Monte Carlo Algorithms : Are there any problems where two opposite Monte Carlo algorithms could solve it?
A Las Vegas algorithm is one which is always correct, but is only efficient in expectation. Given two "opposite" Monte Carlo algorithm, you can create a Las Vegas algorithm by alternating both algorithms (or running them in parallel), halting whenever one of them produces a certain answer.

In the language of complexity theory, this corresponds to the statement $\mathsf{ZPP} = \mathsf{RP} \cap \mathsf{coRP}$. You can find more information on $\mathsf{ZPP}$ on Wikipedia.