Question: One observes that the protocol UMC is based on a nondeterministic protocol that simply guesses the position j where x and y differ, and then verifies if its guess was correct (i.e., if really xj yj). We have converted this nondeterministic protocol into an MC protocol by assigning some probabilities to nondeterministic decisions about acceptance and rejection in situations in which the protocol does not know the right answer.
The idea is to accept with probability less than, but close to, 1/2, and to reject with the complementary probability.
The probability to accept in this uncertain situation must be so close, from below, to 1/2 that one computation accepting with certainty brings the overall probability of acceptance above 1/2. Can one apply this idea for converting any nondeterministic algorithm to an equivalent MC algorithm of the same efficiency?