Probability- boys vs. girls

Factual Questions
41

This ignores the fact that the normal boy-girl sex ratio at birth is 105 boys for every 100 girls. Survival rate is different, especially if assuming a modern society with good healthcare and no wars to thin out the male ranks.

Second, what is the reproduction standard? That one woman is paired with one man? So if there is an imbalance then potential breeders are not breeding.

42

Or perhaps, in the latter case, one would say that the birthrate is 50/50 and the death rate is not. At any rate, other than that, yeah, spot-on.

ETA: Hm, I’m not sure, but I think maybe this is the same point ChinaGuy was making. If not, I’m unsure what his first part is saying. For that matter, I’m not sure about the relevance of his second part either: potential breeders not breeding has no effect on the sex ratio.

43

First of all, birth rate is not 50/50. More males are conceived and are born in most societies. Ignoring sex-selected infanticide, more male infants die. In the US, Japan (and many other ‘modern’ societies) environmental toxins have resulted in more females born than males.

Secondly, due to the differences in sexually based reproductive fitness, high quality mothers tend to produce more sons than daughters. This has been found in many animal species and some studies have suggested it is true in humans as well. Thus, we would expect to see very well-fed, ‘high quality’ women producing more sons (as long as environmental toxins don’t get in the way). And ‘low quality’ women producing more daughters.

Not to mention all the variations based on time of conception, etc.

So, this whole thing is mute. Too many variables.

44

Cute.

Still, I think there is clear precedent for discussing math problems like this, where we speak of real world events but only because, when idealized without real world complications, they are evocative illustrations of the abstract principles we really intend to discuss. We have specifically opened by assuming 50/50 birth rates. That is the context of the discussion. The real world doesn’t impinge on that, except as an interesting side note.

45

Incidentally, one point borschevsky mentioned but which seems to have been lost in the thread is that Blaster Master’s assumption 2 (no twins or other multiple births) is also unnecessary (as long as, in keeping with our cardinal assumption, the probability of giving birth to n boys and m girls all at once is equal to that of giving birth to n girls and m boys all at once. And even that could be weakened, though I doubt anyone would really care to contest it.).

46

WELL EXPLAINED!

47

I think this simple and elegant resolution of the question has been overlooked.

As long as the probability of each birth being that of a boy is 50%, 50% of all births will be of boys. Period. No matter what societal norms have caused this birth to take place. No other considerations* are necessary, nor can they change the outcome.

  • barring selective abortions (although I think this is basically covered by the caveat that “the probability of each birth being that of a boy is 50%”, anyway); and again, I’m talking about births – infanticide, and in general differential survival rates, happen later.
48

Hell, society can adopt whatever norms it wants, but getting biology to cooperate is a bitch.

49

OK, I’m going to have to ask for a cite for this. Any time someone ascribes something to “toxins” my alarm bells go off like gangbusters. I imagine you are referencing the amount of female hormones in drinking water, but I have never heard of that or anything else affecting male/female conception rates.

Also, it’s moot.

50

“It’s moo.”
“What?”
“It’s moo. It’s like a cow’s opinion – it just doesn’t matter.”

51

There’s a simple illustration of the fact that assumption BM#3 isn’t necessary. Consider couples that stop producing children (due to death, infertility, etc.) and the “tree” of children that have been lost (that they would have produced). Note that this looks identical to the original: 50% B, 25% GB, 12.5% GGB etc. IOW, it has an equal number of boys and girls.

So couples that stop having children do not affect the outcome.

52

Thanks Joey.

53

Here’s a simple way to visualize this problem. For simplicity’s sake, let’s assume each family can have only three children before going infertile. Now, let’s consider all of the possible birth combinations. Each of these are in order from first birth to last and have an equal probability of occurring:

GGG
GGB
GBG
BGG
GBB
BGB
BBG
BBB

So, in our hypothetical society of 8 families, each family is “fated” with one of these outcomes. Given their family planning strategies, let’s see where they each end up. The “|” denotes when they stop having children.

GGG
GGB
GB | G
B | GG
GB | B
B | GB
B | BG
B | BB

So, adding up the children that make it into this society, we end up with 7 girls and 7 boys. There’s no reason why you couldn’t scale up these numbers to more realistic fertility possibilities and end up with the same even split.

