# Numeracy #15: Male Population Puzzle

Question: Consider a country where every family wants to have a boy. Each family continues having babies until the arrival of a boy and then they stop reproducing. After years go by, what will be the proportion of boys to girls in the country? Assume that there is a 50% chance of having either a boy or a girl.

As always, try solving the problem yourself or keep reading for the solution.

### Solution

This puzzle is famous for being counter-intuitive. For example, if we imagine that the country only has 10 families, we can imagine a scenario where 9 of those families have a boy and the other has a girl. The family with a girl would try again and maybe they will have a boy this time. The final result would be a country with 10 boys and 1 girl.

It seems intuitive that the final result will be a country that has more boys than girls. However, a scenario where 9 of those families have a girl is just as likely as the scenario where they have a boy. It’s like flipping a coin, every new baby has an equal chance of being a boy or a girl. So what is the most likely outcome for our 10 family country?

Births Boys Girls
Round 1 5 5
Round 2 2.5 2.5
Round 3 1.25 1.25

The first round, we would expect half the families to have boys. The other half would try again and half of those families would have boys. This will continue. So what is the running population total?

Population Boys Girls
Round 1 5 5
Round 2 7.5 7.5
Round 3 8.75 8.75

It becomes apparent that the boys to girls ratio will remain 1 to 1 no matter how long this arrangement lasts. The reason being that each new baby has a 50% chance of being a boy or a girl independently of whether families prefer to have a boy.

### General Solution

Let N equal the number of families in a country. Since every family will continue having babies until they have a boy, we know that the number of boys in the country will eventually be N (each of the N families will have one boy). How many girls will there be?

Boys = N
Girls = N/2 + N/4 + N/8 + N/16 + …

Half of the families will have a girl, then half of those will have a girl, and so on till every family has a boy. We can model this as the infinite sum of an infinite series, which is defined as the limit of the first n terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + 1/32 = 0.96. Extending this to infinity gives us 1.

Girls = N/2 + N/4 + N/8 + … + N/2n
Girls = N – N/2n
Girls = N

Ratio of Boys to Girls = N/N