
Do men or women have more brothers? - networked
http://math.stackexchange.com/questions/1794123/do-men-or-women-have-more-brothers
======
millstone
The question was phrased ambiguously as "do men or women have more brothers."
Here's three possible interpretations.

One interpretation is "if I am the last born (or some other gender-neutral
specifier), do I have more brothers if I am a man or a woman?" As paulmd
answers, the chances are even.

But we can also interpret it at the aggregate level. Say we have a family with
M men and W women, and we decide to count brothers by polling each person for
how many brothers they have, and totaling the result up. If we poll men, we
get M(M-1) brothers, and if we poll women, we get MW brothers. The result
depends on the distribution, but at the expectation value (M = W), women have
more. Same conclusion if we average instead of sum.

The third, most natural interpretation is to "deduplicate:" ask each person to
list his or her brothers, take the count of the union. This is the familiar
sense: we would say there are three Brontë Sisters, even though none of them
individually had three sisters. Here the number of brothers equals the number
of men, except when then there's only one man; then each women has 1 brother
and each man has 0.

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ewjordan
Like tempestn noted in another comment, enumeration is straightforward for
small families, and makes it crystal clear that for any family size N, boys
and girls have the same average number of brothers:

    
    
      Gender:     B  G
      # brothers: 0  0
      B avg # brothers: 0
      G avg # brothers: 0
    
      Gender:     B,B  B,G  G,B  G,G
      # brothers: 1,1  0,1  1,0  0,0
      B avg # brothers:     (2 + 0 + 0) / (2 + 2) = .5
      G avg # brothers: (0 + 2 + 0) / (2 + 2) = .5
    
      Gender:     B,B,B  B,B,G  B,G,B  G,B,B  B,G,G  G,B,G  G,G,B  G,G,G
      # brothers: 2,2,2  1,1,2  1,2,1  2,1,1  0,1,1  1,0,1  1,1,0  0,0,0
      B avg # brothers:     (6 + 6 + 0 + 0) / (3 + 6 + 3) = 1
      G avg # brothers: (0 + 6 + 6 + 0) / (3 + 6 + 3) = 1
    
      Gender:     B,B,B,B  B,B,B,G (x4)  B,B,G,G (x6)  B,G,G,G (x4) G,G,G,G (I collapsed the different permutations here)
      # brothers: 3,3,3,3  2,2,2,3       1,1,2,2       0,1,1,1      0,0,0,0
      B avg # brothers:     (12 + 24 + 12 + 0 + 0) / (4 + 12 + 12 + 4) = 48/32
      G avg # brothers: (0 + 12 + 24 + 12 + 0) / (4 + 12 + 12 + 4) = 48/32
    

In a family of N children, both boys and girls will have, on average, (N-1)/2
brothers.

Arguments for unequal results are slipping in assumptions that make the
observed distribution different from the enumerated one.

It's an exercise for the reader to prove that this equality holds for higher
N, it's quite easy (the way I formatted the arithmetic shows how to match up
terms).

~~~
perilunar
You are assuming that for a family of N kids, each permutation is equally
likely. As others have pointed out, that is not the case.

~~~
ewjordan
This is a late reply because I missed this until now, but if you accept the
typical parameters of a statistical thought experiment, you should assume that
the permutations are equally likely. The real world nastiness of biology and
society fouls that up, but thought experiments like this are never intended to
be about not picky real world details, boy/girl is always assumed to be a coin
flip. (Not to say that real world results are not interesting as well)

And there are a half dozen arguments in this thread that claim to disprove
this result even if they accept the coinflip hypothesis - it's really those
that I take issue with.

------
btilly
The real answer is men. By a small amount.

First of all as many sources will afirm, for example
[http://www.pewresearch.org/fact-tank/2013/09/24/the-odds-
tha...](http://www.pewresearch.org/fact-tank/2013/09/24/the-odds-that-you-
will-give-birth-to-a-boy-or-girl-depend-on-where-in-the-world-you-live/), the
ratio of boys to girls at birth is not 50-50 and is not the same across ethnic
groups, cultures, etc. With some surprising interaction effects.

