The daughter paradox

[tolman]

Ever wonder why wars are fought, negotiations breakdown and lawyers get so rich. People continually interpret what is agreed differently. Even very slight imprecision can lead to hugely different expectations.

One of the main reasons that wars are fought and negotiations break down is because different people desire irreconcilably different things.
Trying to think of wars that have started over accidental misunderstandings about language, I don't come up with a great list.

While lawyers can get rich arguing over the precise meaning of a phrase, that is one of the reasons why legal language has tended towards a form where, used correctly, some of the ambiguity inherent in natural language can be avoided.

[Pebble]

In negotiations with staff and unions, this issue certainly comes up time and again. The management has a different interpretation of what has been agreed with the other side and tensions build when the differences between interpretation and reality begin to become apparent.
In terms of wars, I will leave that mainly to the experts, but the 1921-22 civil war in Ireland was based on different interpretations of what was agreed. The British govt. thought they had safeguarded all their strategic interests and put in place a regime they could control to do their bidding at arms length. The Free state supporters simply regarded it as a first step with implicit agreement to build toward a 32 county republic, the IRA saw the agreement as a total sellout. Each looking at different possible interpretations of what was agreed and what the obligations and responsibilities of the other side was.

[tolman]

In terms of wars, I will leave that mainly to the experts, but the 1921-22 civil war in Ireland was based on different interpretations of what was agreed. The British govt. thought they had safeguarded all their strategic interests and put in place a regime they could control to do their bidding at arms length. The Free state supporters simply regarded it as a first step with implicit agreement to build toward a 32 county republic, the IRA saw the agreement as a total sellout. Each looking at different possible interpretations of what was agreed and what the obligations and responsibilities of the other side was.

So different people had different ideas about what certain organisations and people would do.

Was that based on things actually being written down ambiguously but with clearly different interpretations on different sides, or on people making assumptions that 'X would never do Y' or 'We can always lean on Z to get them to do what we want'?

Do you think that by better clarity of wording, there could actually have been a possible agreement which the British Government and all the various groups in Ireland would all have been happy with?

Seems to me that it might be more a case of irreconcilable differences than linguistic vagueness.

On the other hand, sometimes when there is vagueness, it's there deliberately to give wiggle room, or to allow an agreement to be sold to more people.

When it comes to Northern Ireland and recent agreements, it's certainly arguable that one or more of the armed groups has actually failed in what they wanted to achieve.
In that situation, forms of words vague enough to allow people to save face can actually be rather useful.

[Pebble]

I'm no expert on the NI peace process, but my understanding was that unlike the Israeli/Palastinian process it was highly specific. Prior to public negotiations a specified period of ceasefire was demanded. Renunciation of violence as a legitimate means of pursuing political goals had to be on the table and then delivered as part of the process. Then there was a requirment for an independently verified inventory of arms caches, a process for disarmnement, followed by acceptance and support for the NI police and finally transfer of police control to the NI assembley. With each of these steps came greater invovlement in the affairs of NI, the release of prisioners and the withdrawal of troops in prespecified order. The arguments as I understood them was whether sufficient confidence had been built up by the other side at each stage to support progress to the next stage.

[tolman]

I'm no expert on the NI peace process, but my understanding was that unlike the Israeli/Palastinian process it was highly specific. Prior to public negotiations a specified period of ceasefire was demanded. Renunciation of violence as a legitimate means of pursuing political goals had to be on the table and then delivered as part of the process. Then there was a requirment for an independently verified inventory of arms caches, a process for disarmnement, followed by acceptance and support for the NI police and finally transfer of police control to the NI assembley. With each of these steps came greater invovlement in the affairs of NI, the release of prisioners and the withdrawal of troops in prespecified order. The arguments as I understood them was whether sufficient confidence had been built up by the other side at each stage to support progress to the next stage.

But as to the *wording*, isn't it the case that there was much argument about what people said they were doing, with people desperate to avoid saying things that sounded like they were admitting failure, and other people trying to push them into doing just that?
Hence people announced that armed struggle is over, without saying anything about whether they thought it had actually achieved anything.
People didn't surrender weapons, their weapons were put beyond use.
People agree that peace is better without commenting on whether violence was a waste of time.
Etc.

[Jules]

My feeling is that disagreements may have something to do with your interpretation of what probability actually is. There is a long-running debate among the "frequentists" and the "Bayesians" - is probability a belief conditional on evidence? (Bayesian) or is it concerned with the long run frequency of events? (ie; an inherent property of the event). My view is that both approaches have their uses at times, but some are more dogmatic.

