There's also the (bigger?) question whether randomness helps at all, e.g. whether P is weaker than P + a source of random bits. In practice, randomization does help, but like with P vs NP as far as I know nobody has proven anything definite.
The complexity classes that involve randomness have names like BPP, ZPP, and RP. BPP is usually the one people mean when they talk about practical problems that randomness might help solve. Although the second level of PH contains both NP and BPP, which both contain P, it is still not known if any of those containments are proper, and the relationship between BPP and NP is still open. Even more interesting is the relationship between BPP and its quantum version BQP. The Complexity Zoo has more for the interested reader... http://qwiki.stanford.edu/index.php/Complexity_Zoo
