Got nerd-sniped hard by this. The basic question is simple; given a random person in the population, what are the values of x and y such that the chance that they will give a wrongful vote are 2/3? Obviously, we can see that y = ((2/3) - x)/(1 - x). A simple enough graph shows that the racist population would have to be surprisingly high to fulfill that condition; even a 20% die-hard "convict on race" population would still require the other 80% of people to similarly erroneously convict almost 60% of the time.

It becomes more interesting when we consider juries. Juries, as far as I know (not being American), require unanimous agreement to convict. A jury consists of 6-12 people. To find out if a jury will convict, we have to find the probability of every single person in that jury wrongfully convicting. The formula now is y = ((2/3)^(1/{6|12}) - x)/(1 - x). At this point, either x or y have to be very high for 2/3 of wrongful convictions to be black. Even when x = 66%, y still has to be above 80% for the 6-member jury to convict wrongfully that many times. This is, of course, assuming this is a perfectly mathematical scenario and we aren't dealing with real humans who will vote to convict because everyone else is doing so.

The error, then, is probably more due to humans being likely to fold their convictions under peer pressure than merely a high percentage of them being rabidly racist. If people adamantly voted their conscience in juries, and refused to be swayed by the deliberation of their fellow jurors, we would have many more hung juries and many fewer convictions.

I'll snipe you a bit harder. In the event of a hung jury, the case can be retried (aka the distribution resampled). Practically though, no prosecutor wants to redo the same case over and over. So its safe to assume that 2 hung juries is as good as charges dropped.

I'm glad you enjoyed this problem, both as a riddle and as a neat demonstration of how a societal system can be better (more fair, just) than the underlying society that builds it.

Oh, you and your silly little math. /s

Really, though, things like this should be taught and demonstrated from k-12. A kindergartener can understand the cognitive dissonance when the Monty Hall problem is demonstrated, especially if an adult "cheats" repeatedly when they know the mechanics. After the big reveal, the kid knows the difference between 1 out of 2, and 2 out of 3. They'll play it smart by the time they hit third grade, and by graduation, they'll have an intuitive understanding of posterior probabilities.

Drug tests, statistical sampling used to validate polls, and misleading sales tactics would be neutered by a well informed, Bayesian thinking public.

Exponential thinking should be introduced early and feedback loops explained. If legislators could rationally deconstruct a problem such as racist false convictions they would be equipped to modify the systems they legislate to prevent or mitigate the damages implicit to reality-based statistics.

Prisons for profit would lose 80% or more of their fodder. The broken family, broken community cycle would be effectively ended.