A randomized model verification strategy for RANSAC is presented. The proposed method finds, like RANSAC, a solution that is optimal with user-specified probability. The solution is found in time that is close to the shortest possible and superior to any deterministic verification strategy. A provably fastest model verification strategy is designed for the (theoretical) situation when the contamination of data by outliers is known. In this case, the algorithm is the fastest possible (on the average) of all randomized RANSAC algorithms guaranteeing a confidence in the solution. The derivation of the optimality property is based on Wald's theory of sequential decision making, in particular, a modified sequential probability ratio test (SPRT). Next, the R-RANSAC with SPRT algorithm is introduced. The algorithm removes the requirement for a priori knowledge of the fraction of outliers and estimates the quantity online. We show experimentally that on standard test data, the method has performance close to the theoretically optimal and is 2 to 10 times faster than standard RANSAC and is up to four times faster than previously published methods.