(My ordering)
Your a) and b) aren't fully defined.Originally Posted by John Jackson
If we could have ended up being told about a woman who had a son, then the [simplest] steps would have been:
a) Choose a mother of two (at random).
b) Then choose one of her children (at random) and state their sex.
c) Then state that child's name
In such a setup, only half of the initially selected boy:girl pairs would end up being reported as 'contains a girl', which cancels out the 2:1 bias in favour of boy:girl pairs over girl:girl ones, giving an evens chance that the/a girl of that mother has a brother.
In such a setup, it also wouldn't matter whether the initial selection was from the entire population, or only from the mothers who had at least one girl.
Alternatively, if the method followed was:
a) Randomly choose a mother of two, making sure she doesn't have two boys
b) State that she has a girl
c) Give the name of a girl (randomly chosen if there are two girls)
We do end up with a 2:1 bias in favour of the/a girl of that mother having a brother.
Taking your aproach to its logical extreme, we should not make assumptions about mothers or even parental status let alone humanity when trying to resolve the conundrum!
We are not talking bias here, we are accepting the facts presented, adding as few assumptions as possible and working out the probability compatible with the presented facts.
The art of medicine consists in amusing the patient while nature cures the disease. Voltaire
But your stated interpretation does involve the person providing the information having followed a set of rules in order to have given you the information, and whether you make assumptions about those rules implicitly or explicitly, you're still making them.
You can't reduce the number of assumptions you're actually making simply by either failing to recognise them, or failing to state them.
It's not possible to give three pieces of information and say 'there's no order of primacy', since what that means is that if you wanted to run the puzzle in real life (or a simulation), you would be quite unable to instruct someone (or a computer) about how to provide you with the information.
If you can't instruct someone (or me, or a machine) how to make a deliberate selection here, or a random selection there, then you don't know the overall processes of selection by which you could end up being told 'X' or 'Y', or whether the combined effects of those selections skew the probability of you actually being told 'X' as compared to the sizes of different subsets of the population about which 'X' is a true statement.
For you to be able to directly draw probability estimates from proportions of mothers in the population at large with one or other different kind of family arrangement (number of boys/girls, names of one or more children), you are necessarily making the assumption that the method by which you could come to be given information about a mother's children doesn't skew probabilities one way or another.
For example, when it comes to names, you would have to implicitly be assuming that the method the informant is operating by is as likely to end up telling you 'Emma-Louise' in the case of a boy/Emma-Louise pairing (either birth order) as in the case of a other-girl/Emma-Louise pairing (either birth order).
If one decides that the question was designed from the outset to be about girl-containing pairs (or [children of] mothers of girl-containing pairs, which amounts to the same thing), I've pointed out that a simple, and perfectly possible mechanism (many would say the simplest and most plausible mechanism) someone could have used to give you the information is for someone to choose a girl-containing pair, always tell you it contains a girl, and then name a girl in the pair.
That simple mechanism is not neutral with respect to naming Emma-Louises, since if applied over and over to randomly-chosen girl-containing pairs, it always reports every Emma-Louise who has a brother, but only half of the time does it report Emma-Louises who have a sister.
I wonder:
Choose from the population only two children families.
From this population choose only those families with a mother named Jane and a daughter named Emma-Louise.
From this group count the number of Jane's with two daughters versus daughter son pairs.
Using this simple model any computer programme would end up with approximately a 50/50 relationship.
The problem arises working backwards - have a daughter named Emma-Louise and a mother named Jane; what are the chances a person would have provided you with these specific pieces of information. Now all sorts of possibilities begin to present themselves.
I think this is a bit like religion's problem with science. Science tries to create general rules that can explain the specific. Religion looks only at the specific and assumes that the complexities involved in arriving at a particular point is so extraordinary that only God's will can provide a satisfactory explanation.
The art of medicine consists in amusing the patient while nature cures the disease. Voltaire
But you have to work 'backwards' if you hope to answer the question "Given what I have ended up being told, what can I conclude?", which is effectively what the original questions are asking you, since the mechanism involved in you ending up having been told what you were told is an unignorable part of the puzzle.
