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riddles >> medium >> Who has more sisters?
(Message started by: Grimbal on Nov 29th, 2013, 9:05am)

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Title: Who has more sisters?
Post by Grimbal on Nov 29th, 2013, 9:05am
On average, who has more sisters.   Boys or girls?

Title: Re: Who has more sisters?
Post by antkor on Nov 29th, 2013, 9:45am
Boys. And on average girls have more brothers.

Title: Re: Who has more sisters?
Post by towr on Nov 29th, 2013, 1:08pm
[hideb]*Each family has two children
*boys and girls are equally likely

So then,
BB, BG, GB and GG are equally likely and representative of the population.
The boys have two sisters, the girls have two sisters, both on average 1/2. So in this case it's a draw.

If boys and girls are not equally likely --- suppose we go so far as to say there is only one boy, then that one boy has a sister, and there's one girl without a sister, so then a boy has a very slightly higher average number of sisters (1 vs 1-http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/delta.gif)
Vice versa, if there is only one girl, then that girl has no sister, and boys have a very small chance of having one, so the boys win in that case as well. (http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/delta.gifvs 0)

Ill founded conclusion: it's practically a draw, but only at one point is it actually a draw, and outside of that the boys win by a whisker.
[/hideb]

Title: Re: Who has more sisters?
Post by antkor on Nov 29th, 2013, 1:21pm
i disagree.
This riddle reminds me of the possibilities in another riddle where a man asks his friend about his children and one is a boy. In that case, the chance the 2nd being a girl is 2/3.
I think the riddle should be approached in a way that for a family with N children, one child is of known gender. Then, you start taking all the different combinations regarding all N children with the chosen child remaining constant and the other ones go in all possible combinations.

Title: Re: Who has more sisters?
Post by towr on Nov 30th, 2013, 1:41am
I'm not sure what you disagree with, considering I reach the same conclusion as you and declare at the outset my assumptions are bad (unless you disagree they're bad  :P), but I disagree we should assume one child has a known gender, because if we're talking of a specific family why does the question ask for an average?

The question seems quite plain to me: Is sum(#sister foreach boy in world)/#(boys in world) greater/equal/less than  sum(#sister foreach girl in world)/#(girls in world)
So it's just a bit of demographics. And the puzzle is finding the data or a reasonable model. (Or an unreasonable model that can tell you some bounds.)

Title: Re: Who has more sisters?
Post by Grimbal on Nov 30th, 2013, 3:10am
My idea is that girls have more sisters than boys have.
Actually, they also have more brothers than boys have.
???

Title: Re: Who has more sisters?
Post by rmsgrey on Nov 30th, 2013, 7:35am
If you assume that, for any given parent, the genders of their children are statistically independent, then for any given child, the genders of their siblings (if any) are statistically independent of each other and of that child's gender.

So, on average, they'd be equal.

If they're not statistically independent, then it depends which way the bias goes - if siblings are more likely to be the other gender, then boys have more sisters; if they're more likely to be the same gender, then girls do.

There are differences between male and female sperm that the female body can use to favour one over the other, so certain differences in the mother's body chemistry mean that one family is more likely to have daughters, while another is more likely to have sons. I'm also told there's a statistical tendency for first children to be boys and later children girls. The former effect favours girls having more sisters; the latter boys, and I have no data on the relative scales of the two effects.

@towr: you neglect the cases where there are no girls, or no boys - effectively introducing a slight bias in favour of mixed gender siblings.

Title: Re: Who has more sisters?
Post by Grimbal on Dec 2nd, 2013, 2:50am
Here is my reasoning.

The rule that held in China until recently is that you can have only one child.  If that child is a girl, you can have a second child.

Assuming each couple gets all the children they can, out of 4 families, on average, 2 have one boy, 1 has a girl and a boy and 1 has two girls.  {B, B, GB, GG}
Among these children, 2 girls out of 3 have a sister, but only 1 boy out of 3 has a sister.

Since China is such a big country, and the imbalance is quite large, I would believe it tips the balance for the world population.

Funnily, in the above situation, girls also have more brothers than boys.  Girls have 1/3 brother on average, boys have none.

