# Baby Boomers

by Kevin Lin

The ratio is 50-50.
Intuitively, it may feel that the families are adopting a strategy favoring producing a son, but this is incorrect. The families expected number of sons is one, by the definition of the strategy. But the expected number of daughters doesn't drop. Consider:
There is a 1-in-2 chance that the son is the first born, and so no daughters are born.
There is a 1-in-4 chance that the son is the second born, and so one daughter is born.
There is a 1-in-8 chance that the son is the third born, and so two daughters are born.
And so...
Therefore the expected number of daughters is (.50 * 0 daughters) + (.25 * 1 daughter) + (.125 * 2 daughters)... = 0.5 daughters.