The task of the country with boys and girls

Let’s imagine a country where all parents want to have a son. Each family continues to give birth to children until they have a boy and then stops. What will be the ratio of boys and girls in this country?

You should ignore the situations where twins, triplets, couples who have no children, and couples who died before they make a son. First of all, you need to understand that if every family of this country wants to have a baby, then there will be definitely at least one boy. Why? The matter is that every pair bears children until they get a boy, and then they stop bearing children. In case we exclude the situations of twins-birth, then a boy means the only boy birth. For this reason, it is clear that the number of traditional families is equal to the number of boys. 

However, there can be any number of girls in any family. The good option to continue the analysis is to conduct the enumeration of girls’ number. Imagine that you have invited all the mothers to a giant room. With the help of a special system for speaking to such a huge audience, you say the following: “Every mom having the first baby girl, put your hand up.”

Surely, half of the women will do it. In case there are N moms, then N/2 moms will put their hands up. This number shows the number of girls born first. Specify this number on a special board as N/2.

Then you need to say: “Each mom that has the second baby girl – put your hand up or keep it up”.

The half of the put up hands will down and there will be no new put up hands. (By those moms that have not put hands up after the first message as the first was a child was a boy, and there is the only child in a family). It leaves N/4 put up hands leading to the fact that N/4 born second children were girls. Let’s write down this number on an imaginary board as well. 

“Each mom having the third child girl – put your hand up or keep it up”. You have already understood the provided here approach. Keep going until there will be no put up hands. Per each message, the number of put up hands will be halved. It creates the following sequence of numbers:

(1/2 + 1/4 +1/8 + 1/16 + 1/32 + … ) х N

The sum of an infinite series of such numbers is 1 (x N). Thus, it means that the number of girls is equal to the number of families (N) and to the number of boys (or pretty close to it). Therefore, the ratio of boys and girls we are interested in is 1 to 1. As a result, the ratio will be generally equal.