Although statisticians have the job of making conclusions based on data, for many questions in science and society prior beliefs are strong and may take precedence over data when people make decisions. For other questions, there are experts who could shed light on the situation that may not be captured with available data. One of the appealing aspects of Bayesian statistics is that the methods allow prior beliefs and expert knowledge to be incorporated into the analysis along with the data. One domain where beliefs are almost sure to have a role is in the evaluation of scientific data for extrasensory perception (ESP). Experiments to test ESP often are binomial, and they have a clear null hypothesis, so they are an excellent way to illustrate hypothesis testing. Incorporating beliefs makes them an excellent example for the use of Bayesian analysis as well. In this paper, data from one type of ESP study are analyzed using both frequentist and Bayesian methods.