When you ask a question of the form “What are the chances of X given Y”, your are asking a question of conditional probability. These sorts of questions have come up in this blog before: “What are the chances of cancer given a positive test result?” “What are the chances a monkey prefers blue M&Ms to green M&Ms, given it prefers red M&Ms to blue M&Ms?” “What are the chances of predicting the NCAA tournament perfectly, given perfect predictions for the first two rounds?”
Conditional probability is extremely important for two reasons. First, it occurs all the time: it is a fundamental building block as we aggregate information in an uncertain environment. Second, people are really, really bad at it. In case after case, our intuition misleads us and we badly misestimate conditional probabilities. When a 90% accurate drug test (meaning it is positive 90% of the time for a drug user, and negative 90% of the time for a nonuser) comes back positive, what is the probability the person uses drugs. Our intuition screams “It has to be 90%”! But the probability of “User given positive drug test” is not the same as probability of “positive drug test given user”. If 5% of the population use drugs, then the probability of “User given positive drug test is about 1 in 3. Consider 1000 people: 50 are drug users so 45 will test positive; of the 950 non-users, 95 will test positive; so 45/(45+95) is the probability of user given positive test.
Note that in the argument above, I did not rely on the main theorem in conditional probability: Bayes Theorem. Bayes Theorem states P(A|B) (the probability of A given B) = P(B|A)P(A)/P(B). I could have worked it out that way, but in doing so I would have lost all intuition as to the result. For simple cases, the counting approach is much easier and shows why the result is what it is.
This argument is at the heart of Steven Stogatz’s excellent article “Chances Are”, online at the New York Times (thanks for the pointer, Matt). He gives some excellent examples of conditional probability, including a great riff on the O.J. Simpson murder trial.
The prosecution spent the first 10 days of the trial introducing evidence that O.J. had a history of violence toward his ex-wife, Nicole. He had allegedly battered her, thrown her against walls and groped her in public, telling onlookers, “This belongs to me.” But what did any of this have to do with a murder trial? The prosecution’s argument was that a pattern of spousal abuse reflected a motive to kill. As one of the prosecutors put it, “A slap is a prelude to homicide.”
Alan Dershowitz countered for the defense, arguing that if even the allegations of domestic violence were true, they were irrelevant and should therefore be inadmissible. He later wrote, “We knew we could prove, if we had to, that an infinitesimal percentage — certainly fewer than 1 of 2,500 — of men who slap or beat their domestic partners go on to murder them.”
In effect, both sides were asking the jury to consider the probability that a man murdered his ex-wife, given that he previously battered her. But as the statistician I. J. Good pointed out, that’s not the right number to look at.
The real question is: What’s the probability that a man murdered his ex-wife, given that he previously battered her and she was murdered by someone? That conditional probability turns out to be very far from 1 in 2,500.
Turns out that probability is about 90%.
We are in the midst of a curriculum review at the Tepper School of Business and are considering what we absolutely have to be sure our MBAs know. Being able to work with conditional probability is very high on my list: it is an area where your intuition will almost surely lead you astray. And, as the Strogatz article points out, while Bayes Rule may get you the right results, simple counting arguments are much more convincing.