Fun with Math

[Recursion]

10. So, the generalized answer, given the unequal birth rates we observe, plus the effect on population

EDITED: I screwed up the population calculation at first.

Assuming each childbirth is male with probability p and female with probability q = (1 - p), this is a series of identically distributed independent Bernoulli trials, you have a male on average every 1 / p births. So in the simple case where p = 0.5, you have 1 male every 2 births, and therefore as many males as females.

We observe in most populations that p = 0.5025 or so (presumably we evolved to produce slightly more males to account for higher male mortality), so you would have on average 1.05 males per every female (which is roughly what we have in the US).

The point is, your plan keeps the ratio of males to females at whatever it is "naturally".

Now, what is the expected population? Given that every couple stops after a male, you will have

Half of your couples having 1 child
A quarter having 2 children
An eighth having 3 children
A sixteenth having 4 children,

i.e., the probability that a couple has n children is 2 ^ (-n). So, with X couples, the number of children in the next generation is

0.5 X + (0.25 * 2) X + (0.125 * 3) X + ...

ie, sigma (n = 0 -> infinity) of n * 0.5 ^ n, which in closed form is 0.5 / (1 - 0.5)^2 = 0.5 / 0.25 = 2.

So you will have twice as many children as you have couples, which is a bit under the population replacement rate (since not everyone has children, and not all children survive to adulthood).

With the actual uneven birthrates, instead of 0.5 ^ n you have 0.5025 ^ n, which yields

sigma (n = 0 -> infinity) of n * 0.5025 ^ n, which in closed form is 0.5025 / (1 - 0.5025) ^ 2 = 0.5025 / 0.2475= 2.03 children per couple

Imagine we switch the sexes, and couples had boys until they got a girl?

Then you have p = 0.4975, 1 / p = 2.01, and the series is n * 0.4975 ^ n which in closed form is 0.4975 / (1 - 0.4975) ^ 2 = 0.4975 / 0.2525 = 1.97 children per couple.

So, as you can see, even small perturbations in the gender probabilities can have fairly large impacts on the population size with this scheme.