My post no. 53Originally Posted by Janot
nobody listens to me
Did someone say something?
I think we both realised that the answer to question A was wrong around the same time but I managed to talk myself into believing that there are 2 answers to the question depending on how you conceptualise and so answer it.
Then I decided that if you have one question, one set of information and two answers then it's absurd - one of the answers has to be wrong.
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I think you may have to ga back to tossing coins on this one. (two tosses, exclude two heads).
If a woman has two children, we know that genetically she has a 1/4 chance of 2 males, 1/4 chance of two females and a 1/2 chance of one male and one female.
If one now excludes the 2 male option - then she still has 4 options to have a girl and only two options to have a boy left. But she still is twice as likely to have a boy girl combination as a girl girl combination.
If you now identify one girl as specifically existing, we must look not just at the possible genetic outcomes but the impact of identification on the distibution. As a woman she is still twice as likely to have a boy girl combination as a girl girl combination. But the population of mothers with a girl possessing this identifier is now drawn from the population of mothers of specific girls, not just the mothers of two children. This doubles the chances of a girl girl combination requiring counting.
That is how I finally decided the paradox made sense.
Actually, she's not. That's the trap with this problem. If you look at the numbers of children involved the number of girls in boy+girl is the same as the number of girls in the girl+girl options.
Treating boy-girl and girl-boy as different groups (pairs) is wrong anyway as it artificially introduces birth order as a factor - information that is not present in the question.
There is no paradox. The answer to question A is 1/2 and the answer to question B is 1/2. The paradox is only introduced by using the wrong answer in question A.
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Yes. And the coin tossing analogy makes this obvious. You've got a "heads", toss again, it's 50-50, heads or tails. Nothing else matters.
Mother XX
Father Xy
Offspring possibilities 1 (XX, Xy) 2 (XX, Xy)
Possible combinations: 1/4 (XX, XX) 1/4 (Xy, Xy), 1/2 (XX, Xy) - no order required for that - further validated empirically.
Agreed!
Now, if we rule out the boy-boy possibility (Xy, Xy) we get:
Possible combinations: 1/4 (XX, XX) 1/4 (Xy, Xy), 1/2 (XX, Xy)
There are twice as many (XX, Xy) as there are (XX, XX) BUT there are twice as many XX's in each (XX, XX) pairing as there are in each (XX, Xy) pairing.
This is what evens the numbers up. You end up with the same number of girls in the (XX, XX) group as there are in the (XX, Xy) group - therefore, any girl picked at random has a 50% chance of being paired with another girl; as she does being paired with a boy.
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Absolutely, each girl has an equal chance of having a sister as a brother, but that was not the question, the question was the likelihood of the mother having two daughters.
Given that one of them is already known to be a daughter.
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The question then arises whether you get a different answer by knowing that she does not ahve two sons (in which case the probability is definately 1:3), versus knowing that one of her children is a daughter. From the distribution worked out by Matt, if the daughter has no identifying features the probability is 1:3, but the sub populations of mothers where the daughters have common identifiers, the probability changes to 1:2.
But it isn't. It's 1/2
That's where the apparent paradox comes from.
Seriously, do it with numbers instead of playing with percentages and it becomes clear.
100 families of 2 kids will have an (ideal) distribution like this:
.. (1) .(2)
A) 25 - 25 (boy-boy)
B) 25 - 25 (boy-girl)
C) 25 - 25 (girl-boy)
D) 25 - 25 (girl-girl)
A woman has 2 children at least one of which is a girl
A woman has 2 children and one of them is called Emma.
In both cases you can choose girl/Emma from positions B2, C1, D1, or D2 each with equal probability (0.25) but because there are 2 possible options in row D you're twice as likely to pick a girl from row D than from rows B or C.
If a girl is picked at random there's a 50% chance she's the first born (row C or D) and a 50% chance she's the second born (row B or D). In either case there's a 50% chance she'll be paired with a boy and a 50% chance she'll be paired with a girl.