Let's take the old classic: You meet a mathematician on the street, and she happens to mention that she has given birth to two children on two separate occasions. You ask: "Is at least one of your children a boy?" The mathematician says, "Yes, he is."
What is the probability that she has two boys? If you assume that the prior probability of a child being a boy is 1/2, then the probability that she has two boys, on the information given, is 1/3. The prior probabilities were: 1/4 two boys, 1/2 one boy one girl, 1/4 two girls. The mathematician's "Yes" response has probability ~1 in the first two cases, and probability ~0 in the third. Renormalizing leaves us with a 1/3 probability of two boys, and a 2/3 probability of one boy one girl.
But suppose that instead you had asked, "Is your eldest child a boy?" and the mathematician had answered "Yes." Then the probability of the mathematician having two boys would be 1/2. Since the eldest child is a boy, and the younger child can be anything it pleases.
Likewise if you'd asked "Is your youngest child a boy?" The probability of their being both boys would, again, be 1/2.
Now, if at least one child is a boy, it must be either the oldest child who is a boy, or the youngest child who is a boy. So how can the answer in the first case be different from the answer in the latter two?

http://lesswrong.com/lw/oj/probability_is_in_the_mind/

[Janot]

Let's take the old classic: You meet a mathematician on the street, and she happens to mention that she has given birth to two children on two separate occasions. You ask: "Is at least one of your children a boy?" The mathematician says, "Yes, he is."
What is the probability that she has two boys? If you assume that the prior probability of a child being a boy is 1/2, then the probability that she has two boys, on the information given, is 1/3. The prior probabilities were: 1/4 two boys, 1/2 one boy one girl, 1/4 two girls. The mathematician's "Yes" response has probability ~1 in the first two cases, and probability ~0 in the third. Renormalizing leaves us with a 1/3 probability of two boys, and a 2/3 probability of one boy one girl.
But suppose that instead you had asked, "Is your eldest child a boy?" and the mathematician had answered "Yes." Then the probability of the mathematician having two boys would be 1/2. Since the eldest child is a boy, and the younger child can be anything it pleases.
Likewise if you'd asked "Is your youngest child a boy?" The probability of their being both boys would, again, be 1/2.
Now, if at least one child is a boy, it must be either the oldest child who is a boy, or the youngest child who is a boy. So how can the answer in the first case be different from the answer in the latter two? /

After 12 pages, it looks like this thread has now become a loop, because this is back to the OP. The solution to the paradox is of course that the probability is in the first case still 1/2, not 1/3, as argued by my post no. 51.

This thread could run and run in an ever-decreasing loop until it disappears.....

[tolman]

After 12 pages, it looks like this thread has now become a loop, because this is back to the OP. The solution to the paradox is of course that the probability is in the first case still 1/2, not 1/3, as argued by my post no. 51.

This thread could run and run in an ever-decreasing loop until it disappears.....

You can only really define a probability if you define the mechanism by which you think a pair has been selected, and by which someone has decided to tell you something about it.

If a pair was chosen at random and someone had decided to randomly choose a child from the pair and tell you about their gender, you get a different answer to if someone had decided to ask a question about girls (named or un-named) having sisters, and chosen a pair accordingly.

[Admin]

This is an old one but it was published in the ASKE newsletter with a twist I hadn't come across before:

1. The first part of the puzzle is a classic problem: ‘Jane has two children. One is a daughter. What’s the probability that she has two daughters?’
   .
2. The second part is: ‘Jane has two children. One is a daughter, Emma-Louise. What’s the probability that she has two daughters?’

The answers given are:

Part 1: 1 in 3

It's the standard way of working it out. There are 4 ways of having 2 children:

Boy - Boy
Boy - Girl
Girl - Boy
Girl - Girl

If at least one is a girl then the boy-boy option is ruled out so the girl is twice as likely to have a brother than a sister.

Part 2: 1 in 2

This is the answer given:

In the second part of the puzzle the child who is a daughter is uniquely identified by her name, Emma Louise (we make the reasonable assumption that Jane does not give both her children the same name). The ‘other child’ now is the child who is not Emma Louise and can be a boy or a girl. Hence the probability now that Jane has 2 daughters is 1 in 2!

Another way of uniquely identifying the daughter in the question could be to say that she is the older of the two or the taller, and so on. Or by being introduced to her. Again the answer would be 1 in 2.

I think that's wrong.

It doesn't matter what the girl's name (or any other attribute for that matter) is, the answer is 1 in 3 unless the birth order is known - in which case it becomes 1 in 2.

Any thoughts on this puzzle?

I can expand on my reasoning but I'll wait for now.

