Two Children 2

$\frac{1}{2}$21​

When you first find out that your colleague has 2 children, the possibilities made known to you are:

1. Both children are boys
2. The older child is a boy, and the younger child is a girl
3. The older child is a girl, and the younger child is a boy
4. Both children are girls

Each outcome comes at equal chance.

This time, knowing the gender of one child does not tell you the gender of the other. Knowing the older child is a girl narrows it down to 2 possibilities (outcomes 3 and 4), only 1 of which involves 2 daughters. Thus, the chances of him having 2 daughters is $\frac{1}{2}$21​.