How Likely Is Simpson's Paradox?

What proportion of all 2 x 2 x 2 contingency tables exhibit Simpson's Paradox? An exact answer is obtained for large sample sizes and extended to 2 x 2 x l tables by Monte Carlo approximation. Conditional probabilities of the occurrence of Simpson's Paradox are also derived. If the observed cell proportions satisfy a Simpson reversal, the posterior probability that the population parameters satisfy the same reversal is obtained. This Bayesian analysis is applied to the well-known Simpson reversal of the 1995-1997 batting averages of Derek Jeter and David Justice.