Empirical evidence suggests that the brain during perception and decision-making has access to both point estimates of any external stimulus and to the certainty about this estimate. This requires a neural representation of entire probability distributions in the brain. Two alternatives for neural codes supporting such representations are probabilistic population codes (PPC) and sampling-based representations (SBR). We examined the consequences of an SBR and its implications in the context of classical psychophysics. We derive analytical expressions for the implied psychophysical performance curves depending on the number of samples collected (i.e. the stimulus presentation time), which constitute the theoretical limit for optimal performance. This time-dependence allows us to contrast SBR with PPC, in which probability distributions are represented explicitly and near-instantaneously as opposed to successively over time as for sampling. We compared our predictions with empirical data for a simple two-alternative-choice task distinguishing between a horizontal and a vertical Gabor pattern embedded in Gaussian noise. Recasting the decision-making process in the sampling framework also allows us to propose a new computational theory for endogenous covert attention. We suggest that the brain actively reshapes its representation of the posterior belief about the outside world in order to collect more samples in parts of the stimulus space that is of greatest behavioral relevance (i.e. entails rewards or costs). We show that compared to using the veridical posterior, the benefit of such a mechanism is greatest under time pressure – exactly when the largest effects due to endogenous attention have traditionally been seen. We present experimental data for a task in which attention has been manipulated by varying the behavioral relevance of two stimuli concurrently on the screen, but not their probabilities as traditionally done.