We tested three fundamental properties of Bayesian Decision Theoryaccuracy, additivity, and sufficiency. In Experiment1, observers were shown a sample of dots from a bivariate Gaussian and estimate the probability that an additional samplewould fall into specified regions. There were three types of regions: symmetric around the mean (S), the upper andlower halves of the symmetric region (SU and SL). In Experiment 2, the same observers were asked to maximize theexpected rewards based on jointly sufficient statistics for given the sample (herein, mean and covariance of a Gaussian).In Experiment 1, We found that the observers estimates of symmetric region P[S] were close to accurate. However, theyshowed a highly patterned super-additivity: the sum of P[SU] + P[SL] ¿ P[S]. In Experiment 2, the observers violatedsufficiency by assigning too much weight to a feature of the sample rather than jointly sufficient statistics.

People overestimate the conjunctive probability of independent events (Bar Hillel, 1973). We examined conjunctive per-formance in a task involving motor uncertainty and binomial sampling. Human probabilistic judgment is typically near-optimal with either of these sources of uncertainty alone. Four subjects attempted to earn rewards by reaching to circulartargets. They chose between a single smaller target and one of N larger targets. Hitting the single target always earned areward but only one on the N larger targets was rewarded: they chose between P[Smaller] and the conjunctive probability(1/N)*P[Larger] as we varied N and the sizes of the targets. The ideal observer should be indifferent when P[Smaller] =(1/N)*P[Larger]. We also asked observers to estimate the probability of hitting targets of different sizes to verify that theycould do so accurately. Remarkably, three out of four observers ignored numerosity N in their preferences.