have been given any name in the case of a girl/girl pair, then in the case of a girl/Emma-Louise pair, only half the time would you hear ''Emma-Louise', and half the time the other girl's name.

Now, it's certainly true that, following the same rules, in the case of a boy/Emma-Louise pair, if one child from the pair had been randomly chosen to be talked about, you'd also only hear about Emma-Louise half the time.
That method equates to answering the question
"As a result of a someone choosing a pair from the population by a totally gender-blind and name-blind selection process, and then by a further gender-and name-blind selection process choosing a child to tell me the gender and name of, if I am told 'girl, Emma-Louise', what are the odds that the pair contains two girls?"
and has the odds of 1 in 2.

However if actually trying to be as consistent as possible between the two questions, in the case of the first question, one would have to interpret it as:
"As a result of a someone choosing a pair from the population by a totally gender-blind and name-blind selection process, and then by a further gender-and name-blind selection process choosing a child to tell me the gender of, if I am told 'girl', what are the odds that the pair contains two girls?"
which also has the odds of 1 in 2

One could view the questions as I initially did, as being about specifically girl-containing pairs of children and coming up with 1 in 3 chances of a girl-girl pair.
However, the more I think about that approach, the less good I think it is, since it effectively makes the assumption that we could never have ended up being told 'boy', which does seem hard to defend when considering the necessary rules to permit a realistic simulation.
Just as your approach effectively double-counts Emma-Louise-containing girl/girl pairs compared to an unbiased selection from the population, that approach effectively double-counts boy/girl pairs.

However, it just seems like a stretch too far to have to assume in the case of the second question that one could only ever have ended up being told about an Emma-Louise.
If anything, seeing the implications of the assumptions you made me much less happy about the assumptions I had made.

Even from a simple linguistic point of view, if one is actually trying to interpret the questions as fairly and consistently as possible, given "...One is a daughter..." and "...One is a daughter, Emma-Louise..." the most consistent reading seems to be that in each case the person giving us the information has a specific girl in mind, except that in one case we are given more information than the other.

[Pebble]

Focusing on the children as you suggest should always give a 1/2 relation. This is because any real child in this scenario can only have a brother or a sister. However, one rephrases the question that will be the inevitable conclusion.

However, if one reconsiders the question as I believe is intended "What are the chances that a hypotheticial mother of two, who has at least one girl of having a pair of girls?" Now you move the focus back to the genetics, hence the 1/3. The impact of naming that offspring is to negate the consideration of genetics by forcing the inclusion of a factor that is unique to the offspring, thus even in a hypothetical situation each possible girl has an equal chance of contribuiting to the final calculation.

The appropriate simulation is as previously suggested - tossing two coins and only counting the result if there is at least one tails. In the second scenario, the same two coins are tossed, but after tossing several thousand times and keeping pairs that only include a tails, one now randomly assigns a unique identifier to every fourth or perhaps tenth tails and marking it with an 'E'. Counting now only pairs that include the 'E' marked tails, it is evident that one cannot avoid excluding head/tail pairs that do not include a tails with an E, changing the outcome of the tally. This is of course 'unrealistic' but the whole question is fanciful from the beginning.

So given that the whole scenario seems a little mad, why do I persist? Because, if you follow the rules presented (without assumptions as to the intent or mechanism), the answers reached accord with reality. This therefore gives insight into why slight variations in 'inclusion and exclusion' criteria from the same population can give unexpected susbtantial changes in the selected study populations and render them markedly different to each other. This renders comparisons between apparently similar study populations meaningless.

[tolman]

The thing is, your approach seems to be making more assumptions about a simulation agent than is necessary, by not answering the question:
"Given an assumed normal population, what are the various ways I could have been told what I have been told, by an unbiased agent."
but
"Given an assumed normal population, and the assumption that I could only ever have been told what I have been told, what are the various ways I could have been told it."

The more specific the things you are told, the less secure the extra assumption seems to be, if the question is still framed in a potentially general way.

[tolman]

Had you been told something that you couldn't statistically examine:

>> "The second part is: ‘Jane has two children. One is a daughter, who is five foot six tall. What’s the probability that she has two daughters?’"

Would you dismiss the extra information as irrelevant, or conclude that you could not possibly give an answer?

[Matt]

Had you been told something that you couldn't statistically examine:

>> "The second part is: ‘Jane has two children. One is a daughter, who is five foot six tall. What’s the probability that she has two daughters?’"

Would you dismiss the extra information as irrelevant, or conclude that you could not possibly give an answer?

That's an interesting one.

For me the issue is that by naming one of the daughters you've uniquely identified her.

In case one we deal with the probability that the identified daughter could actually be one or other of a pair of daughters, each of which are the "other" daughter. To put it Pebbles way we're selecting sibling pairs where the first child fits the criteria or the second child fits the criteria or where both children fit the criteria.

In case two we deal with the probability that the identified daughter can only be one of the pair even if they are both daughters. The assumption is made we will not have both daughters named Emma-Louise. Again to put it Pebbles way we're selecting sibling pairs where the first child fits the criteria or the second child fits the criteria but there are assumed to be no instances where both children fit the criteria.

In your example the child is not uniquely identified. It is possible for both children to be daughters, five foot six tall but the probability of that occurrence is not known. The pragmatist within me is tempted to approximate the probability to zero just so that a definite answer can be given. (As we ignored previously the possibility that a perverse parent would give both her daughters the same name) The mathematician wants to declare the variable and give an answer in the form of an expression. This is complicated by the emprical observations about sibling height. Height is more likely to be similar if the siblings are fully grown or if they are twins (esp, identical). Twins of course are more likely to be the same gender as one another as their number includes monozygotics. The actual height you've chosen, 5 foot 6, may well be a fully grown height especially for a female.

On balance I don't think an answer can be given.