I know: as headlines go, this one is not exactly Man Bites Dog. Let me be a bit more specific. Either the *New York Times* or trial lawyers don’t understand probability. (This, incidentally, is a good example of the inclusive “or”.)

The *Times* has an interactive feature illustrating the process by which lawyers decide whether to allow someone to be seated on a jury. For those who don’t know, in most if not all US courts, lawyers are allowed to have potential jurors stricken from jury pools, either for cause, if there’s evidence that a juror is biased, or using a limited number of “peremptory challenges” to remove people that the lawyer merely suspects will be unfavorable to his or her side. The Times piece asks you a series of questions and indicates how your answers affect the lawyers’ opinion about you in a hypothetical lawsuit by an investor suing her money manager for mismanaging her investments.

The first two questions are about your job and age. As a white-collar worker, I’m told that I’d be more likely to side with the defendant, but the fact that I’m over 30 makes me more likely to favor the plaintiff. A slider at the top of the screen indicates the net effect:

So far so good. Question 3 then asks about my income. Here are the two possible outcomes:

So if I’m high-income, there’s no effect, but if I’m low-income, I’m more likely to side with the plaintiff. This is logically impossible. If one answer shifts the probability one direction, the other answer must shift it the other direction (by some nonzero amount).

Before the lawyers found out the answer, they knew that I was either low-income or high-income. (A waggish mathematician might observe that the possibility that my income is *exactly* $50,000 is not included in the two possibilities. This is why no one likes a waggish mathematician.) The lawyers’ assessment of me before asking the question must be a weighted average of the two subsequent possibilities, with weights given by their prior beliefs about what my income would turn out to be. For instance, if they thought initially that there was a 70% chance that I’d be in the high-income category, then the initial probability should have been 0.7 times the high-income probability plus 0.3 times the low-income probability.

That means that if one answer to the income question shifts the probability toward the plaintiffs, then the other answer *must* shift the probability in the other direction.

So either the lawyers the reporter talked to are irrational or the reporter has misunderstood them. For what it’s worth, my money is on the first option. Lots of people don’t understand probabilities, but it seems likely to me that the *Times *reporters would have asked these questions straightforwardly and accurately reported the answers they heard from the lawyers they talked to.

If that’s true, it seems like it should present a money-making opportunity for people with expertise in probability. Lawyers who hired such people as consultants would presumably do a better job at jury selection and win more cases.