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.

What's important?

Computer scientist and mathematician Richard Hammond -- he of the Hamming Code -- used to ask his colleagues two questions. In order:

1. What the most important issues in your field?, and the follow-up
2. Why aren't you working on those?

Often this induced a cold-shoulder response, but others were grateful for the nudge.

I have been meditating on what the equivalent question should be in a business, as opposed to Science. My tentative questions are:

1. What are the most important issues facing our customers?
   
2. Why aren't we working on those?
   

From this it follows that deeply understanding our customers' issues is of paramount importance.