»  Solutions to puzzles in my National Review Online Diary

  October 2009


In my October diary I posed the following brain-teaser:

The Androsocian tribe practices the following population limits: Each couple is entitled to have one boy child. No limit on number of girls. However, as soon as a boy is born, no more children! So, what will be the male/female ratio in the population, and what will be the average family size? (Assume zero childhood death rate, no twins, etc.)



A numerical example shows the main points. Let's suppose we have a population of 1,048,576 couples. Every couple has a baby. On the simplest assumption, half the babies will be boys, and those 524,288 families will then be complete. The remaining 524,288 couples have a second child. Half of these second children, which is to say 262,144, will be boys, and those families are then complete … and so on.

In table form, with totals in the bottom row:

This number
of couples
will have
this number
of girls
this number
of boys
524,288   0   1  
262,144   1   1  
131,172   2   1  
65,536   3   1  
32,768   4   1  
16,384   5   1  
8,192   6   1  
4,096   7   1  
2,048   8   1  
1,024   9   1  
512   10   1  
256   11   1  
128   12   1  
64   13   1  
32   14   1  
16   15   1  
8   16   1  
4   17   1  
2   18   1  
1   19   1  
1   20   0  
1,048,576   1,048,575   1,048,575  

I have assumed that 20 is the childbearing limit for all the females. You can, if you like, let the remaining family keep on trying for a boy up to some higher limit …

Thus the number of boys and girls is precisely equal. To get the average number of kids per family, you add up the number of kids (answer: 2,097,150) and divide by the number of families. The answer is a teeny tad less than 2, about 1.999998. The median number of kids is 1.5, half the families having less than that, half more.

That of course is all fiction. In practice most people will give up way before 20 births. This makes a difference to the total number of kids, of course, but not to the sex ratio. If nobody wants more than six kids, for example, you end up with 1,032,192 girls and the same number of boys, though the average number of kids is further below 2, actually 1.96875.

And even that is still remote from reality, as a statistically sapient friend points out:

There will be more girls than one would otherwise expect, because propensity-to-have-boys varies among families, and families more likely to have boys will tend to have their first boy sooner and so stop sooner … We should also account role of human nature in any practical demographic prediction. The empirical answer, alas, is that there will be lots more boys because propensity-to-have-boys does not vary symmetrically around a biologically-determined norm …