As soon as you have a mix of families with _different_ sex ratios, the odds of
siblings being brothers is now positively correlated (even if ever so little)
with your own gender. And the result of that is that the average man has more
brothers than the average woman does.

However the theoretical effect is very small, and you would need a large
random sample to reliably detect its existence.

------
mudbungie
I chose to answer the question empirically.
[http://pastebin.com/1g1bAEPt](http://pastebin.com/1g1bAEPt) Turns out it's
equal.

~~~
micheljansen
While that's a nice way to approach a mathematical problem, it does not take
into account the social aspects of the problem. For example, it used to be
(and perhaps still is?) quite common for families to aspire to have a boy and
therefore have another child when they have a girl, but not when they have a
boy.

Interestingly, these sort of things are much easier to model empirically the
way you did (all you would need is a small tweak to the "haveAKid" condition
in the while loop around line 35), whereas the math gets pretty complicated
fast.

~~~
dhimes
Well, it _was_ posted in _probability_ and not is some social science board.
So I thought it was safe to assume a purely mathematical question. Not that I
can answer it...

~~~
mudbungie
Yeah, I wrote that after staring at it for a few minutes, and giving up any
pretense that I could model it with confidence.

------
rcar
With the assumptions that 1) boy/girl is always random but 50/50, and 2)
people are choosing to have 1-10 kids ahead of time and going for it (that's
how people plan their families, right?), girls do have very slightly more
brothers than boys.

The notebook below shows a simulation of 50k towns of 100 families. It also
holds for different max numbers of children and more families.

[https://gist.github.com/rcarneva/7baac666bd65f487df73378a90c...](https://gist.github.com/rcarneva/7baac666bd65f487df73378a90c0d0b3)

~~~
waqf
Nice, it took me a while to find the fallacy. (I really hope you did that
deliberately, it's masterly.) What you have done is a variant of the
observation that _within any family_ the sisters have more brothers than the
boys, except that you averaged over 100 families to make the effect more
subtle.

But if you make the towns larger, the discrepancy will disappear (for example
at 100 towns of 50k families I would expect it to be lost in the noise).

------
paulmd
Seems like a straightforward Gambler's Fallacy situation. Assuming the chances
of each sex are independent and equally probable, then each subsequent child
has a 50% chance of each sex. Your previous streak is irrelevant (it doesn't
matter whether you're male or female), the chance of having a brother is
equal. As the number of families approaches infinity the number of brothers
will even out.

Or to rephrase the question - after flipping a coin and getting 50 heads in a
row, what's the chance of the 51st coin coming up heads? (answer: 50%)

Now in reality the chances of each sex may not be equal - some people have a
genetic predisposition to have more sons than daughters or vice versa. And of
course there's family planning and so on that may artificially alter the
number of brothers/sisters (eg to carry on the fmaily name).

[https://www.sciencedaily.com/releases/2008/12/081211121835.h...](https://www.sciencedaily.com/releases/2008/12/081211121835.htm)

~~~
jakobegger
Many people really want either a son or a daughter. So they'll keep trying
until it works. An acquaintance of mine had four sons before the fifth one
finally was a daughter.

I'm pretty sure that effect is a lot stronger than any genetic
predispositions.

~~~
squidfood
Actually, if everyone has babies of sex A until you have a baby of sex B, you
get a 50/50 mix.

    
    
      - Half the parents will have an only child of sex B.
      - One quarter will have one A, one B.
      - One eighth will have A, A, B.
      - One sixteenth A,A,A,B.
    

This converges on 50/50!

~~~
xadhominemx
Well if you assume some sort of practical limit to the number of children a
couple can have, then you end up with more B.

~~~
squidfood
Actually, no. If you have a hard limit, then some people will end up A,A, ...
A with no B. That evens it up again.

~~~
xadhominemx
True

------
aaron695
My understanding is women have a higher death rate than men when young due to
when there are limited funds they get worst heath care in families.

So brother and sister goes to boy. Brother brother stays brother brother.

Not sure on percents though. It might not override the issue of correlated
data and selective breeding.

(Edit - actually now I try think about it this is wrong since now there's also
one less women in the world, the mens average also changes)

------
serge2k
Classic stack overflow, top answer is a giant mass of "look how awesome I am!"

sensible answer is 3 down with half as many upvotes.