Title: Re: Who has more sisters?
Post by antkor on Dec 2nd, 2013, 6:46am
Grimbal I agree that since China has a population over 1M, it can be a good representation for the whole population on earth. However, there is something I don't understand very well. The fact that law enforces/affects the births and the gender of the children a family can have is introduced by government. I mean introducing a human factor isn't a way of having bias towards a gender of the two (in this case boys)? Would that be acceptable? Wouldn't it be more accurate if the calculations are based only on the number and the gender of the children a family may have? If this bias is acceptable how would you justify your opinion? Would you include all factors that would affect births within your experimental conditions?

Title: Re: Who has more sisters?
Post by Grimbal on Dec 2nd, 2013, 7:36am
I never said human factors introduce a bias towards a gender or the other.

The way it works is that the regulations introduce a correlation between the sex of the first child and the size of the family.  Therefore girls live in larger families and therefore have more brothers and more sisters.

There are other factors which I don't know.  That is why I said "I would believe that..." and not "It is proven that...".

Title: Re: Who has more sisters?
Post by rmsgrey on Dec 2nd, 2013, 7:41am

on 12/02/13 at 06:46:23, antkor wrote:
 Grimbal I agree that since China has a population over 1M, it can be a good representation for the whole population on earth. However, there is something I don't understand very well. The fact that law enforces/affects the births and the gender of the children a family can have is introduced by government. I mean introducing a human factor isn't a way of having bias towards a gender of the two (in this case boys)? Would that be acceptable? Wouldn't it be more accurate if the calculations are based only on the number and the gender of the children a family may have? If this bias is acceptable how would you justify your opinion? Would you include all factors that would affect births within your experimental conditions?

The Chinese law doesn't (directly) affect the gender ratio of births, but it does affect the division of those births into families - in every two-child family, the elder child is a girl.

There may be a reverse bias from Chinese parents taking steps to ensure that their second child is a boy, but it would have to be strongly expressed to offset the natural 2:1 advantage girls possess.

Title: Re: Who has more sisters?
Post by towr on Dec 2nd, 2013, 10:33am
According to wikipedia: "The sex ratio at birth (between male and female births) in mainland China reached 117:100  and remained steady between 2000 and 2013"
Though it's still a question where this bias falls. You'd think the first child doesn't matter much if they can have a second when it's a girl (unless they can only afford one financially). If so, there is a pretty strong bias to ensure the second child is a brother (I get ~ 61%).

I'm also not sure we can just ignore the rest of the world. And we have to consider the one-child policy only applies to a third of the Chinese.

Title: Re: Who has more sisters?
Post by Grimbal on Dec 4th, 2013, 5:55am
Indeed, if the second child is always a boy, then only boys have sisters.  I calculated that if the bias is on both children, you need a ratio of 0.618 boys per birth (the golden ratio) to make the number of sisters equal between boys and girls.  If the bias is on the second child only, you need a ratio of 2/3.

But clearly, the pressure for boys reduces the difference I found in the number of sisters (the 2/3 for girls vs. 1/3 for boys).

Title: Re: Who has more sisters?
Post by rloginunix on Dec 5th, 2013, 7:42pm
Beautiful question, Grimbal.

If a brother has as many brothers as sisters and a sister has twice as many brothers as sisters how many brothers and sisters are there in this family?

Let B be the number of brothers and S be the number of sisters in that family. Translating brother has as many brothers as sisters from English into algebraic we get:

B - 1 = S

Why minus one? Because a brother who has brothers is also a brother to other brothers and must be counted out. Translating sister has twice as many brothers as sisters from English into algebraic we get:

2*(S - 1) = B

Why minus one? Because a sister who has sisters is also a sister to other sisters and must be counted out. Solving this system of equations we get that B = 4 and S = 3.

The reason I introduced this goofy sub puzzle is to see what happens if the number of brothers and sisters becomes the same. Say it's 3. Then each of 3 brothers has 3 sisters. But, according to Grimbal's question, each of 3 sisters has only 2 sisters. And so on for any N. Please tell me if I am failing my English again.