Right.... Now that we've worked through this, I'll go back to the original questions as they were stated.

The first part of the puzzle is a classic problem: ‘Jane has two children. One is a daughter. What’s the probability that she has two daughters?’

The answer is 1/3. The question is about Jane (the parent) so she should be the focus of the solution to the problem.

The second part is: ‘Jane has two children. One is a daughter, Emma-Louise. What’s the probability that she has two daughters?’

The answer again is 1/3. That's because the question is about Jane (the parent) and not Emma-Louise. So the answer provided (1/2) is actually wrong as it pertains to how the question was stated.

The error creeps in because instead of focusing on the problem from Jane's point of view we start considering it from Emma-Louise's point of view (in which case the answer would be 1/2).

[tolman]

Right.... Now that we've worked through this, I'll go back to the original questions as they were stated.

The first part of the puzzle is a classic problem: ‘Jane has two children. One is a daughter. What’s the probability that she has two daughters?’

The answer is 1/3. The question is about Jane (the parent) so she should be the focus of the solution to the problem.

The second part is: ‘Jane has two children. One is a daughter, Emma-Louise. What’s the probability that she has two daughters?’

The answer again is 1/3. That's because the question is about Jane (the parent) and not Emma-Louise. So the answer provided (1/2) is actually wrong as it pertains to how the question was stated.

The error creeps in because instead of focusing on the problem from Jane's point of view we start considering it from Emma-Louise's point of view (in which case the answer would be 1/2).

It still comes down to how Jane ends up being chosen as a subject, and how we end up being told one child is a girl.

If our informant is choosing a parent at random, and choosing something to say about gender at random, either choosing a child from the pair at random and giving their gender, or choosing a gender at random and then choosing a child of that gender (if possible otherwise then choosing a child of the other gender), the odds do end up as 1 in 2.
Though there are twice as many parents with boy+girl as girl+girl, if the parent and gender choice are both fully random, we only end up hearing 'girl' for half of the boy/girl pairs, but all of the girl/girl ones.

If our informant is choosing something to say something about parents with at least one girl, choosing a mother accordingly and then telling us that mother has a girl, then the odds are 1 in 3.

Really, it's down to whether we're being asked about 'mothers of pairs of children at least one of which is a girl', or possible genders of children of randomly selected mothers.

In the absence of clarity, either interpretation is defensible, but neither interpretation leads to a paradox, since the naming of a daughter doesn't actually provide us with any extra information.

[Pebble]

If the only information provided is that Jane has two children, then the chances of having two girls is 1/4.

If an addtional piece of information is provided 'at least one is a daughter' then the population of mothers from which Jane could be derived is reduced (two male offspring removed) therefore the chances of two daughters is 1/3.

Provide a further piece of information 'name of daughter' then the group of mothers that Jane could be chosen from is severely restricted - the pivotal change is that every daughter in two parent families is assumed to have an equal chance of being given this particular name, hence the same number of Janes' with boy girl pairs will have daughters with this name as the number of Janes' with girl girl pairs. In other words, the offspring's name becomes the dominant feature of the selection choice - hence 1/2 chance of a girl girl combination.

[tolman]

If the only information provided is that Jane has two children, then the chances of having two girls is 1/4.

If an addtional piece of information is provided 'at least one is a daughter' then the population of mothers from which Jane could be derived is reduced (two male offspring removed) therefore the chances of two daughters is 1/3.

There is the issue of how we come to be told the information.
If the process being followed was (process 1):
a) Randomly choose a woman with 2 children who are not both sons.
b) State that this woman has a daughter
Then we would have the 1/3 probability that the woman had two daughters.

If the process being followed was (process 2):
a) Randomly choose a woman with 2 children
b) Randomly choose one of her children
c) Give that child's gender
Then if we had ended up being told 'the woman has [implicitly at least one] girl', there would be a 1/2 probability that the woman had two daughters.
That's because we'd only end up with a 50% chance of having been told 'girl' in the case that a boy/girl pair was selected in stage a), but a 100% chance of being told 'girl' in the case that a girl/girl pair had been selected in stage a).

Provide a further piece of information 'name of daughter' then the group of mothers that Jane could be chosen from is severely restricted - the pivotal change is that every daughter in two parent families is assumed to have an equal chance of being given this particular name, hence the same number of Janes' with boy girl pairs will have daughters with this name as the number of Janes' with girl girl pairs. In other words, the offspring's name becomes the dominant feature of the selection choice - hence 1/2 chance of a girl girl combination.

I don't think that works, for reasons similar to the logic for the second of the examples given above. Let's assume the name given to us is 'Emma'

Take process 1, and add the (very reasonable) step:
c) If the woman has only one girl, give her name, if two girls, choose one at random and give her name.

