The average probability estimate of J > 1 judges is generally better than its components. Two studies test 3 predictions regarding averaging that follow from theorems based on a cognitive model of the judges and idealizations of the judgment situation. Prediction 1 is that the average of conditionally pairwise independent estimates will be highly diagnostic, and Prediction 2 is that the average of dependent estimates (differing only by independent error terms) may be well calibrated. Prediction 3 contrasts between- and within-subject averaging. Results demonstrate the predictions' robustness by showing the extent to which they hold as the information conditions depart from the ideal and as J increases. Practical consequences are that (a) substantial improvement can be obtained with as few as 2-6 judges and (b) the decision maker can estimate the nature of the expected improvement by considering the information conditions.