Percentage of Sons
We choose to analyze these questions theoretically, using the basic theory of probability together with knowledge about geometric sums and series. We could have explored these questions in other ways -- for example, experimentally using simulations.
Suppose that p is the probability that a given birth is male. Then the fraction of the female population that gives birth to a son in their first pregnancy is p and the fraction of the female population that gives birth to a daughter in their first pregnancy is (1 - p).
The fraction of the original female population that has a second pregnancy is
(1 - p) and, thus, the fraction of the original female population that has a second pregnancy and has a son in the second pregnancy is
and the fraction of the original female population that has a second pregnancy and has a daughter in the second pregnancy is
Thus, after two pregnancies the number of sons as a fraction of the original female population is
and the number of daughters as a fraction of the original female population is
Assuming an infinite number of potential pregnancies for each female, the number of sons as a fraction of the original female population is
and the number of daughters as a fraction of the original female population is
Simple calculations of the two geometric series show that assuming an infinite number of potential pregnancies for each female the number of sons as a fraction of the original female population is 1 and the number of daughters as a fraction of the original female population is
If p = 1/2 there will be an equal number of sons and daughters and the population will replace itself in each generation.
A straightforward analysis of the results when p = 1/2 and each female has two children yields the same predictions.
- What will happen if p < 1/2, that is, if for each birth a son is less likely than a daughter?
- What will happen if p > 1/2, that is, if for each birth a son is more likely than a daughter?
- What will happen if the number of potential pregnancies for each female is less than infinite?