Allan Oaten (1977, Theor. Pop. Biol. 12, 263–285) has argued that stochastic models of optimal foraging may produce results qualitatively different from those of the analogous deterministic models. Oaten's model is very general and difficult to understand intuitively. In this paper a simple, tractable model is considered in which the predator searches each patch systematically (without going over the same area twice) until he exhausts the patch or decides the patch is not very good. It is assumed that each patch contains a fixed number of bits, each of which may contain a prey. The number of prey per patch is assumed to have a binomial distribution with n equal to the number of bits and p being a random variable having a beta distribution. After searching each bit the predator decides whether to leave the patch or not according to how many prey it has found. In this paper the best strategy is determined and the long-term rate of feeding is compared with that of the naive animal that searches each patch completely. The advantage of being a Bayesian is determined for a variety of environmental conditions.