 Even if numerical thresholds cannot be used in the daily business of trial proceedings, they can still serve as theoretical concepts for understanding the role of proof standards in the justice system, such as regulating the relative frequency of false positive and false negative decisions or minimizing expected costs. A more stringent threshold will decrease the number of false positives (say false convictions) at the cost of increasing the number of false negatives (say false acquittals), and a less stringent threshold will increase the number of false positives while decreasing the number of false negatives. This trade-off has been described, among others, by Justice Harlan in his concurring opinion in re Winship, 397 U.S. 358, 397 (1970). As shown below, the formal apparatus of probability theory, in combination with expected utility theory, can make this point more precise.