Drunkard's Walk - Leonard Mlodinow [59]
One way to understand this, if it still seems puzzling, is to imagine that we gather into a very large room 75 million families that have two children, at least one of whom is a girl. As the two-daughter problem taught us, there will be about 25 million two-girl families in that room and 50 million one-girl families (25 million in which the girl is the older child and an equal number in which she is the younger). Next comes the pruning: we ask that only the families that include a girl named Florida remain. Since Florida is a 1 in 1 million name, about 50 of the 50 million one-girl families will remain. And of the 25 million two-girl families, 50 of them will also get to stay, 25 because their firstborn is named Florida and another 25 because their younger girl has that name. It’s as if the girls are lottery tickets and the girls named Florida are the winning tickets. Although there are twice as many one-girl families as two-girl families, the two-girl families each have two tickets, so the one-girl families and the two-girl families will be about equally represented among the winners.
I have described the girl-named-Florida problem in potentially annoying detail, the kind of detail that sometimes lands me on the do-not-invite list for my neighbors’ parties. I did this not because I expect you to run into this situation. I did it because the context is simple, and the same kind of reasoning will bring clarity to many situations that really are encountered in life. Now let’s talk about a few of those.
MY MOST MEMORABLE ENCOUNTER with the Reverend Bayes came one Friday afternoon in 1989, when my doctor told me by telephone that the chances were 999 out of 1,000 that I’d be dead within a decade. He added, “I’m really sorry,” as if he had some patients to whom he would say he is sorry but not mean it. Then he answered a few questions about the course of the disease and hung up, presumably to offer another patient his or her Friday-afternoon news flash. It is hard to describe or even remember exactly how the weekend went for me, but let’s just say I did not go to Disneyland. Given my death sentence, why am I still here, able to write about it?
The adventure started when my wife and I applied for life insurance. The application procedure involved a blood test. A week or two later we were turned down. The ever economical insurance company sent the news in two brief letters that were identical, except for a single additional word in the letter to my wife. My letter stated that the company was denying me insurance because of the “results of your blood test.” My wife’s letter stated that the company was turning her down because of the “results of your husband’s blood test.” When the added word husband’s proved to be the extent of the clues the kindhearted insurance company was willing to provide about our uninsurability, I went to my doctor on a hunch and took an HIV test. It came back positive. Though I was too shocked initially to quiz him about the odds he quoted, I later learned that he had derived my 1-in-1,000 chance of being healthy from the following statistic: the HIV test produced a positive result when the blood was not infected with the AIDS virus in only 1 in 1,000 blood samples. That might sound like the same message he passed on, but it wasn’t. My doctor had confused the chances that I would test positive if I was not HIV-positive with the chances that I would not be HIV-positive if I tested positive.
To understand my doctor’s error, let’s employ Bayes’s method. The first step is to define the sample space. We could include everyone who has ever taken an HIV test, but we’ll get a more accurate result if we employ a bit of additional relevant information about me and consider only heterosexual non-IV-drug-abusing white male Americans who have taken the test. (We’ll see later what kind of difference this makes.)
Now that we know whom