I just finished Leonard Mlodinow's new book The Drunkard's Walk: How Randomness Rules Our Lives. It was very interesting, though not really what I expected. I expected a sort of Freakonomics for probability and statistics, and to a certain extent it was. But it was very historical. Mlodinow discusses how each of the concepts was originally discovered, and by whom, in some detail. This was a pleasant surprise, because while I didn't learn any new concepts (nor was I expecting to), I didn't know much at all about the history of probability and statistics. Overall, it was highly entertaining, and one of the problems actually tripped me up the first time I read it (the same "girl named Florida" problem that intruiged Alex Tabarrok). I'm always a little weirded out after I read a book by a probabilist, like Normal Accidents, or Struck by Lightning: The Curious World of Probabilities, or even Fooled by Randomness. I've always been acutely aware of the powerful role played by randomness and chance, but the probabilistic world view assigns an uncomfortably large role to chance. No serious person actually thinks that markets are perfect, but probabilists tend to almost categorically reject that market outcomes are indicative of anything. For example, Mlodinow implicitly takes aim at markets:

The cord that tethers ability to success is both loose and elastic. . . . It is easy to believe that ideas that worked were good ideas, that plans that succeeded were well designed, and that ideas and plans that did not were ill conceived. And it is easy to make heroes out of the most successful and to glance with disdain at the least. But ability does not guarantee achievement, nor is achievement proportional to ability. And so it is important to always keep in mind the other term in the equation — the role of chance. ... It is a tragedy when a belief in the judgment of experts or the marketplace rather than a belief in ourselves causes us to give up.People routinely make determinations that have market consequences based on way too few data points, and that obviously leads to significant inefficiencies (the less talented employee getting the promotion, the less efficient company winning the contract, etc.). But I think competition does a pretty good job of diminishing the effects of randomness as much as possible. Chance is extremely important, but not overwhelming.

## 41 comments:

I think the "girl named Florida" example is tricky because it matters how you discover this fact about her. This is true of the original thought experiment, too.

So to start at the beginning, take the situation where you know that someone has 2 children, with at least 1 daughter (assume that each child's sex is independently set to M or F with p = 50% of each). The probability of two daughters (FF) depends on exactly what you know. If you happen to see one of the kids, and it's a female, then the probability of FF is 0.5. This is because there are two children, one observed and one unobserved. The unobserved child will be M 50% of the time and F 50% of the time.

On the other hand, if you elicit the sex of one of the children in a different way, p = 1/3. So for instance, you ask the mother, "Do you have at least one daughter?" In that case, there is no "observed" child, so the traditional explanation works.

Now, how do you find out that there is a girl named Florida? I think the answer might determine the probability. Say you ask your friend, "Do you have at least one daughter?" "Yes." "All right, now tell me something really unusual about that daughter." "Ummmm... her name is Florida" (or "she was born between midnight and 12:07 a.m. on June 14," which is, changing the date and time, something about each person that is rare).

In this instance, I don't think p = 1/2, I think it still equals 1/3, since there is something unusual about everyone, and you specifically elicited that information about this particular girl.

On the other hand, if you start out by asking, "Do you have any girls named Florida (or whatever)?" then p = ~1/2. Picture looking at all families with a child named Florida, narrowing it down to 2-child families where Florida is a girl, and then seeing what proportion are FF. Essentially, "Florida" is standing in for observing one of the children.

So yeah, I guess I think the question is posed in such a way as to muddy the crucial factor, which is how you came to learn of the child's name.

Ah, I see that AntiAntiCamper made the same point in comments on Alex Tabarrok's post.

Mlodinow's error is in the last phrase of the sentence in the second full paragraph on p. 113: "That leaves us with just (boy, girl-F), (girl-F, boy), (girl-NF, girl-F), and (girl-F, girl-NF), which are, to a very good approximation, equally likely." In fact, the odds for (B, GF) and (GF, B) are 1/3 each, while the odds for (GNF,GF) and (GF, GNF) are 1/6 each. Let's look at it step-by-step.

First, we are told that a woman is pregnant with fraternal twins. There are four equally likely outcomes: (B,B), (B,G), (G,B), and (G,G). So the odds that both twins are girls is 1/4.

Second, we are told that at least one of the twins is a girl. There are now three equally likely outcomes: (B,G), (G,B), and (G,G). So the odds that both twins are girls is 1/3.

Third, we are told that some girls are named Florida, but we don't yet know how many girls are so named. Let's assume for a moment that half of all girls are named Florida. (B,G) thus has two equally likely outcomes: (B,GF) and (B,GNF). Each is thus 50% of 1/3, or 1/6 likely. Similarly, (G,B) has two equally likely outcomes, (GF,B) and (GNF,B), each of which has a 50% of 1/3, or 1/6 chance of happening. (G,G) has four equally likely outcomes, (GF,GF), (GF,GNF), (GNF,GF), and (GNF,GNF), each of which is 25% of 1/3, or 1/12 likely to occur.

Fourth, we are told that at least one of the twins is a girl named Florida. (B,G) can thus only be (B,GF) and has a 1/3 chance of being the outcome. Similarly,(G,B) can only be (GF,B), and has a 1/3 chance of being the outcome. (G,G) can thus only be (GF,GF), (GF,GNF), or (GNF,GF), each of which has a 33% of 1/3, or 1/9 chance of happening.

Finally, we are told that Florida is a very rare name, and nobody (except maybe George Foreman) is going to name both twin girls Florida. So, (G,G) can now be only (GF,GNF) or (GNF,GF) each of which has a 50% of 1/3, or 1/6, chance of happening. The 1/3 chance that both twins are girls has not changed because of any additional information. Simple, eh?

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