This paper is about a century-old yet durably-opaque constitutional doctrine that has the potential to disrupt every field of US law that uses legal presumptions. That doctrine demands, for any legal presumption, at least “some rational connection between the fact proved and the ultimate fact presumed.” The paper shows how that doctrine is best read to require likely-accurate presumptions, i.e., in the run of cases in which the presumption can apply, the probability of the presumed fact, given the proven fact, must provably exceed fifty percent. If so, however, lawyers and judges applying this doctrine face distinctive selection bias and reference class problems. The paper illustrates these and other issues with five examples: the McDonnell-Douglas presumption in employment discrimination law; the Bail Reform Act's presumption about a bail applicant's level of danger to others based on being charged with certain drug-sale crimes; the tort law doctrine of res ipsa loquitor; a presumed damages amount for emotional distress; and presumptions about whether a worker is an “employee” or “independent contractor.” In so doing, this paper contributes to the legal and philosophical literature on presumptions in adjudication, as well as ongoing controversies in the US law of tort, work, damages, and bail.