Are you sure? Assume the lawyers have access to a table like the following, of previous results.

              Plaintiff    Defense
    Under 50k        75         25
     Over 50k        50         50

What's wrong with the Times's conclusions?

If this was the first question, and both incomes are equally common, then before they asked your income they would estimate your chances of siding with the plaintiff as 62%. They ask the question, and either your chances go up to 75% or down to 50%.

If one of two answers gives "no effect" then so must the other.

Think about it like point system. Based on the answer, the plaintiff gives you a point and the defense gives you minus a point; or nobody gives you any points. Theoretically, maybe you should have started with half a defense point, but I don't think that's how it's really run.

Lawyers usually don't understand Bayesian inference.

If that was the table and you were the defence, wouldn't you strongly prefer the Over 50k person?

This isn't a choice between the under 50k person and the over 50k person.

Huh?? If that was the table and you were the defense, wouldn't you strongly prefer any over $50k person?

The problem there is that if that table is accurate, the baseline assumption isn't a 50/50 split, it's that you're more likely to side with the plaintiff than the defense. (How much more likely depends on the prevalence of "under 50k" people relative to "over 50k" people.) So learning that you make "over 50k" moves you towards the defense from the position of no information (no information other than the table, that is).

No, it doesn't. The baseline presumption is that the question doesn't reveal anything useful about the juror's leanings.

A salary below $50k indicates a strong preference for the plaintiff, but the reverse is not true (i.e., that a salary above $50k indicates a strong preference for the defense). Rather, the table indicates that a salary above $50k doesn't provide any useful information one way or the other about a prospective juror's leanings.

That isn't true, though. As other sibling comments have explained, a salary above $50k indicates a stronger preference for the defense than would be expected. If counsel for the defense is choosing between (a) a person known to make over $50k, and (b) a person about whom nothing is known, then that table will inform them that person (a) is the correct choice, as they are biased in favor of the defense relative to person (b). Similarly, the table informs counsel for the prosecution that person (b) is biased in favor of the prosecution relative to person (a), and that a person known to make under $50k would be even more so.

This is starting to turn into a poorly phrased, under specified version of the three door problem.

It depends on how you interpret these phrases. This is valid logic if

"More likely to side with X" => The lawyers believe that you are more likely to side with the plaintiff than with the defendant on the basis of this information (Pr(P | Over $50k) - Pr(D | Over $50k) > epsilon)

"No effect on either lawyer": The lawyers believe that this information provides no insight into your decision (|Pr(P | Under $50k) - Pr(D | Under $50k)| < epsilon).

People who are likely to be poor are more likely to side with the investor rather than the investment bank.

But perhaps there are just as many people, who are not poor, who do not trust investment bankers as there are who do trust investment bankers.

Let's make a very simple model:

    poor? side with investor 100% of the time
    rich? equally likely to side with either

And again for simplicity, before they ask you what your income is say they think you're equally likely to be poor or rich, and that this is the first question.

Before they ask the question they think there's a 50% chance you're poor and will definitely side with the investor and a 50% chance you're rich and will side with either. In other words they think you're 75% likely to side with the investor. Then they ask you the question. If you say you're poor then they both update your probability upwards from 75% to 100%, if rich then downwards from 75% to 50%.