# Social Engineering (The Birthrate Problem)

The distant country of Toomalia has a birthrate of 67 boys for every 33 girls. In an effort to restore a 50:50 gender ratio in the country, Toomalian policymakers institute a mandatory new social policy: Couples seeking to have children may continue to do so as long as they bear only girls, but must stop as soon as they bear one boy.

Assuming the Toomalian people i) rigorously adhere to the policy, and ii) do not selectively abort their offspring, what will the net effect on the boy:girl birth ratio be when the new policy is put into effect?

Note that you may assume the gender of each conceived child is independent from the gender(s) of its siblings, but you should not assume that every couple wishes to conceive as many children as possible. Many couples will voluntarily stop having children after having 2 or 3, for example.

• Why are we assuming that the children are independent of each other? The whole point of a country doing something like this is knowing they are dependent. A couple that has a boy first has a greater chance of having another boy... Would be better to factor that into this question to some degree. – blankip Oct 25 '14 at 6:46

The ratio will stay the same. I'm not even sure how to prove that, since it seems so self-evident.

One simple approach:

Since the gender of a child is independent from their siblings, we can ignore who the parents are and ask under this procedure, what will the gender of the next child born to any couple in the country be? Well it'll still have a 67% chance of boy, 33% chance of girl. And since the parents/siblings don't matter in that number, those chances will be constant, so over time as more children continue to be born, it will be in the same ratio.

• You are correct, of course. However I believe most people won't see the answer as so straightforward. If you want a more challenging problem, you can compute the precise correlation between back-to-back sibling genders needed to restore a 50:50 birthrate under the effect of the policy. In this case you can assume that couples will continue to have children indefinitely until they have a boy. – COTO Oct 25 '14 at 3:17
• +1. I prefer this answer because of its simplicity. Three birth ratio isn't changing so nothing else matters. – Joel Rondeau Oct 25 '14 at 4:37
• @JoelRondeau: Assuming that there would be no correlation between the sex of a particular parent's earlier and later offspring, the policy would have no effect. If such a correlation existed, however, it could have a huge effect. Suppose every male individual had a 2/3 chance of inheriting a gene that would cause all offspring to be male, and a 1/3 chance of inheriting a gene that would cause all offspring to be female. If the number of children each father would sire was independent of the children's sex, then the 2/3 male-female ratio would be observed, but... – supercat Oct 29 '14 at 18:00
• ...if anyone whose first child was a boy refrained from having any more, while those who a girl would then proceed to have more, the male/female ratio could be equalized or even reversed. – supercat Oct 29 '14 at 18:02
• @supercat Yes, that's an interesting side angle not specifically covered by the question. Gender is independent of siblings but nothing stated it was independent of parents. – Joel Rondeau Oct 29 '14 at 18:05

Theroretically, I believe, the ratio would go up. Explanation will follow as soon as it is confirmed.

EDIT: I think the ratio will stay the same. The proof is outlined below.

If families want 3 kids, there are 8 possible outcomes. BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG, with B for boy and G for girl. However, these are not all equal scenarios. I will change the birthrate to 2/3 and 1/3 for boys and girls, respectively, as this helps with calculations but won't change the outcome.

• BBB = 8/27
• BBG = 4/27
• BGB = 4/27
• BGG = 2/27
• GBB = 4/27
• GBG = 2/27
• GGB = 2/27
• GGG = 1/27

If each family is to stop after having a boy, the outcomes are changed. 4xB, 2xGB, GGB, and GGG. After adding up the percentages, it comes out to:

• B = 18/27
• GB = 6/27
• GGB = 2/27
• GGG = 1/27

This means that the birth rate would be 26 boys for every 13 girls - so the ratio will stay the same. It seems my hypothesis was wrong in the first place.

EDIT again: I still think this is right, but I am starting to doubt my proof. I'll see if there is a better way to explain it.

2/3 = One boy, no girls. 2/3 of 1/3 = One boy, one girl. 2/3 of 1/3 of 1/3 = One boy, two girls. 2/3 of 1/3 of 1/3 of 1/3 = One boy, three girls.

As one can see, this can be extended infinitely. Simplified mathematically, there is a $\frac{2}{3^{G+1}}$ chance of getting G girls and 1 boy. This shows that the birth rate will stay the same.

If the sex of each child is independently randomly determined with a 2:1 ratio favoring boys, then the stated policy will have no effect on male versus female birth rates. If, however, the events are not independent, then the policy might have an effect. Suppose, for example, that that one third of all males produced only XY chromosomes while the other two thirds produced an even mix of XX and XY. In that scenario...

• 1/3 of the males generate only XY chromosomes; they would sire one son and stop.
• 1/3 of the males (half of 2/3) would generate mixed chromosomes, but their first child would be a son, so they'd stop.
• 1/6 of the males (half of the remainder) would generate mixed chromosomes, and sire a girl followed by a boy and then stop.
• 1/12 of the males would generate mixed chromosomes and sire two girls and a boy, then stop.
• 1/24 would sire three girls and a boy, then stop.
• 1/24 would sire four girls; assume for simplicity that they all stop.

Every man would thus sire one boy; 1/3 would sire a least one girl; 1/6 at least two, 1/12 at least three, and 1/24, four.

The average number of girls sired by each man would thus be 2/3. Not enough to manage a 1:1 male/female ratio, but yielding a 3:2 ratio which is less lopsided than the 2:1 ratio that would exist without the policy.

This (pretty) graph is the best illustration I could provide:

 1st batch: B (1/3)   B (1/3)   G (1/3)
//\
2nd batch: ―         ―       BB  G


At first batch, we have 2/3 boys and 1/3 girls.

Second batch has the same ratio.

It would be the same for next batches.

As all batches have the same ratio, if we merge batches the ratio remains the same.