## Monday, November 7, 2011

### 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.

1. is there any online tutoring for learning Conditional Probability ?? i don't wanna go to school lectures so please give me....

2. @nick: Chapter 5 of Think Stats presents conditional probability. Also, http://en.wikipedia.org/wiki/Conditional_probability

3. If I'm not mistaken, the answer to this one turns out to be pretty complicated. I got an answer that reduces to the sensible value of 1/3 in the extreme case where everyone has red hair. In the limit as the probability of having red hair approaches zero, my solution reduces to the surprising (to me, anyway) value of 7/15. I expected it to approach 1/2 in that limit (like the Florida case), but I think I've managed to convince myself in hindsight that the 7/15 value makes sense. Using the given value of 2% redheads, I get an answer of about 45.6%.

I'll be interested to see if I've got it right.

4. @Ted: I believe that is the correct answer. Nicely done!