New Bayesian estimates are proposed for multinomial probabilities, when the prior distribution is a mixture of Dirichlet distributions. They are based upon a distributional approximation for the χ2statistic and may be contrasted with the frequentist approximations recommended by other authors. An alternative is also suggested to the χ2goodness-of-fit test, when it is employed as a preliminary test of significance. The new procedure possesses remarkably different properties; it is used to reanalyze some data from Mendel's pea breeding experiment. The results suggest that the χ2test may have no particular relevance in preliminary testing situations.