The Problem of the Unfinished Game

Today's post is a simple question.

Let's say, hypothetically speaking, you met someone who told you they had two children, and one of them is a girl. What are the odds that person has a boy and a girl?

Consider your answer carefully, without doing a web search, or reading the comments to this post. Don't cheat -- but be prepared to explain your reasoning, because the solution might surprise you.

It's almost like some kind of conspiracy or something.