Choice rules based on probability thresholds are common in several disciplines. The most well-known application of such a threshold rule is the standard of reasonable doubt. Accordingly, a rational juror prefers to convict a defendant if and only if the probability that she attaches to the defendant being guilty is above a given threshold. In this paper we prove that generically such a threshold exists if and only if the juror reasons only about two events, viz., the defendant's guilt and innocence. This result implies that threshold rules are usually inconsistent with individual rationality. Thus, if we insist on using a threshold choice rule, we will have to accept some irrational convictions (false negatives) or some irrational acquittals (false positives) or both. We subsequently characterize each probability threshold in terms of the irrationalities that it induces. Finally, we discuss the empirical implications of our theory.