The laws and court rules of each State and the Federal system provide for a certain number of peremptory challenges in civil and criminal cases. Peremptory challenges remove potential jurors from a case without the necessity of justification or explanation. In a Federal criminal trial, for example, the defense has 10 such challenges and the prosecution has 6. In any particular case, however, the judge has the authority to increase the number of peremptory challenges to ensure a fair trial. Presented herein is a model for deciding how many such challenges should be allowed each side as a general rule, whether more peremptory challenges should be allowed according to the circumstances, and how many more should be allowed. The principle that the number of peremptory challenges be determined with the goal of ensuring a fair trial can be expressed in terms of the probabilities that each side will have sufficient challenges to eliminate all prospective jurors not removed for cause who are biased against it. Those probabilities should be the same and should also be considered in the formulation. It is presupposed that all jurors not removed for cause may be thought of as falling into one of three categories: biased against the prosecution, biased against the defense, or fair. It is proposed that a fair number of peremptory challeges for a side is that number sufficient to challenge all jurors biased against it, with a certain probability. Given estimates for the various parameters in the model, an exact algorithm, and an approximation are given and applied. Extensive statistical formulas and 11 references are provided in the article.