If not then it seems that to answer Grimbal's question properly we have to realize that:

1). It discriminates against only one variable and as such
2). A girl can not be her own sister (unless you are shooting a comedy in which case your heroine when asked if she is her sister's mother will say yes) and hence we have these primitive cases:

if N(brothers) > N(sisters) -> boys
if N(brothers) == N(sisters) -> boys
if N(brothers) + 1 == N(sisters) -> neither
if N(brothers) + 2 <= N(sisters) -> girls

Was too tired last night, dangers of trying to think after you've been writing code all day. This morning I realized that it's not the number of brothers but the number of sisters that matters. Say there's 1 brother and 10 sisters. 1 brother can claim all 10 females as his sisters. But each sister can claim only 9 females as her sisters. 10 > 9, boys. So it looks like as long as there at least one male and one female offspring its boys (B is the number of boys, G is the number of girls):

default is neither
if B > 0
if G > 0
boys
else if B == 0
if G > 0
girls
[/edit]

These must be fed further into statistical and probabilistic analysis - your turf.

This was a look at the problem through the microscope. In the next post if you don't mind I will look at it through the telescope. Both posts should be probably read at the same time.

Title: Re: Who has more sisters?
Post by rloginunix on Dec 5th, 2013, 8:15pm
With Grimbal's permission I will generalize his question by finishing it thus "... over long term across many species" (rabbits have brothers and sisters too).

Why boys and girls exist in the first place?

Earth is about 3.5-4 billion years old. The first sexual intercourse defined as exchange of genetic material happened a couple of billion years ago (scientific speculation not a proven fact) between two primitive cells swimming in primordial soup. The first sperm and egg intercourse happened (scientific fact) 600 million years ago at the bottom of the ocean between corals. So mother nature invented two genders and a sexual reproduction because it provides the variance of the genetic material which helps the species fight the bad things off - viruses, etc.

There are exceptions of course. There's this lizard dwelling in the deserts of New Mexico, US, that manages to reproduce via eggs. All species are females. Consequently all daughters are identical clones of their mothers. Which may be a good thing. But if they get hit with some disease that their genes can't fight off - they are doomed.

So the little brothers and sisters of today become mommies and daddies of tomorrow. And there is a rather hard statistical metric out there called Average Fertility Rate (AFR). If AFR == 2.1 then mommy and daddy produce future mommy and daddy which produce future mommy and daddy, etc. and the population of species is stable. If AFR < 2.1 population is dwindling down. If AFR > 2.1 population grows.

So my reasoning is that unlike imagining flipping a coin many many times trying to understand what an infinity is we have a sort of historical luxury to witness this very long and very statistically significant experiment - nature says majority of species have boys and girls and the AFR must be 2.1 for the population to at least not die out.

So:
if AFR == 2.1
boys
else
depends - see table from previous post.

Title: Re: Who has more sisters?
Post by BMAD on May 23rd, 2014, 3:27pm
The theoretical distribution shows that it is equal as to who would have more sisters.  This can be shown using induction.

But just to start the proof:
[hide]
N=number of children in household

N=1
Either it is a
Boy with no sisters
Girl with no sisters
Equal

N=2
Either it is
2 Boys (No sisters)
2 Girls (each girl has 1 sister) 25% likely
1 Boy 1 Girl (Boys have 1 sister)  50% likely
Equal

N=3
Either it is
3 Boys (No sisters) p=1/8
2 Boys 1 Girl (boys have 1 sister) p=3/8
1 Boy 2 girl (Boy has 2 sisters, Girls have 1 sister) p=3/8
3 Sisters (girl has 2 sisters) p=1/8
equal

and so on[/hide]

Title: Re: Who has more sisters?
Post by Grimbal on May 23rd, 2014, 4:01pm
You assume that the decision to have one more child is independent of the sex of the existing child or children.  This assumption is not valid.

Title: Re: Who has more sisters?
Post by BMAD on May 23rd, 2014, 4:24pm
I only assume that a pregnant woman has a 50/50 shot to have either a girl or a boy.

But I do acknowledge that I am ignoring many cultures mores on male preference.

Title: Re: Who has more sisters?
Post by Grimbal on May 24th, 2014, 4:22am
The assumption that a child has equal chances to be male or female is not enough.
If, as it was the case in China, it is forbidden to have a second child if the first is a boy the odds change.  You will never have a 2-boy family.
The odds for N=2 become:
boy-boy 0%, boy-girl 0%, girl-boy 50%, girl-girl 50%.
In this case, a boy always has 1 sister, a girl has 2/3 chances to have 1 sister, 2/3 sisters on average.

Title: Re: Who has more sisters?
Post by BMAD on May 24th, 2014, 7:03am
I think we are having a theoretical vs empirical discussion.   I am saying that biologically it should be balanced.  Obviously by practice humans skew the odds.

