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Comment by paulmd | Hacker News Reader

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...

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Hm... I'm not sure you're answering the correct question.

Assume you're a man and you have a sister. Your next sibling is about to be born.

There is a 50% chance that it's a boy! In which case, he will have one brother (you), you will have one brother (him), and your sister will have 2 brothers. She wins.

There is a 50% chance that it's a girl! In which case, both she and your other sister will both have 1 brother (you), whereas you won't have any. You loose again!

"Assume you're a man and you have a sister."

You just put your finger on the scale. All you can assume is that you're a man, not that you have a sister.

4 siblings... 2 boys, 2 girls... 50/50...

The girls each have 2 brothers...

The boys each have 1 brother.

6 siblings, 3/3... 8 siblings, 4/4... etc

The problem is that you're not considering all the other cases. Here are the possibilities with 2 siblings:

    Gender:     B,B  B,G  G,B  G,G
    # brothers: 1,1  0,1  1,0  0,0

Average # of brothers for boys: 0.5

Average # of brothers for girls: 0.5

You can see that in the families with a mix of boys and girls, the girls do indeed have more brothers. But this is balanced out by the families with more (or in this case all) of one or the other.

Sure. Limit families to four children.

There's one way to have zero daughters. In BBBB, each of the four boys has 3 brothers. Score 12 for the boys here.

There's four ways to have one daughter. In each of them, she's got 3 brothers, while the 3 boys each have 2 brothers. Score 12 for the girls, 24 for the boys.

There's six ways to have two daughters. In each of them, the 2 girls each have 2 brothers, while the 2 boys each have 1 brother. Score 24 for the girls, 12 for the boys.

There's four ways to have three daughters. In each of them, the 3 girls each have one brother, and the boy has none. Score 12 for the girls, zero for the boys.

There's one way to have four daughters, but obviously nobody has a brother.

Add up the scores, and you'll see a tie at 48.

(EDIT: Bogus expected value removed after re-reading.)

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 likelihood, you find that males and females have the same number of brothers on average. (You can test this pretty easily with the n=1 and n=2 cases, and with a bit of work for the n=3 case.)

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.

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."

Nice. Thanks.

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.

My mom was one of four daughters (no sons), their mom was one of three daughters (no sons), two of my cousins only have girls, and my sister only has girls... my brother is the only one of myself and cousins that have any sons.

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!

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.

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.

True

That's true, but it doesn't imply boys and girls have the same number of brothers. As a commenter in the op pointed out, if families keep having children until they have a son, no boys have any brothers and all girls have exactly one brother. I think if they keep having kids until they have a girl, though, for gors and boys you have a .5 chance no brother, .25 1 brother, and so on.

If you kept flipping coins until you got heads - would you then beat the casino on average?

My question is: does the ratio of famale-to-male matter? What if the chance of a girl was 60% and male was 40%? My intuition is that it wouldn't, but I'm not sure.

If the simple model is in exact equipoise then the more complex considerations become very important for answering the question. If the a model taking into account all the complexities was still in equipoise than it would come down to noise. It is very unlikely that of the roughly 7 billion people on the planet the women have exactly as many brothers as the men. There's an empirical answer here.

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