The daughter paradox

[Admin]

So there must be an error in the process that forces us to look at the number of sisters, instead of looking at the problem from the mother's perspective

The way I finally understood this was:

...(1) .(2)
A) 50 - 50 (boy-boy)
B) 50 - 50 (boy-girl)
C) 50 - 50 (girl-boy)
D) 50 - 50 (girl-girl)

If we have an initial set up like that and we ask the first question: "Jane has two children. One is a daughter. What’s the probability that she has two daughters?"

Then we are forced to choose between rows (B), (C), and (D) so we get the 2/3 - 1/3 split.

If we ask the second question: "Jane has two children. One is a daughter, Emma-Louise. What’s the probability that she has two daughters?"

Emma-Louise could be in any position occupied by 'girl' and so we're forced to choose a girl from cells (B2), (C1), (D1), and (D2) which leads to a 1/2 - 1/2 split.

In question 2 where the name Emma-Louise is a very popular name (i.e. a high percentage of girls are called Emma-Louise) the split tends away from 1/2 - 1/2 towards 2/3 - 1/3 simply because in row (D) Emma-Louise cannot be paired with another Emma-Louise.

The information in the two questions is subtly different in that by talking about a 'girl' we think of her a one of a pair, but when she's named (e.g. Emma-Louise) we think of her as an individual. So we end up choosing from pairs (selecting info from rows) or from individuals (selecting from cells) which gives 2 different answers.

It's not unusual actually as the probability of what we can know or predict depends crucially on the information contained in the question. The almost invisible difference between the two questions is that 'girl' and 'Emma-Louise' are different types of information (I certainly made the mistake of assuming they were equivalent!)

[Pebble]

The information in the two questions is subtly different in that by talking about a 'girl' we think of her a one of a pair, but when she's named (e.g. Emma-Louise) we think of her as an individual. So we end up choosing from pairs (selecting info from rows) or from individuals (selecting from cells) which gives 2 different answers.

It's not unusual actually as the probability of what we can know or predict depends crucially on the information contained in the question. The almost invisible difference between the two questions is that 'girl' and 'Emma-Louise' are different types of information (I certainly made the mistake of assuming they were equivalent!)

My interpretation of the question is "What is Janes probability of having two daughters?" One of them is named Emma Louise! Unless we know that Emma Lousie is more likely to occur in D2, then the probability of two daughters cannopt change.

[Janot]

The almost invisible difference between the two questions is that 'girl' and 'Emma-Louise' are different types of information (I certainly made the mistake of assuming they were equivalent!)

Well, the difference remains invisible to me, and I still think they are equivalent.

Suppose you meet the women with a child, the child dressed in a boiler suit with a bag on its head. During the conversation it emerges that the mother has another child. You compute on the spot that there is a 50-50 chance of it being a sister. The child you meet is then introduced to you as Emma-Louise. The probability of the sibling being a girl remains the same.

I think my problem (one of them) is that I can't see why you would treat the children as anything other than individuals. It is my interpretation of the statement 'The woman has two children, one of them is a girl' which is the statement I could make after the above encounter. By then, of course, I have labelled one daughter, the one I met, which is equivalent to giving her a name. I think the statement 'one of them is a girl' has an inherent ambiguity, hence this thread.

[polomint38]

Well, the difference remains invisible to me, and I still think they are equivalent.

Suppose you meet the women with a child, the child dressed in a boiler suit with a bag on its head. During the conversation it emerges that the mother has another child. You compute on the spot that there is a 50-50 chance of it being a sister. The child you meet is then introduced to you as Emma-Louise. The probability of the sibling being a girl remains the same.

I think my problem (one of them) is that I can't see why you would treat the children as anything other than individuals. It is my interpretation of the statement 'The woman has two children, one of them is a girl' which is the statement I could make after the above encounter. By then, of course, I have labelled one daughter, the one I met, which is equivalent to giving her a name. I think the statement 'one of them is a girl' has an inherent ambiguity, hence this thread.

Janot, I have been reading this thread with interest and I think you are correct, people are trying to over-intellectualise this, the name of one sibling is irrelevant on gender of other sibling.

Also as far as I know there is no law to stop me from giving the same given name to all my children (regardless of gender or order they are born), which would make all this thing about names irrelevant.

P.S. I do accept that calling all your children the same name would be problematic and extremely stupid.

ETA:
I was talking to this girl from Croydon, she had 4 sons, I aksed (that how it is spelt there) the names of her sons.
She said, "that one is Bob, so is that one and the other 2 are Bobs as well, it makes it easy to call for them to come into the house."
I asked, "what if you only want one of them to come in?"
"Easy", she said, "I use their surnames".

