You meet two arbitrary children of a married couple the odds are $50\%$ that they both have blue eyes. How many children does the family have?

Question

I'm stuck on the following riddle: you meet two arbitrary children of a married couple the odds are $50\%$ that they both have blue eyes. How many children does the family (the married couple) have?

The possible answers are:

• A: $3$

• B: $4$

• C: $5$

Answer 1

There can't be $3$ children, since then there would be $3$ possibilities for which two children you meet, so you would need $1.5$ of those possibilities to give two blue-eyed children.

It can be $4$ (I'll leave you to work out how many of the four need to have blue eyes).

It can't be $5$. There are $10$ options for which two children you meet, so you need $5$ of these to give two blue-eyed children. But if there were $3$ blue-eyed children there would $3$ ways to meet two of them, and if there were $4$ blue-eyed children there would be $6$ ways, so you can't have exactly $5$.

So the answer must be $4$.

(edit: of course this assumes that you know the answer is one of those three options. But the next value that works mathematically is $21$, which is possible but rather unlikely.)