Chinese Children

I don't think this problem is to be taken literally (that is the females are aborted, what about twins triplets etc).


I thought it was fair to assume a couple has a child and continues to have children until they have a boy, and they only have one child at a time.

I'll try the clever way first.


First a lemma: If a family has exactly one boy, then the expected number of children in the family is 2.


general concept of proof: if that were not the case then half of the chilren born would not be boys, which violates our assumption.


So we have 1 billion families and they all will have exactly 1 boy. Therefore on average a family has 2 children. The answer is then 2 billion children.


I can also sum the infinite series, but that is boring.


1B * (1*(.5)+2*(.5)^2+...)


or


1B * x * Sum(k*x^k-1) for x=.5

=1B*x*d/dx[Sum(x^k)]

=1B*x*d/dx[1/(1-x)] - geometric series

=1B*x*1/(1-x)^2

=2B when you plug in .5 for x

Apparently, 1 in 5 people in the world are Chinese.


There are 5 people in my family, so it must be one of them.


It's either my mom or my dad. Or my older brother

Colin. Or my younger brother Ho Cha Chu.


But I think it's Colin.

Half the total children born will be boys, and each couple will have one boy, so there will be one billion children.


This problem is very similar to the "bee and trains" problem I heard a while ago.


Two trains start 100 miles apart and approach each other at 20 miles per hour. A bee, starting at the first train, flies towards the second train at 100 miles per hour. Upon reaching the second train, it turns around and flies back towards the first train also at 100 miles per hour. It continues to do this back and forth until the trains meet. When the trains meet, how far has the bee traveled?


The word on the street is that John von Neumann was asked this question and immediately gave the correct answer. The guy who asked him the question said, "wow... you sure found the trick quickly," to which von Neumann responded, "what trick? I summed the series..."

pretty funny

It seems to me that if we use the geometric distribution, we haven't really gotten away from summing the series, we've just hidden it inside some very intuitive and easy to understand probability concept. You can say 'geometric distribution', but finding the expectation of that is basically summing a series. (At least, I can't think of another way.) Of course, I don't think there's anything wrong with this. All of math (good math) is an attempt to put difficult or intimidating problems into a previously known context so the solution seems natural, and the geometric distribution seems to do that in this case.