or 10 points, a Monte Carlo algorithm is what kind of nonÂ­deterministic algorithm that may run on a probabilistic Turing machine?
This kind of algorithm creates an ErdősÂ–RÃ©nyi graph, which Fagin's zeroÂ­one law describes. One algorithm of this kind, in big O of k log cubed of n, halts if a to the n minus 1 isn't 1 mod n, which is called a witness. An algorithm of this kind can check a proof of any problem in NP by the PCP theorem. An algorithm of this kind in class BPP gives an exponentially decreasing Chernoff bound. MetropolisÂ–Hastings, (*)primality tests like MillerÂ–Rabin, simulated annealing, and skip list insertion are algorithms of this kind. Because of how they pick the pivot to avoid the n squared worst case, most versions of quicksort are this kind of algorithm. Markov's inequality can turn algorithms of this kind that always succeed, called Las Vegas, into ones that always halt. For 10 points, a Monte Carlo algorithm is what kind of nonÂ­deterministic algorithm that may run on a probabilistic Turing machine?
This kind of algorithm creates an ErdősÂ–RÃ©nyi graph, which Fagin's zeroÂ­one law describes. 
randomi  zed algorithm [or  probabilistic  algorithm until read; or  Monte  Carlo  or  Las  Vegas  until read;accept Markov chain  Monte  Carlo accept  random  sampling; prompt on "nonÂ­deterministic algorithm" until read;antiÂ­prompt for a "general kind of algorithm" on "primality test" or "checking if a number is prime," etc. until read]