How many boys? How many girls?

I hadn’t heard this one before, but I like it a lot:

In a country in which people only want boys, every family continues to have children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop.

What is the proportion of boys to girls in the country?

Apparently used as a Google interview question (not that I’m a great fan of puzzle-based interviewing).

My long solution first

Distribution of families as proportions of all families in the country:

½: B
¼: GB
1/8: GGB
1/16: GGGB
etc.

Let F be the number of families.

How many boys?

Since every family stops after they get a boy the number of boys is F.

Alternatively, we can count the contributions of the different families:

Total boys = F x (½ + ¼ + 1/8 + 1/16 + ...)

This demonstrates that the infinite sequence

½ + ¼ + 1/8 + 1/16 + …

sums to 1, which is also apparent if you stand 1 meter from a wall and 50 cm then 25 cm etc., etc.

How many girls?

Again we sum:

Total girls / F = ¼ x 1 + 1/8 x 2 + 1/16 x 3 + …

= 1/4 + 1/8 + 1/16 + …
+ 1/8 + 1/16 + …
+ 1/16 + …
+ …

= 1/2 x (½ + ¼ + 1/8 + 1/16 + ...)
+ 1/4 x (½ + ¼ + 1/8 + 1/16 + ...)
+ 1/8 x (½ + ¼ + 1/8 + 1/16 + ...)
+ …

= 1/2 x 1
+ 1/4 x 1
+ 1/8 x 1
+ …

= ½ + ¼ + 1/8 + 1/16 + ...

= 1

So, Total girls = F

Solution

Equal numbers!

Simplifying assumptions?

Birth rate of 50-50

• No multiple births
• Large population
• No account made of multiple generations

Even without these assumptions, I would guess that the solution roughly holds.

Simple solution

Each child born has a 50-50 chance of being a boy or a girl. Each birth is independent, and not influenced by decisions of this, or any other family, so naturally half are boys and half are girls.