Puzzle #3: Boys and Girls

Solution:
The right answer to this question can be determined by intuition. However, does your intuition tell you that there will be more boys and less girls eventually? That is what most people would say. However, let’s see if we can solve it mathematically.

We know that for each birth the probability of a boy or a girl must be equal and is thus fifty percent for each. It is perhaps best to think of this as flipping a coin with half a chance on heads and the other half on tails.

Let’s first break it down by analysing what must be true at all times:

1. All families always have at least 1 boy.
2. All families must have somewhere between (inclusive) 0 and ^[1] girls.

We can further break down the second question in infinitely many sub-questions of the form: “How many families have girls?”. Where ranges between 0 and . A more visual way to express this is by coding girls with a G, boys with a B and then determining the likelihood of existence of each of the following family compositions (not counting the father and mother):

• B
• GB
• GGB
• …

Well, we already established that the probability of having either a boy or girl is fifty percent. Hence, the probability of a family with only a boy is fifty percent (equivalent to a family with zero girls), a family with one girl and one boy: twenty five percent, and a family with two girls and one boy: twelve point five percent, et cetera. Based on this we can calculate the number of girls as follows:

If you find this difficult to follow, keep in mind that each family that does not have a boy yet will, at the next birth, again have a fifty/fifty chance on a boy. However, as there are more births in the same family the probability of not having a boy goes down with each girl that joins the family. Realise that this is not because the independent probability of having a girl or boy changes, but the probability of having ever longer strings of girls being born in the same family goes down as the family grows larger. This problem is in fact equivalent to flipping a fair coin and determining the probability of heads or tails over an infinite set of trials also known as a Bernoulli trial.

People familiar with mathematics and probability have probably noticed that the above approximation can more accurately be written as an infinite series:

where is the number of families.

So, what is the outcome of this? The number of girls approaches 1 as we let approach infinity. We also know the number of boys in each family must be at least 1. Hence, the counter-intuitive answer is that most families will consist of one boy and one girl.

Some people will now draw the conclusion that the ratio must be 1:1 as well. However, that conclusion is controversial. Economist Steven Landsburg went as far as to offer a public bet on this. His main point is that even if the main variables (boys and girls) have an expected difference of zero, you can not conclude that they have a ratio of 1 to 1.

Mathematically the expected value of boys and girls are both one:

However, the question is what is the expected ratio of girls to the child population:

The approximation given by Landsburg is:

where is the number of families.

This seems like a more satisfactory answer then saying the ratio is 1:1, as it captures the value of very explicitly. Based on this, we can safely say that the right answer is: for a very large population the ratio will approach 1:1, but never actually reach it.

There are many other people that have given perspectives on this problem, and more thorough mathematical background, like here and here. However, some of them also make the original problem much more complicated than it was. Other sources I used to compile this article are here and here.