1. ## Probability problem

I'm sure some of you have heard some variant on the below problem:

A dictator wanted more soldiers in his army so he decided to pass a new law whereby he thought he could increase the male population. The law was that women were allowed to have as many boy children as they liked but once they had a girl, they were not allowed reproduce after that. So for example, some mothers were lucky and got to have 10 boys and other were unfortunate and they only had one girl.

A census was done 30 years later and the dictator wanted to see if his plan had worked. To his surprise, the ratio of males to females was still roughly 50:50!

Now, I understand that babies come out one at a time and that a baby always has a 50:50 chance of being either sex.

but if you look at it another way:

Some mothers had 10 boys, some had 8 and some had 5. However, no mother had more than 1 girl. For the population to be 50:50, there had to have been much more mothers with a firstborn girl than there was mothers with a firstborn boy. If the chances of the firstborn child is 50:50, then why is this the case?!

2. I think the ratio boy/girl was 49,51/50,49.
Perhaps I got that wrong, but I can't be far off (lol).

3. To see how it works out, consider a pool of 128 mothers. Each one has one child, and let's suppose that exactly 64 of these turn out to be girls. Now, we have a child population of 64 boys and 64 girls, with 64 mothers left over to reproduce. So, we repeat the process, this time with 64 mothers, resulting in 32 girls and 32 boys. After this second round of reproduction, we have 96 girls and 96 boys, with 32 mothers left to reproduce. So you can see that each round of reproduction leaves the gender ratio unchanged, no matter how many times you repeat it.

The "babies arrive one at a time" part is key. It implies that for every mother who has, say, 8 boys, there are (on average) 8 other mothers who had 1 girl.

4. Originally Posted by John Connellan
For the population to be 50:50, there had to have been much more mothers with a firstborn girl than there was mothers with a firstborn boy.
Not so!
50% of first-borns were girls.
50% of second-borns were girls.
50% of third-borns were girls...

Focusing on the mothers is a red-herring.

To see how it works out, consider a pool of 128 mothers. Each one has one child, and let's suppose that exactly 64 of these turn out to be girls. Now, we have a child population of 64 boys and 64 girls, with 64 mothers left over to reproduce.
You are implicitly assuming that a family will keep having children until they are forced to stop. Some of those 64 mothers who had a boy will stop with one boy.

This doesn't matter, however. Looking at the mothers is, as Pete said, a red herring. Pete nailed it: Half of the first-borns will be girls. Whether all or just 1% of the families who have an eldest son go on to have a second child, is irrelevant. Of those that do go on, half of the second children will be girls and half boys. And so on.

The census would of course have not only revealed that the ratio of males to females was still roughly 50:50, but also that the population had dwindled because

$\sum_{n=1}^{\infty} \frac n {2^n} = 2$

So even if every family continued having babies until they had a girl (some poor mother who have to have had 30 boys) the fertility rate would be less than a sustaining level (about 2.1 or 2.2 with modern medicine).

To see how it works out, consider a pool of 128 mothers. Each one has one child, and let's suppose that exactly 64 of these turn out to be girls.
But this is wrong, the whole point is that some mother can have more than one child

7. Originally Posted by Pete
Not so!
50% of first-borns were girls.
50% of second-borns were girls.
50% of third-borns were girls...

Focusing on the mothers is a red-herring.
But why? It still logically must be the case that if no mother had more than one girl (whereas many mothers had more than one boy) - that the only way the next generation can be 50:50 is if there were more first born girls than boys.

I can't see a way around this logic and saying it's a red herring is not helping me

8. Originally Posted by John Connellan
But this is wrong, the whole point is that some mother can have more than one child
So what? You obviously have the wrong answer, so ignore the mothers. Look instead at the male-to-female ratio of the first-born, second-born, and so on.

Some families stop at one child, either because they decided one child was enough or because their first child was a girl. Whatever the reason they decide to stop at one, 50% of these first-born children are girls.

Some families will go on to have two children (and stop at two). 50% of these second-born children are girls. Some families will go on to have three children (and stop at two). 50% of these third-born children are girls. And so on.

So, the ratio of males to females of the first-born is 50:50; second-born; 50:50; third-born, 50:50, ..., nth-born, 50:50. The overall ratio of males to females is a weighted average of the ratio for first-born, second-born, etc. Since the ratio for each is 50:50, the overall ratio can only be 50:50.

9. OK, I have finally reconciled the problem in my own head. I can see that although there are many mothers with more than one son and although there are no mothers with more than one daughter, the distribution of sons and daughters is what explains the 50:50 ratio in the end.

The thing is, half the mothers have no sons at all and it is this characteristic of the distribution that allows the 50:50 ratio even though there are mothers with many sons

10. Originally Posted by D H
You are implicitly assuming that a family will keep having children until they are forced to stop.
Indeed. My reading of the OP was that it (implicitly) included such an assumption.

Originally Posted by John Connellan
But this is wrong, the whole point is that some mother can have more than one child
Right, I didn't say that each mother has ONLY one child. I just said that they each have one child to start with. Hopefully it was clear from reading the remainder of my post that more children could follow that first round.

Also, I think the insistence on ignoring the mothers is misplaced. It's true that they're a red herring, but the entire point of the OP was to understand why that's the case.

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