That step doesn't actually give us any extra information.
Even though there are equal numbers of boy:girl and girl:girl pairs which include an 'Emma', and an Emma in the general populations has an evens chance of having a brother or a sister, we have only a 50% chance of being told 'Emma' if an Emma/non-Emma girl/girl pair is selected in step a), but a 100% chance of being told 'Emma' if a boy/Emma pair is selected, so the odds of us being told 'Emma' are twice as high in the case of a boy/girl pair being selected in step a), so we still have a probable brother:sister ratio of 2:1 for any Emma we end up being told about.

On the other hand, take process 2, and add the (very reasonable) step:
d) Give that child's name.

Now we have a situation where the female child had already been selected in step b) before we were told the name.
Before being told the name, we had a 50% probability that the girl that had been selected in step b) was the girl in a boy/girl pair, or a girl in a girl/girl pair.

Being given that child's name doesn't actually change anything.
If the odds of a randomly selected girl in the population being called Emma are 1 in 10 or 1 in 100, all that being told the child's name is Emma means is that the number of pairs in the wider population that we might be dealing with is 10 or 100 times smaller, whether or not that selected child has a brother or a sister.

You could argue that if we preselected the population to include only those pairs including an Emma that that biases things towards girl:girl pairs, since such pairs have two ways to include an Emma.
However, such pairs have two girls in, and in the case of one of those pairs having been initially selected, we will only hear about the Emma in it half of the time, which precisely cancels out the 'double'-dip bias that girl:girl pairs possess.

[Admin]

the offspring's name becomes the dominant feature of the selection choice - hence 1/2 chance of a girl girl combination.

But that's exactly where the error occurs!

The dominant feature, or focus of the problem, should be the parent, not the offspring (at least in the examples used here - wording is important).

A woman is picked at random
We find out she has at least one daughter
That daughter is called Emma-Louise

The woman is chosen before we know any info about her children or their names.

There are twice as many parents in the sample space with B-G than G-G including when a small percentage of girls are called Emma-Louise. So you do get this 2/3 vs 1/3 split when you focus on the parent as the primary focus of the problem.

The problem needs to be solved by considering the distribution of parents, not by gender or names.

[Pebble]

I think trying to decide the order in which the information has been selected - mothers, mother's of two, mother's of two not including two boys or mother's of daughters with a given name - introduces too many assumptions. We have three prices of information - mother of two, not two boys, a girl named Emma-Louise, we have no reason to assign an order of primacy.
Making the minimum number of assumptions, one then looks at the proportions of mothers with a child named Emma-Louise who have girl-girl pairs versus boy-girl pairs.

[tolman]

I think trying to decide the order in which the information has been selected - mothers, mother's of two, mother's of two not including two boys or mother's of daughters with a given name - introduces too many assumptions.

But you end up making assumptions anyway.

Given the presence of two different but quite reasonable sets of assumptions as to the nature of the process that ends up with us being given the information in the question, each of which results in a different 'correct' answer, if one is not explicit about what assumptions one is making, one cannot really give a 'correct' answer.

We have three prices of information - mother of two, not two boys, a girl named Emma-Louise, we have no reason to assign an order of primacy.

In which case, you can't actually answer the question with reliability.

Making the minimum number of assumptions, one then looks at the proportions of mothers with a child named Emma-Louise who have girl-girl pairs versus boy-girl pairs.

If what you mean is "one then looks at the proportions of mothers [in the population as a whole] with a child named Emma-Louise who have girl-girl pairs versus boy-girl pairs.", that would mean that:

a) You're implicitly assuming that you would always be told 'she has a girl' even if the woman concerned has a girl and a boy. (That is, you're assuming a total bias in the informant to only talking about girls, and that a woman with two sons could never have been selected.)

and

b) You're also implicitly assuming a total bias in the informant to talking about Emma-Lousies. (That is, in the case of a mother of a girl/girl pair having been chosen, you'll somehow always be told 'this woman has a girl called Emma-Louise', and never the name of the other girl.)

a) Is somewhat understandable - the question could indeed only ever have been about women who have at least one girl, and that's actually the way I was first tempted to interpret it.

However, b) does look like a bit more of a stretch, especially for someone making the minimum number of assumptions.

In reality, the assumptions in a) and b) cancel out, the first overrepresenting boy/girl pairs among the pairs one ends up being told about, and the second overrepresenting Emma-Lousie/other pairs to the same extent, but only if both assumptions are actually being made.
Even then, it's arguably more luck than judgement that the cancellation happens.