The red-haired girl named Florida

In The Drunkard's Walk, Leonard Mlodinow presents "The Girl Named Florida Problem":

"In a family with two children, what are the chances, if one of the children is a girl named Florida, that both children are girls?"

I like this problem, and I use it on the first day of my class to introduce the topic of conditional probability.  But I've decided that it's too easy.  To give it a little more punch, I've decided to combine it with the Red-Haired Problem from last week:

In a family with two children, what are the chances, if at least one of the children is a girl with red hair, that both children are girls?

[Edit: I clarified the wording to say that at least one of the children is a girl with red hair.  To be even more precise, I am looking for the conditional probability P(two girls | at least one girl with red hair).] 

Just like last week, you can make some simplifying assumptions:

About 2% of the world population has red hair.  You can assume that the alleles for red hair are purely recessive.  Also, you can assume that the Red Hair Extinction theory is false, so you can apply the Hardy–Weinberg principle.  And you can ignore the effect of identical twins.

Here is my solution.

For more fun with probability, see Chapter 5 of my book, Think Stats, which you can read here, or buy here.