This article is concerned with the statistical detection of copying on multiple-choice exams. As an alternative to existing permutation- and model-based copy-detection approaches, a simple randomization p-value (RP) test is proposed. The RP test, which is based on an intuitive match-score statistic, makes no assumptions about the distribution of examinees' answer vectors and hence is broadly applicable. Especially important in this copy-detection setting, the RP test is shown to be exact in that its size is guaranteed to be no larger than a nominal a value. Additionally, simulation results suggest that the RP test is typically more powerful for copy detection than the existing approximate tests. The development of the RP test is based on the idea that the copy-detection problem can be recast as a causal inference and missing data problem. In particular, the observed data are viewed as a subset of a larger collection of potential values, or counterfactuals, and the null hypothesis of "no copying" is viewed as a "no causal effect" hypothesis and formally expressed in terms of constraints on potential variables.