## October 2009

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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.)

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*Solution*

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:

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 …