Yao proved that in the decision-tree model, the average complexity of the best deterministic algorithm is a lower bound on the complexity of randomized algorithms that solve the same problem. Here it is shown that a similar result does not always hold in the common model of distributed computation, the model in which all the processors run the same program (which may depend on the processors’ input). We therefore construct a new technique that together with Yao’s method enables us to show that in many cases, a similar relationship does hold in the distributed model. This relationship enables us to carry over known lower bounds on the complexity of deterministic computations to the realm of randomized computations, thus obtaining new results. The new technique can also be used for obtaining results concerning algorithms with bounded error.