The mathematical model originally developed by Simeon Poisson has been refined and extended, principally by Gelfand and Solomon, who have used it with data from the 1950's on jury verdicts in the United States. Among other things, Gelfand and Solomon purport to show (with varying degrees of confidence) that the probable guilt of an accused brought before a jury in the United States is between .66 and .76, that there is 'essentially no difference' in the probability of conviction by a 6-person as opposed to a 12-person jury, and that the probabilities of both false convictions and false acquittals are substantially greater with the smaller jury. Conclusions such as these are of obvious interest in connection with the issue of the constitutionality of departures from the traditional 12-member jury. After reviewing the Poisson model to identify its many assumptions, this discussion explains why these assumptions may introduce serious errors into the probabilities calculated according to the model. To emphasize the difficulties involved, a more plausible but intractable model is exhibited. The concluding section gives one reason why even a relatively error-free model would have limited usefulness in resolving the constitutional issue posed by juries composed of fewer than 12 persons. That reason is that even if a model's mathematical result is trustworthy, it tempts a policymaker to forget that the result is a mean figure and that important differences might b revealed by a more fine-grained analysis. Overall, the discussion's primary conclusions are that major problems in the collection of data and perhaps in further refinement of the Poisson model remain to be solved and that obstacles to drawing policy conclusions from research efforts must be addressed through a more constructive dialogue between the disciplines of law and statistics. Mathematical equations of the model, and 39 footnotes are provided.