Abstract For Bernoulli random variables sequential tests of the simple hypotheses p=p 0 vs. p = p 1 are considered. In particular a comparison is made between the performance of sequential probability ratio tests (SPRTs) and curtailed single sampling plans (CSSPs), when both tests have the same error probabilities. In [1] it is shown that the CSSP has a strong optimality property. Nevertheless, if one admits randomized SPRTs, the latter are better. Numerical examples are considered in detail for p 0 = 1/2 and the CSSP with n = 2. The method of finding the randomized SPRTs with the correct error probabilities is indicated. These rules have positive probability of deciding without taking any observations.