Drunkard's Walk - Leonard Mlodinow [55]
We also make Bayesian judgments in our daily lives. A film tells the story of an attorney who has a great job, a charming wife, and a wonderful family. He loves his wife and daughter, but still he feels that something is missing in his life. One night as he returns home on the train he spots a beautiful woman gazing with a pensive expression out the window of a dance studio. He looks for her again the next night, and the night after that. Each night as his train passes her studio, he falls further under her spell. Finally one evening he impulsively rushes off the train and signs up for dance lessons, hoping to meet the woman. He finds that her haunting attraction withers once his gaze from afar gives way to face-to-face encounters. He does fall in love, however, not with her but with dancing.
He keeps his new obsession from his family and colleagues, making excuses for spending more and more evenings away from home. His wife eventually discovers that he is not working late as often as he says he is. She figures the chances of his lying about his after-work activities are far greater if he is having an affair than if he isn’t, and so she concludes that he is. But the wife was mistaken not just in her conclusion but in her reasoning: she confused the probability that her husband would sneak around if he were having an affair with the probability that he was having an affair if he was sneaking around.
It’s a common mistake. Say your boss has been taking longer than usual to respond to your e-mails. Many people would take that as a sign that their star is falling because if your star is falling, the chances are high that your boss will respond to your e-mails more slowly than before. But your boss might be slower in responding because she is unusually busy or her mother is ill. And so the chances that your star is falling if she is taking longer to respond are much lower than the chances that your boss will respond more slowly if your star is falling. The appeal of many conspiracy theories depends on the misunderstanding of this logic. That is, it depends on confusing the probability that a series of events would happen if it were the product of a huge conspiracy with the probability that a huge conspiracy exists if a series of events occurs.
The effect on the probability that an event will occur if or given that other events occur is what Bayes’s theory is all about. To see in detail how it works, we’ll turn to another problem, one that is related to the two-daughter problem we encountered in chapter 3. Let us now suppose that a distant cousin has two children. Recall that in the two-daughter problem you know that one or both are girls, and you are trying to remember which it is—one or both? In a family with two children, what are the chances, if one of the children is a girl, that both children are girls? We didn’t discuss the question in those terms in chapter 3, but the if makes this a problem in conditional probability. If that if clause were not present, the chances that both children were girls would be 1 in 4, the 4 possible birth orders being (boy, boy), (boy, girl), (girl, boy), and (girl, girl). But given the additional information that the family has a girl, the chances are 1 in 3. That is because if one of the children is a girl, there are just 3 possible scenarios for this family—(boy, girl), (girl, boy), and (girl, girl)—and exactly 1 of the 3 corresponds to the outcome that both children are girls. That’s probably the simplest way to look at Bayes’s ideas—they are just a matter of accounting. First write down the sample space—that is, the list of all the possibilities—along with their probabilities if they are not