If you're going to try and argue that the brother:sister ratio of Emma-Louises in the general population is a correct guide to the brother:sister ratio, you're implicitly making the assumption that the your chances of being told about an Emma-Louise who has a sister are the same as your chances of being told about one who has a brother, and that looks like it could actually be a rather large assumption.
Not sure of your point there, unless you're arguing that you're taking a religious approach to a mathematical question.
In this case, there must be some specific (as in defined) rules, but those rules are also general in the sense that they can be used to process the entire possible population, and hence could be used in a simulation, with the rules being applied over and over to a given population.
If not enough was known (or at least assumed and clearly stated) to enable a simulation to be run, I can see no way that a probability could be meaningfully given.
Working backwards:
a) The mother's name is irrelevant unless there's some unusual, unanticipatable (and unexplained) correlation between mother's name and child gender balance or naming. If there is such an unexplained correlation, answering the question does not merely require us to state assumptions, but to state assumptions we can't possibly guess, and therefore the question would be unanswerable. Therefore we must be allowed to ignore the mother's name in order to progress further.
b) The specific girl's name is irrelevant, unless there's some unusual, unanticipatable (and unexplained) correlation between a specific child's name and the gender of their other sibling, and/or there is an unusual and unanticipatable asymmetry in the procedures being followed, making them biased arbitrarily in favour of certain girl's names over others. If there are such unexplained correlations or asymmetries, they would make the question unanswerable. Therefore we must be allowed to ignore the specific child's name in order to progress further.
After b), we end up with both questions (with and without the named girl) having the same answer.
Therefore even without knowledge of the general mechanism by which we came to be given the information, and without stating various assumptions we made about such a mechanism which we would have to give to justify an assertion of a specific probability, we can assert the absence of a paradox.
(For points a and b, just imagine whether we could possibly give a different answer to the question if we'd been told the mother's name was Julie, or Anne, or any other woman's name. If not, then the name must be irrelevant.
Likewise, if we'd been told that a daughter of her had any other girl's name, rather than Emma-Louise, that couldn't affect the answer we gave, so that name must be irrelevant as well.
Last edited by tolman; 23rd August 2010 at 12:19 AM.
The unique thing about any girl's name is that only one daughter in any family will have that name. The assumption being made is that each female offspring has an equal chance of being given that name. This fact and the following assumption are all that is required to end up with the paradox.
If you have a valid reason for adding further assumptions, then one can create additional scenarios associated with different probabilities. Assuming that the name provided was an accidental occurrence at the end of a selection process, would give a 1/3 ratio, but despite your protestations that a logical process (guiding mind) must have existed to get to this particular point I remain unconvinced.
The art of medicine consists in amusing the patient while nature cures the disease. Voltaire
The 'following assumption' seems to be missing (presumably an editing issue?).
Given the initial questions, and that there must be *some* mechanism being followed to select a pair from a population to tell you about, do you think that basically the same selection mechanism is being followed in the case of each question, but with you being given extra information in the case of the second question, or do you think that two different mechanisms are being followed?
If you think the pair-selection mechanisms are the same (that is, in the second question a pair has already been chosen, and you are simply additionally given the name of the daughter or a randomly-chosen daughter) then there cannot be a paradox since the specific name given adds no information - you could always be told some girl's name, and whatever name you are given has no bearing on the sex of the unnamed child.
That is true unless the mechanism is arbitrarily biased towards or away from specific names, but if such an arbitrary bias exists and you are not told what it is, then you cannot answer the question at all.
If you think the mechanisms are different, then there cannot be a paradox.
If I said "By one [undescribed] method of selecting pairs from a population I get a certain distribution of different types of pairs, but by a different [undescribed] method I get a different distribution", I can't see anyone seeing that as remotely paradoxical.
Effectively, the claimed existence of a paradox seems to place limits on the assumptions one can reasonably make about the underlying selection and description mechanisms.
