Interview with Leonard Mlodinow (part 14)
MLODINOW As I start talking about events in the world around us and looking at the psychological components–and I dealt with that, I greatly expanded that part–they were fascinating studies and I was just so interested I just kept putting more and more into the book.
ROBERTS Yes, that’s when you decided to ask me for help. “Oh, I wasn’t planning on this.” How did you learn about the lottery winner who won twice–the Canadian?
MLODINOW It was in a book somewhere, an academic book. A lot of those interesting stories came from academic papers or books.
ROBERTS That’s interesting.
MLODINOW Sometimes I’ll find something in the newspaper that was really interesting and I would track it down but a lot of it was in academic research. I don’t know why they found it.
ROBERTS Yes, who knows where they got it, but that’s where you got it. How did you learn about the Girl Named Florida stuff? Some professor told you?
MLODINOW My friend Mark Hillery that I mentioned from Berkeley.
ROBERTS A physics professor.
MLODINOW He heard it somewhere… It wasn’t quite this problem but then I kind of tweaked it and made it the Girl Named Florida Problem. That’s a great problem for the book.
ROBERTS Yes, I loved that. So he got it from some physicist . . .
MLODINOW I’m not sure; probably. I took a few days to figure out how to make it into this problem; I don’t remember exactly the problem he told me but I tweaked it into this problem. Just to show you how much work goes into the book, I even spent a whole afternoon deciding on the name Florida. I went back into the records–I needed a rare name–and I looked up different names and tried to find one that would be colorful, interesting, but that was rarely used, and I wanted to know the percentage that it was used; I dug up percentages of names. Everything in the book . . . if you read it, it might just sound like, ‘Oh, you know’ . . .
Not a thing is just tossed out there. Or very little; there’s an amazing amount of thought and work that goes behind every little detail.
ROBERTS That’s a very memorable detail I must say. I like it better than the Monty Hall Problem.
MLODINOW I do, too. I think it’s interesting; I found in the reactions to the book that the Monty Hall Problem has gotten more press and in some ways more reactions, which I found interesting given that it has been talked about before and this problem was completely new. I think this problem is in some ways even more striking than the Monty Hall Problem, more counterintuitive and more difficult to believe and certainly closer to something you might actually encounter. And yet I’ve gotten a lot more response based on the Monty Hall Problem and a few places have said that I gave the best explanation they’ve seen. I think the New York Times review said that, too. The New York Times did mention the Girl Named Florida Problem and said that they still find it hard to believe even though they followed the explanation.
ROBERTS I thought your explanation of the Girl Named Florida problem was very clear.
February 27th, 2009 at 12:36 pm
[…] Interview with Leonard Mlodinow (part 14) […]
August 16th, 2009 at 10:43 pm
I am very familiar with the Girl Named Florida problem — I have read Mlodinow’s book and other sources that cited it. I agree with the answer, though I don’t really find the explanation all that satisfying either in Mlodinow’s book or on other websites. I came up with my own explanation that arrives at the same result and works better for me. I’m hoping that my alternate explanation will help others to understand the problem and the result.
As Mlodinow explains, the key to solving many difficult problems in probability is ensuring that you arrive at the correct sample space. Usually in confusing problems like this one, there are multiple “filters” on a larger sample space that must be applied to screen out irrelevant outcomes. The order in which these filters are applied doesn’t really matter from a mathematical standpoint, but it can make a huge difference in whether the explanation helps to clear up the confusion. In this problem, I think this is a key factor and for me (and I’m guessing many others) it helps to apply the filters in a different order than that chosen by Mlodinow.
Mlodinow’s explanation starts with an original sample of all families with children. This is reduced to all families with two children, then all families with two children one of whom is a girl, then to all families with two children one of whom is a girl named Florida.
Now here is my approach. Start with all families with two children. In this particular problem it was useful for me to think of the family as being represented by a particular set of parents rather than a particular set of children. The reason is that the child’s name is an attribute that tells you more about the parents than the child since they are the ones who chose it. So let’s then reduce the set of all parents of two children to the set of all parents of two children for whom the name Florida ranks in their top 2 favorite names for girls. Further dissecting this set we observe that there are twice as many families with a boy and a girl than there are families with two girls. (This should be obvious since it just involves counting the four possible outcomes, discarding the two boy families, and looking at the proportion of the remaining possible outcomes: BG, GB, GG). Now, all of the GG families will have at least one Florida. I’ll assume as Mlodinow did that the number parents who chose to name both of their girls Florida is immaterial. For the families with only one girl, it depends on whether Florida was the parents’ first choice or second choice. Let’s assume that this is a 50/50 split. It now should be clear that a girl named Florida with one sibling is equally likely to have a sister as she is to have a brother.
Not only do I find my own explanation easier to follow. I think it provides some additional subtle insights that were missing in Mlodinow’s answer: (1) By focusing on parents preferences, it is more clear that the random distribution of names is between, but perhaps not within families. The chances that two girls will both be named Florida in the same family is probably much less than the probability of one girl named Florida squared. (2) When you use my approach, another important assumption becomes evident. Are parents more likely to give the first child or the second child an unusual name? A hidden assumption in Mlodinow’s solution is that parents are equally likely grant an unusual name to the first and second born. With my approach, this assumption is more exposed. This is probably a very testable assumption, though I have no idea whether it is true or not. I wouldn’t be surprised if the evidence showed a bias one way or the other greatly altering the result.