Has any government ever investigated the rate of false positives in a jury system by faking trials? ... Would this be a worthwhile thing to do?
It's probably worthwhile, but the purpose of a jury was not to improve accuracy. After all, you could just have one or more judges, who would have more experience and thus be less prone to error, determine guilt instead of a jury. The purpose of a jury was to make corruption more difficult (harder to secretly bribe a dozen randomly selected citizens), and to provide a way for the people to negate laws they don't like (can't punish juror for declaring someone innocent).
So even if there were a more accurate way to produce verdicts, there are other factors to consider.
Today an Italian court has declared that Amanda Knox is, once again, guilty. She did not attend that trial (is not required to in Italy), so her final verdict will be either by appeal to the Supreme Court of Italy or the US extradition court. Extradition requests might be impeded due to the fact US does not have double jeopardy.
Previously on LessWrong, in The Amanda Knox Test: How an Hour on the Internet Beats a Year in the Courtroom there was some complaint that it actually took more than an hour on the internet to thoroughly research the case. Of course, the courts have been at this since 2007...
Her co-defendant, Raffaele Sollecito, who did show up at the trial, got sentenced to 25 years, but I don't know for sure where he is now because apparently he's totally unimportant and who cares (the media's opinion, not mine). I'm fairly sure he's in Italy though. So far it seems the plan is to revoke his passport but not arrest him.
Anyone want to take their hand at making predictions?