What proportion of all 2 × 2 × 2 contingency tables exhibit Simpson’s Paradox? An approximate answer is obtained for large sample sizes and extended to 2×2×ℓ tables. Several conditional probabilities of the occurrence of Simpson’s Paradox are also derived. Given that the observed cell frequencies 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.