Title: Re: Who has more sisters?
Post by rmsgrey on May 26th, 2014, 7:17am

on 05/24/14 at 04:22:17, Grimbal wrote:
 The assumption that a child has equal chances to be male or female is not enough.If, as it was the case in China, it is forbidden to have a second child if the first is a boy the odds change.  You will never have a 2-boy family.The odds for N=2 become:boy-boy 0%, boy-girl 0%, girl-boy 50%, girl-girl 50%.In this case, a boy always has 1 sister, a girl has 2/3 chances to have 1 sister, 2/3 sisters on average.

But how many 1-child single girl families are there compared to the number of one-child single boys?

If you assume that all families have a second child if possible, and no families have a third child, then, in 4 families, there are 3 boys with 1 sister between them and 3 girls with 2 sisters between them.

Title: Re: Who has more sisters?
Post by dudiobugtron on May 26th, 2014, 3:21pm

on 12/02/13 at 02:50:52, Grimbal wrote:
 Assuming each couple gets all the children they can, out of 4 families, on average, 2 have one boy, 1 has a girl and a boy and 1 has two girls.  {B, B, GB, GG}Among these children, 2 girls out of 3 have a sister, but only 1 boy out of 3 has a sister.

Title: Re: Who has more sisters?
Post by Grimbal on May 30th, 2014, 3:41am
And yet I forgot it in my last post...

Inded, you have to consider all families, it doesn't mean much to compare boys and girls in just 2-child families.

Title: Re: Who has more sisters?
Post by EdwardSmith on Jul 16th, 2014, 12:22am
I think the question being asked is :-
If a couple have a girl child, does it make it more likely that their next child is a girl.
I think the answer to this is yes.
So therefore Girls have more sisters.
Its more biology than maths.

Title: Re: Who has more sisters?
Post by Grimbal on Jul 16th, 2014, 5:53am
To be precise, what I meant is:
If you compute
a = the average number of sisters over all boys of the planet
and
b =  the average number of sisters over all girls of the planet,
which one of a and b is largest?

Title: Re: Who has more sisters?
Post by wakiza33 on Sep 25th, 2014, 12:49pm
[hide]With the slightly more girls than boys in the world, and an average of 2.4 kids--

Boys have more sisters, because of the '.4'. Because a girl can't be her own sister, boys will have more sisters. [/hide]

Title: Re: Who has more sisters?
Post by rmsgrey on Sep 26th, 2014, 4:37am

on 09/25/14 at 12:49:00, wakiza33 wrote:
 [hide]With the slightly more girls than boys in the world, and an average of 2.4 kids--Boys have more sisters, because of the '.4'. Because a girl can't be her own sister, boys will have more sisters. [/hide]

That's only true in families with at least one of each - in a family with no girls, nobody has any sisters; in a family with no boys, the girls have at least as many sisters each as the boys.

Even within mixed families, the families with more sisters per child have more girls, so, for example, if you take two 7-child families: one with 5 girls; one with 5 boys, then the 7 girls have 22 sisters between them, while the 7 boys have 20, meaning that, despite it being true that each boy has more sisters than his sisters do, it's still true that, in those two families, girls have more sisters than boys.

The effect whereby a group can win in every category but lose on aggregate is known as Simpson's Paradox if you want to look into it more.

In the case of numbers of sisters, families with roughly even numbers of each are more common than families with more extreme splits, and, on aggregate, do give boys more sisters than girls, but that's balanced by the more extreme splits giving girls more sisters than boys, and, so long as there's no correlation between an individual's gender and the number and genders of their siblings, it all cancels out (in theory - I haven't actually run the numbers for a general case) and you end up with equal numbers.

In practice, you have places like China, where social policy has introduced a bias, and you have various biological mechanisms that also mess with the assumption that knowing a person's gender tells you nothing further about their siblings' genders, and there's always plain old random noise which will tend to prevent perfect equality, so the answer in the real world will probably be that one gender does have more sisters than the other, but it may not always be the same gender...

Title: Re: Who has more sisters?
Post by wakiza33 on Sep 26th, 2014, 12:16pm
Thanks for the info.

I knew my logic was incomplete, but couldn't progress further--I was waiting for someone to incur a deeper direction.