No not missing, I simply have no use for that assumption. The simplist format is to accept the question as a hypothetical scenario, designed to test logic. This requires no assumptions about selection. The next option is to regard the question as applicable at a population level, giving a specific answer to a proposed example. Only then need one consider adding additional assumptions.
The art of medicine consists in amusing the patient while nature cures the disease. Voltaire
If you treat the questions as entirely independent, assuming that the mechanism by which you are told information in the first question could be entirely different to the mechanism in the second, there aren't any reasonable grounds for claiming a paradox, since you're effectively saying "Two [potentially quite different] selection mechanisms give a different spread of results when selecting from a given population".
Why would anyone even bother asking the questions if that was going to be the outcome?
On the other hand, if you assume that the questions are actually connected, in the sense that they share a common process but differ only in the amount of information you are given you have to make (explicitly or otherwise) unusual assumptions to justify giving different answers to the two questions.
The natural reading of the questions is that in each case you actually are being asked questions about possible pairs of children.
If you were actually being asked merely about girls with (or without) an arbitrarily chosen name in a population of two-child families, then the questions are actually being framed in an extraordinarily poor way.
Even if you think you are being asked about girls with (or without) an arbitrarily chosen name in a population of two-child families, there is still no difference between the answers to the two questions.
To support a 'paradox' (different answers to the two questions), you must interpret:
‘Jane has two children. One is a daughter. What’s the probability that she has two daughters?'
as meaning something like:
'What are the chances of a pair of children containing a girl actually containing two girls?'
and
‘Jane has two children. One is a daughter, Emma-Louise. What’s the probability that she has two daughters?’
as meaning:
'What are the chances that a girl called Emma-Louise in the population has a sister?'
That is, you choose for no obviously explained reason in the second instance to ignore the fact that you're being told/asked about the children of a chosen woman, and instead switch all your attention to the name of a girl, as if everything else is not merely unimportant, but is actually a distraction from the true answer, whereas in the first instance, you do not ignore the rest of the framing of the question and interpret it to mean:
'What are the chances that a girl the population has a sister?'
Of course, the questions
'What are the chances that a girl called Emma-Louise in the population has a sister?'
and
'What are the chances that a girl in the population has a sister?'
necessarily have exactly the same answer in a 'normal' population.
I don't see how it's defensible to chuck away everything but the girl's name in interpreting the second question while failing to chuck away everything but the fact that there is a girl in the first one.
I am making no assumptions whatsoever about how the questions were constructed, I am simply addressing them as logical scenarios which give rise to probabilities.
In the first question you have two pieces of information (Jane mother of two; one a daughter). Here the daughter could be one of 4 possible daughters, but possessing no unique identifiers, the probability of all the Jane's mothers of two children having two girls in this instance is 1:3.
In the second question we have three pieces of information (Jane mother of two; one a daughter; this daughter has a unique identifier). When one empirically tests this in the population, there are just as many Jane's with older and younger daughters that possess this unique identifier.
Hence using the same process one ends up with two different answers. Importantly, if you try the process on the same population but without any identifier for the daughters, you get 1/3 - this is because the named daughter selects out a different population of Jane's from the total population of Janes mothers of two.
The art of medicine consists in amusing the patient while nature cures the disease. Voltaire
For each question, it is certainly possible to interpret the question in at least two ways.
Firstly, a question can be interpreted 'pairwise' as being about pairs of children.
(This could be visualised as someone following a set procedure to choose a pairs at random and tell you something about it, and you having to work back from what you are told to calculate a probability.)
Secondly, one could interpret the question 'specifically' as if it is primarily about individual children.
(This could be visualised as someone following a set procedure to choose a child from the whole population at random and tell you something about it, and you having to work back from what you are told to calculate a probability.)
What you seem to have done is decided to interpret each question using different interpretations.
Just as you decided to interpret the second question specifically, as effectively being about the probability of a particularly-named girl in the population (who has a particularly-named mother, though the mother's name isn't actually relevant) having a sister, you could quite easily have also chosen to interpret the first question specifically, as being about the probability of an unnamed girl (ie any girl) in the population (who has a particularly-named mother, though the mother's name isn't actually relevant) having a sister, which clearly has the same probability, but you decided to interpret the two questions in different manners.
