A detailed quantitative kinetic model for the polymerase chain reaction (PCR) is developed, which allows us to predict the probability of replication of a DNA molecule in terms of the physical parameters involved in the system. The important issue of the determination of the number of PCR cycles during which this probability can be considered to be a constant is solved within the framework of the model. New phenomena of multimodality and scaling behavior in the distribution of the number of molecules after a given number of PCR cycles are presented. The relevance of the model for quantitative PCR is discussed, and a novel quantitative PCR technique is proposed.