We obtain a large class of significant examples of n-random reals (i.e., Martin-Löf random in oracle $\varphi ^{(n - 1)} $ ) à la Chaitin. Any such real is defined as the probability that a universal monotone Turing machine performing possibly infinite computations on infinite (resp. finite large enough, resp. finite self-delimited) inputs produces an output in a given set O ⊆(ℕ). In particular, we develop methods to transfer $\Sigma _n^0 $ or $\Pi _n^0 $ or many-one completeness results of (...) index sets to n-randomness of associated probabilities. (shrink)