Two new, efficient probabilistic methods are presented for evaluating the reliability of an elastic/perfectly plastic continuous medium exhibiting randomly varying yield strength. Both the Tresca and Mohr-Coulomb yield criteria are considered in the present study. The two probabilistic methods developed are based on the upper and lower bound theorems of plastic limit analysis. By extending these two theorems to probabilistic cases the upper bound and lower bound reliability indices of a safety problem can be calculated for the elastic/plastic medium. Both two and three-dimensional problems are studied. These include a two-dimensional (2-D) wedge loaded on one face and the bearing capacity of a strip footing under constant normal pressure. Both the classical Prandtl and Hill mechanisms are considered for this strip footing problem and the Hill mechanism is found to be more critical probabilistically. The three-dimensional (3-D) problem considered is the extension of the 2-D problem wedge problem. In the 3-D problem the failure length in the longitudinal direction and the resisting strength provided by two vertical end sections play important roles in the reliability calculations, and numerical results are given to illustrate these effects. The spatially varying yield strength is modeled as a Gaussian random field in two or three dimensions, depending on which type of problems is analyzed. Several existing methods to discretize the random fields are reviewed, and their advantages and disadvantages when applied to the plastic random media problems are addressed. It is shown that the limit state functions in the upper bound reliability method can be formulated as linear ones and for the lower bound reliability method the limit state function is formulated as a linear programming problem. The proposed methods provide efficient analytical tools for the probabilistic analysis and design for continuous load-carrying media whose failure is defined by their plastic limit state. The reliability methods presented in this study can be applied to several important classes of problems in geotechnical engineering and are potentially applicable to the plastic failure of plates and shells.