~~~
hinkley
That's one of my productivity secrets. Always read the top 4 answers before
deciding on a course.

Answer #3 is right far too often for my liking. And sometimes answer #4 brings
up corner cases the others missed.

~~~
stcredzero
I think the above structure is eerily reflected in large corporations and in
government. The actual work/know-how resides in tier 3 and 4, with the top two
tiers actually just being better at politics and self promotion.

------
tehwalrus
If every family only has one child, nobody has any brothers; the results are
equal at 0 average brothers.

If we take some distribution over size _n_ of the families, say a distribution
where family size is either 1 or 2, then the 1-size families don't contribute
(they add a zero to the fraction, dragging the probability down in a
systematic way equal for both sexes), and the 2-size families have four
equally probable outcomes.

those four outcomes (bb bg gb gg) yield:

two females with no brothers. two females with one brother. two males with no
brothers. two males with one brother.

i.e. the average numbers of brothers are equal (1/2 for males, and the same
for females.)

The next layer, where _n_ = 3, there are (ggg ggb gbg bgg gbb bgb bbg bbb) 8
combinations, yielding:

three females with no brothers. six females with one brother. three females
with two brothers. three males with no brothers. six males with one brother.
three males with two brothers.

Average number of brothers for females is (3 _0 + 6_ 1 + 3 _2) / 12 = 1.
Average number of brothers for males is (3_2 + 6 _1 + 3_ 0) / 12 = 1.

for _n_ = 4, (gggg gggb ggbg gbgg bggg ggbb gbgb gbbg bggb bgbg bbgg gbbb bgbb
bbgb bbbg bbbb), there are 16 combinations (yes, these map to binary integers
quite nicely):

four females with no brothers. twelve females with one brother. twelve females
with two brothers. four females with three brothers. four males with no
brothers. twelve males with one brother. twelve males with two brothers. four
males with three brothers.

Again, this comes out at an even probability - 48 / 32 = 2 brothers, on
average.

By induction, we can say that for any _n_ , the probabilities are equal (if
the probability of male and female children is equal).

Thus, for any distribution of family sizes, the probabilities are equal.

\---

Real world criticisms:

* males are more likely (~51% of births naturally, higher thanks to sex-selective terminations in China and India.) Thus, outcomes with more males are more likely, which skews towards male-heavy families. While this means both males and females have more brothers, there are more males to count with higher numbers of brothers, so their average goes higher.

* males die more frequently at younger ages, so older people will have fewer brothers than younger people.

~~~
qbrass
> * males are more likely (~51% of births naturally, higher thanks to sex-
> selective terminations in China and India.)

In China, there may be more sons, but they're not likely to have brothers.

> * males die more frequently at younger ages, so older people will have fewer
> brothers than younger people.

Most people still consider dead siblings to be siblings.

------
tlrobinson
> I think women have more as no man can be his own brother

The obvious rebuttal: no woman can be her own brother either.

------
PascLeRasc
I've often wondered if a certain restaurants is has more groups or individuals
dining - this seems pretty similar.

------
optimuspaul
my favorite... "it's unclear whether the question even makes sense"

------
jakobegger
What a great example of a seemingly simple question that can't be answered.

The question sounds like an elegant and abstract question straight out of a
mathematical textbook, but to answer it you need to start making assumptions
about the the distribution of family sizes, which leads you to family planning
and cultural differences. There is no way to solve this problem on a napkin.

Why does everyone try to solve this problem from first principles? The best
way to solve this problem would be to look at actual statistics!

So now I wonder: Is there any publicly available census data that's detailed
enough to answer this question? Or are there any public genealogical databases
that might be helpful in answering this?

~~~
acbabis
> Why does everyone try to solve this problem from first principles?

Trying to solve the problem from first principles is probably how a lot of the
posters came to the conclusion that it wasn't possible. If the problem _were_
possible to solve from first principles, it would have been a waste of time to
find/create data, so trying to solve it from first principles is the correct
approach given no prior knowledge of the answer.