What is interesting, though, is how larger families tend to have far more girls in them than smaller families. It would be interesting to find out how that would impact the cultural norms of this hypothetical society. Since each individual boy would get more attention on average and thus be more likely to succeed, it’s not hard to see how this bias towards boys could be self-perpetuating. But that is beyond the scope of this question.

54

It could perhaps be seen as a bias toward boys. It could also be a bias against them, along the lines of “Girls are good - have as many as you want. Boys are a problem - once you’ve had one, you don’t get to stink up the show with any more.”

55

What about small culture? I think the below example should work for larger cultures too, but it’s easier to look at the numbers with a smaller sample. If there are 128 couples existing in this culture:

64 have B
32 have G B
16 have G G B
8 have G G G B
4 have G G G G B
2 have G G G G G B
1 has G G G G G G B
1 has G G G G G G G +they will try again and have a (B or G)
This gives us a:
50% chance the culture will have 128 boys, 127 girls (last family is G G G G G G G + B).
25% chance the culture will have 128 boys, 128 girls (last family is G G G G G G G + G B).
12.5% chance the culture will have 128 boys, 129 girls (G G G G G G G + G G B)
6.25% chance the culture will have 128 boys, 130 girls (G G G G G G G + G G G B)
3.125% chance the culture will have 128 boys, 131 girls (G G G G G G G + G G G G B)
etc.

The odds that the ratio will favor boys is 1:2
The odds that the ratio will favor girls is 1:4
The odds that the ratio will be 1:1 are only 1:4.

56

here’s a script to run simulation:

society where families stop having children once they have a boy

10/9/07

repeat experiment 10 times

10.times do
boys=0
girls=0

#1000 generations
1000.times do
while rand>0.5
	girls+=1
end
boys+=1

end
puts “#{boys} boys”
puts "#{girls} girls

"
end
…and the output…
>ruby bg.rb
1000 boys
1080 girls

1000 boys
1078 girls

1000 boys
1016 girls

1000 boys
959 girls

1000 boys
974 girls

1000 boys
1039 girls

1000 boys
934 girls

1000 boys
916 girls

1000 boys
1004 girls

1000 boys
934 girls

>Exit code: 0
…expected ratio seems to be 1:1

57

Well, this isn’t really the right sort of thing. You seem to be taking it as given that exactly 64 couples will have a boy on their first try, the exactly 32 couples will have a boy on their second try, and so on, with variability only introduced at the last stage, because you can’t have exactly 1/2 a couple getting a boy on their 8th try. But this isn’t quite right: the very fact that your methodology would indicate “exactly 1/2” except for your ad hoc workaround at that point indicates the need for reformulation. If you use a limited population but account for the probabilities that “the last” couple could have any number of girls, you really ought to account for the probability that not exactly 1/2 of them will have a boy on the first try, and so on. After you account for all those probabilities, the odds that the ratio will favor boys and the odds that the ratio will favor girls will be precisely equal.

As in the first couple of posts, think of this as coin flips. Whenever any family conceives, God flips a fair coin to determine the gender of the baby. God also keeps a logbook of all his coin flip results. To determine the birth gender ratio at any time, it suffices to look at the head/tails ratio in God’s logbook. But, of course, this head/tails ratio is equally likely to favor heads as tails: it was a fair coin, after all, and whatever humans’ sex practices, they can’t do anything about that. They could make God flip it faster, slower, only on Thursdays, but a fair coin is still a fair coin.

58

This would seem to say that there is a resulting preference for boys, but that is not true. There is a 50% chance there will be one more boy. There is a 25% chance they will be equal, and there is a 25% chance they will favor **one or more **girls.

The “or more” part is critical to understanding that the expected value at the end is 50/50.

59

What flight says is true as well, but I reiterate that timtupp’s method of analysis is misguided nonsense, for the reasons I outlined in my last post (there’s no guarantee that exactly 64 out of 128 couples will have boys on the first try; in fact, it’s extraordinarily unlikely that exactly 64 will. And once you account for this sort of thing, the odds of there ending up being more boys are the same as the odds of there ending up being more girls)