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Brilliant analysis, at last we have clarity.
Wait - sounds a bit familiar
Then we need to go back to this example.
72 pairs of siblings, of which 54 contain at least one girl, 18 of which (1:3) contain 2 girls.
The issue as you had previously demonstrated is that if you take the total population of mothers of two children then the boy girl pairs are twice as common as girl girl pairs. Looked at from the female childs perspective the chances of a female sibling are certainly 1:2, but that doesn't look at it from the mothers perspective.
So the issue is whether we are looking at a random population of mothers of two, or a specified subpopulation. Excluding two males is a clear selection procedure which increases the probability of a female pair from 1:4 to 1:3, specifying a common identifying characteristic of one of the girls creates a further subselection which increases the probability of the 'mothers' within this group of families having a pair of daughters to 1:2. The reason for this is that the identifying feature is unique to the girls in that selection (not just the whole population of girls) and each girl in each family is assumed to have an equal chance of possessing this identifier.
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As for the first part of the puzzle, I don't see the argument about a girl having a 50% chance of having a sister as working, because we're not talking about picking a random girl from the population in general and asking:
'If this girl had one sibling, what are the odds that that sibling would be a girl?'
The question is about pairs, and is effectively asking:
'Of all the possible pairs of children which include at least one girl, what are the odds that both are girls?'
And the answer there is 1 in 3
Even though (in the case of pairs), the numbers of girls chosen randomly who have a sister is the same as the number who have a brother, from a 'pair' perspective that's because there's double-counting in the girl-girl pair (each has the other as a sister).
That double-counting doesn't happen if the unit of inspection is the pair, and we're asking questions about sibling pairs that have at least one girl in them.
As for the names/birth order, etc, if I was told that a woman had two children (of unknown gender) and I asked her "Is the eldest a girl?", and she said it was, I could then state with some confidence that the odds that both were girls was 1 in 2.
However, when it comes to information being offered unprompted, it's not immediately clear whether it changes anything - if I knew at least one child is a girl and I'm then told that the oldest (or youngest) is a girl, is that actually additional information, or is it just duplication - given that one child is a girl, it would always be possible for more data to be given away about that child (their age, birth order, etc) without necessarily giving me any more clues about their sibling.
Let us imagine that there was an algorithm to be gone through to inform the victim (eg me) of the relevant information:
a) Choose a mother of two who doesn't have two sons
b) Tell the 'victim' that the woman has a daughter
c) Give the name of a daughter (if the woman has two daughters, choose the name at random)
Given a really simple population with 2 boy's names and 2 girl's names, the possible combinations (ignoring birth order) would be:
Albert+Bob (25%)
Albert+Diane (12.5%)
Albert+Emma (12.5%)
Bob+Diane (12.5%)
Bob+Emma (12.5%)
Diane+Emma (25%)
We're left in a situation where the odds of an 'Emma' having a sister are 50%
However, think back to the algorithm above.
Given that algorithm, on being told a woman has a daughter named Emma, what odds should I give that that Emma has a sister?
Clearly, the only sibling pairs that could have ended up with me being told there is an Emma are
Albert+Emma (12.5%)
Bob+Emma (12.5%)
Diane+Emma (25%)
However, if the girl's name I am told is being picked randomly in the case of a pair of girls, then only half the time will the initial choosing of the Diane+Emma pair actually result in me being told 'One of the children is a girl called Emma!', which effectively 'dilutes' the Diane+Emma case down to be effectively no more common than the other two.
That seems to suggest that if I am told that a woman has a daughter called Emma, then there's only a 1 in 3 chance that that Emma has a sister, and a 2 in 3 chance that she has a brother.
Even generalised to having more names, unless there's something special about the name 'Emma', in the case of a boy:Emma pair I'll always end up knowing Emma's there, but in a girl:Emma pair, only half the time will I end up being given Emma's name.
I may well be missing something, but at the moment I'm not sure what.
Also I've just got back after a drive across too much of Europe, so am not really at my best. Do please make appropriate allowances.
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