Judges often give statutes strained interpretations. What makes an interpretation strained? What, if anything, justifies a strained interpretation? Is there always a point past which an interpretation would be too strained, or are there some interpretations which judges are entitled to adopt, whatever a statute says? I claim to answer all of these questions. Using the formula about conditional probability known as Bayes’ theorem, I offer a precise definition of a strained interpretation, and identify two reasons for strained interpretations. I claim that the case for strained interpretations is often stronger than is realised, and illustrate that claim with a novel defence of the House of Lords’ decision in Anisminic Ltd v Foreign Compensation Commission. Finally, I suggest that even if a legislature is sovereign, there may be meanings which a legislature is unable to enact into law. Thus, there may be limits on even a sovereign legislature’s powers.