[Croydon Bob]

P.S. I do accept that calling all your children the same name would be problematic and extremely stupid.

I used to work with a woman called "Rose", she had an older sister called Rose (they were NOT from Croydon).

Michael Jackson gave several of his kids the same name (he is NOT from Croydon).

It seems to be more common than a sane person would expect.

[chaggle]

I was talking to this girl from Croydon, she had 4 sons, I aksed (that how it is spelt there) the names of her sons.
She said, "that one is Bob, so is that one and the other 2 are Bobs as well, it makes it easy to call for them to come into the house."
I asked, "what if you only want one of them to come in?"
"Easy", she said, "I use their surnames".

Although it pains me to say so - that wasn't bad

[Pebble]

Sorry to intellectualise this, but I think understanding where the fault in the logic has arisen helps understand how statistics can fool us.

A mother of two children both of whom are not boys, has a one in three chance of having two girls. Of the four possible girls she might have in these combinations each has an equal chance of being named Emma Louise. Emma Lousie as a name for a daughter is therefore just as likely in a daughter with a brother as a sister.

The apparent paradox is created because from the mother's perspective we are looking at combinations (2xBG, 1xGG), from Emma Louise's perspective we are looking at permutations - where order of birth is relevant so 4 possible outcomes among the three combinations.

[Janot]

A mother of two children both of whom are not boys

Well, I still think this statement is ambiguous because you can arrive at one in three or one in two depending on the argument. I must be missing something.

[hughcumber]

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?’

I still think that the way the question is worded the name is irrelevant.

JJ and others reasoning seems sound at first but to me doesn't apply to this question. The name is different information but I am seeing it as irrelevant information.

I think the confusion is that you are answering the question "Jane has two children. One is a daughter, Emma-Louise. What's the probability that Emma-Louise was born first?"

I think.

My brain has enjoyed the workout though

[Matt]

Suppose you meet the women with a child, the child dressed in a boiler suit with a bag on its head. During the conversation it emerges that the mother has another child. You compute on the spot that there is a 50-50 chance of it being a sister. The child you meet is then introduced to you as Emma-Louise. The probability of the sibling being a girl remains the same.

Both Correct,

Before the naming of the child you don't know the sex of either the child you met or the other child. The 50-50 odds you quote are correct both for the other child and for "bag-head"

If however you're told that "one of her children is a girl" but not whether it's "bag-head" or "the other one" then that's a whole different ball game. The information that you're given might then apply either to Bag-Head being a girl or "the other one" being a girl.

However after the naming of "bag-head" as Emma Louise then the information that "At least one or other of the children are girls" is not news. The odds of the other child being a girl or boy are 50-50. That at least is correct.

[Pebble]

Both Correct,



However after the naming of "bag-head" as Emma Louise then the information that "At least one or other of the children are girls" is not news. The odds of the other child being a girl or boy are 50-50. That at least is correct.

But that is not the question. The question is still, what is the probability of the mother having two daughters, not the probability of the other child being a girl - these two questions give different answers.

[Janot]

If however you're told that "one of her children is a girl" but not whether it's "bag-head" or "the other one" then that's a whole different ball game. The information that you're given might then apply either to Bag-Head being a girl or "the other one" being a girl.

Good point - therein lies the difference between the 1 in 2 and 1 in 3 probabilities.

[Harryprice]

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

Is this not simply the same as: "If a mother has two children, one of which is a daughter, what is the probability that the second child also a girl?"

In which case the answer is 1 in 2, since the only options available for the second child are girl or boy. I guess I must be missing something given the number of replies here but what is it?

[Pebble]

Well, I still think this statement is ambiguous because you can arrive at one in three or one in two depending on the argument. I must be missing something.

If I ask the question, a mother with at least one daughter named Emma Louise has one other child, what is the probability that the other child is a girl? The answer is 50%.

2 x B+Emma Louise and 2x Girl + Emma Louise. The reason for this is that one pair is counted twice. GG becomes G followed by Emma Louise and Emma Louise followed by G. But these are a sinlge instance of GG, described differently because of naming.

If however, I ask the question, a mother has at least one daughter named Emma Lousie has one other child, what is the probability of her having two daughters? Now the answer is 33%.

The reason for this is that any one mother can have 3 possible outcomes BG; GB; GG - she may have Emma Louise in either GG position in the last option this does not change the probability of her having two daughters, only the probability of one of them being named Emma Louise - since she happens to have two daughters to choose from.

[Harryprice]

I suspect the point of the "paradox" is a lot simpler than this thread suggests! I think it is simply saying that apparently irrelevant information, the sex of one child, actually IS relevant as it changes the question. It reminds me of those physics exam questions where they supplied the values for a lot of 'physical constants'. It was always worrying when you didn't need them all!