Gender Control

In a certain country, having a daughter is traditionally prefferable to having a son. Therefore, a couple will continue to have children until a daughter is born. Once they have a daughter, they will stop. (Assume that it is possible for a couple to have an unlimited number of children). If the chance of a boy or a girl being born is equal, what would be the ratio boys to girls in this country?


Answer:


Assume the country has N families. Since every family will have children until a daughter is born, they will have N daughters. But there is a 1/2 chance that the first child will be a boy, making for N/2 "firstborn" boys. There is also a 1/4 chance that the second child will be a boy as well, for a total of N/2 + N/4 boys among the first- and second-born. It is easy to see that after each child, the chance that all the children are boys is reduced by 1/2, so the total number of boys is equal to N/2 + N/4 + N/8 + ... an infinite series that adds up to N. So the ratio of boys/girls in the country will actually be 1 to 1, unaffected by the strange custom.