Not exactly,
What I am doing is counting the number of Janes that meet the subsequent criteria. Thus the focus remains the mother in both analysis, the outcome changes because of the modifiers.
Q1.
Step 1
Count all Janes with at least one daughter in the population
Step 2
Divide this population of Janes into those with daughter son, and daughter daughter pairs
Step 3
Calculate probability
Q2.
Step 1.
Count all Janes with a daughter named Emma-Louise.
Step 2
Divide this population of Janes into those with daughter son, and daughter daughter pairs
Step 3
Calculate probability
The art of medicine consists in amusing the patient while nature cures the disease. Voltaire
Actually I think this is precisely the point of the questions.
Both questions look at the same population, minor differences in how we select subgroups can lead to unexpectedly large biases in the outcome. Thus for example many dietary studies are constructed to assess the relationship between longevity or disease burden and particular dietary intake. These are virtually always positive suggesting that whatever richer more educated people consume reduces disease burden. Attempts are then made to 'control' for the impact of known confounders (social class) yet the results remain positive. Then once a randomised trial is undertaken the benefit disappears.
Why is this? I would suggest that the selection procedure inadvertently identifies compliers not just education and wealth, thus the correction is applied to the wrong confounders.
As in these questions it is assumed that associated confounders (daughter, named daughter) give similar extracted populations.
The art of medicine consists in amusing the patient while nature cures the disease. Voltaire
But you're not being asked about Janes (and even with your approach, the Jane 'information' is not relevant to the probabilities compare to being told 'a woman').
In the most direct interpretation of the question, you're being asked whether this girl in a pair of children (named or unnamed) has a sister.
It's fairly clear that for any given pair chosen from the population, if that pair contains a girl, if you are asked "Does this girl have a sister?" then whether the girl is named or not is irrelevant, and indeed, in the case of a girl/girl pair, whether a specific girl has even been chosen or the question is about an arbitrary one of the girls is also irrelevant.
In the case of a named girl, what you are choosing to do equivalent to listing all the pairs which contain a girl of that name and then assume that in the case of such a pair being selected, you'd always be told about that particular girl.
In other words, to justify your answer, if you wanted to run a simulation, your instructions to the computer would be would be something like:
a) Choose an Emma-Louise and her sibling
b) Tell me you have selected a pair of children
c) Tell me that the pair contains a girl, and then tell me the girl's name is Emma Louise.
Or
a) Choose a pair containing an Emma-Louise
b) Tell me that the pair contains a girl, and then tell me the girl's name is Emma Louise.
Which seems a strange set of instructions - why implicitly assume that the mechanism by which you end up being told is so totally biased towards the name Emma Louise?
If you think the second question could only ever have been about Emma-Louises, if you wanted to be as symmetric as possible in your approach to the two questions, you could have decided that the first question could only ever have been about unnamed girls, and interpreted the first question as:
"Jane has two children. I have selected one, which is a girl. What are the chances that this girl has a sister?"
and doing exactly what you did in the case of the second question - simplifying it to how many girls in the population have a sister, just as with the second question you simplify it to how many Emma-Louises in the population have a sister.
Which are, of course, equivalent questions.
To the extent you chose not to do that, you chose to create a paradox.
In the OP, Jane has two children, one of them is a daughter, we are then asked to calculate the probability that she has two daughters (not that the daughter has a sister).
Probability calculations apply to populations, so we must first decide what the population is - is it the population of Janes mothers' of two children, or the population of Jane's offspring. As Jane has only two offspring, and we know that one is a daughter then we are forced to focus on her other child - we have only a population of one to consider. Being restricted to a population of one that actually exists - probability ceases to have relevance - since that offspring is either male (zero percent chance of being female) or female (100%) chance of being female. Such considerations are meaningless.
Thus the only relevant populations to consider are 'Janes mothers' of two' or 'Janes mothers' of two, one named Emma Louise' - that is what I have done.
The art of medicine consists in amusing the patient while nature cures the disease. Voltaire
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