I quoted the following passage from Richard Posner in my recent article on Scotland’s three-verdict system:

When . . . judges and juries are asked to translate the requisite confidence into percentage terms or betting odds, they sometimes come up with ridiculously low figures-in one survey, as low as 76 percent, see United States v. Fatico, 458 F. Supp. 388, 410 (E.D.N.Y. 1978); in another, as low as 50 percent, see McCauliff, Burdens of Proof: Degrees of Belief, Quanta of Evidence, or Constitutional Guarantees?, 35 Vand. L. Rev. 1293, 1325 (1982) (tab. 2). The higher of these two figures implies that, in the absence of screening by the prosecutor’s office, of every 100 defendants who were convicted 24 (on average) might well be innocent.

See if you can spot the error in this reasoning.

I’m not entirely sure what Posner means by “screening by the prosecutor’s office” but it is not true that a jury system that convicts everyone whose guilt they are at least 76 percent sure of will convict an average of 24 innocent defendants for every hundred convictions. The marginal convict will have a 24 percent chance of innocence, but the average convict will generally have a lower chance of innocence than that.

For instance, suppose that there are only two convicts, one who was caught red handed (100 percent chance of guilt), another who was 76 percent likely to be guilty. Then the average chance of a convict being guilty is (76+100)/2=88 percent.

It’s only in the case where all convicts have a 76 percent chance of being guilty that there will be an average of 24 innocent convicts out of every 100. That’s the extreme lower bound of all possible distributions of probable guilt. In all likelihood, there will be fewer than 24 innocent convicts out of every 100, even if jurors convict everyone with a 76-percent-or-higher chance of guilt.