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Do men or women have more brothers?

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Re: Do men or women have more brothers?

#51

Post by mudbungie »

kqr wrote: post #47
mudbungie wrote: post #2

I chose to answer the question empirically. http://pastebin.com/1g1bAEPt Turns out it's equal.

I'm surprised you managed to make it that complicated! http://pastebin.com/fck8qkMc

I like your solution! Good use of iter().

You actually bring up an interesting question of design philosophies. I thought that the script that I posted was extremely explicit, and therefore clear. Yours is far more terse, and implicit, which means that there's less to understand, but there is some onus on the reader to decode what is happening.

I'm not sure which is better, as a policy. Obviously, solving problems with fewer lines is desirable, but it's also good to make solutions that are intrinsically obvious.

Re: Do men or women have more brothers?

#52

Post by 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).

Re: Do men or women have more brothers?

#53

Post by waqf »

rcar wrote: post #22

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/7baa…

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).

Re: Do men or women have more brothers?

#54

Post by 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..., 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.

Re: Do men or women have more brothers?

#55

Post by dhimes »

tempestn wrote: post #24
dhimes wrote: post #9

Earlier quoted context omitted.

But by the wording, however, it seems different. (I really, really suck at this stuff though.) If I have 50 men and 50 women in a family, the men each have 49 brothers, and the women each have 50. The coin-flip comparison would be "how many heads are there not counting a particular head" or something.

By assuming a starting family with equal numbers of males and females, you bias the result. You have to look at the distribution of all families. In families with similar numbers of boys and girls, the girls with have more brothers. But in families with significantly more of one gender, the boys will, on average, have many more brothers. When you combine all possible families of a given size, weighted by their likeli…

Very cool- thank you for getting me to think about this. I get your results if you answer the question for any given sibling group (say, M M F, for example)

  Number of male brothers of males = 1 each * 2 Males = 2
  Number of male brothers of females = 2 each * 2 Females = 2
and proceed to add all of those. In other words, if you weight the count by the number of siblings that have that count.

If you just add up, for any group, how many brothers a female in that group has vs how many brothers a male in that group has, then the females win. In the case above, for example, males have 1 brother, females have 2. So, said another way, odds are that a given female has more brothers than a given male.

But your argument led me to a result that was not at all intuitive (for me)- which is very cool.

Re: Do men or women have more brothers?

#56

Post by 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 (30 + 61 + 32) / 12 = 1. Average number of brothers for males is (32 + 61 + 30) / 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.

Re: Do men or women have more brothers?

#57

Post by serge2k »

get2zpointer wrote: post #41
serge2k wrote: post #3

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.

Why is three better? In any family with k brothers, each male will have k-1 brothers, while each female (if any exist) will have k brothers. So the number of brothers in S(x) is gender dependent, despite the the third answer's claim that it isn't.

> Why is three better?

Because you're wrong.

Re: Do men or women have more brothers?

#58

Post by 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)

Re: Do men or women have more brothers?

#59

Post by tempestn »

dhimes wrote: post #55
tempestn wrote: post #24

Earlier quoted context omitted.

By assuming a starting family with equal numbers of males and females, you bias the result. You have to look at the distribution of all families. In families with similar numbers of boys and girls, the girls with have more brothers. But in families with significantly more of one gender, the boys will, on average, have many more brothers. When you combine all possible families of a given size, weighted by their likeli…

Very cool- thank you for getting me to think about this. I get your results if you answer the question for any given sibling group (say, M M F, for example) Number of male brothers of males = 1 each * 2 Males = 2 Number of male brothers of females = 2 each * 2 Females = 2 and proceed to add all of those. In other words, if you weight the count by the number of siblings that have that count. If you just add up, for an…

In any given family of siblings that has both boys and girls, by definition the females will have more brothers than the males, yes. But it doesn't follow that "odds are that a given female has more brothers than a given male". The reason is that not all families have both. Families with multiple boys and no girls will bring the average up for boys. Families with multiple girls and no boys will bring it down for girls. The end result is that any given person has the same expected number of brothers as any other.

That also makes sense if you think about it intuitively. As the best (IMO) answer to the SO post put it:

"I'll write p for the gender ratio, i.e. the proportion of all people who are men. In real life p is close to 0.5, but this doesn't make any difference. In any random set of n persons, the expected (average) number of men is n⋅p.

1. Take an arbitrary child x, and let n be the number of children in x's family.

2. Let S(x) be the set of x's siblings. Note that there are no gender-related restrictions on S(x): It's just the set of children other than x.

3. Obviously, the expected number of x's brothers is the expected number of men in S(x).

4. So what is the expected number of men in this set? Since x has n−1 siblings, it's just (n−1)⋅p, or approximately (n−1)÷2, regardless of x's gender. That's all there is to it."

Re: Do men or women have more brothers?

#60

Post by perilunar »

ewjordan wrote: post #52

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:…

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.
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