Question

In an attempt to reduce the growth of its population, China instituted a policy limiting a family to one child. Rural Chinese suggested revising the policy to limit families to one son. Assuming the suggested policy is adopted and that any birth is as likely to produce a boy as a girl, and the successive births are independent, explain the following:

A) What would be the average family size?

B) What would be the ratio of newborn boys to newborn girls?

Answer 1

A) Average family size will be 2 children + 2 parents or a total of 4 people. B) Ratio of newborn boys to girls will be 1:1 For this, I will assume that each family will continue to have girl babies until a boy is born, then they will stop having children. So let's determine the number of girls that are born. 1/2 of all families will have a girl as their first child. 1/4 of all families will have a girl as their second child. 1/8 of all families will have a girl as their third child. The average number of girl children per family will be the limit of the infinite sequence 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/(2^n) where n approaches infinity. The limit to the above equation is 1. So on average there will be 1 girl per family. Now each family continues to have children until they get a boy. So each family will have 1 boy. So the average family size will be 2 children. Half the families will only have 1 boy, 1/4th will have 1 girl, and 1 boy, 1/8 will have 2 girls and 1 boy, etc. So the average family size is only 2 children. And the ratio of boys to girls will be 1 to 1 as expected if the birth of a boy is just as likely as the birth of a girl.