"Monte Carlo" method(s)

What is a "Monte Carlo" (factorization) method?
Essentially one where luck can play a major part - hence the rather exciting-sounding connection with gambling! This is the case whenever the process is seeded by some (pseudo-or-otherwise) random number generation (eg for the benefit of parallelization), or when there is inherent unpredictability in the algorithm itself.
Note also, that for the Rho method, for example, this Monte Carlo nature of the algorithm results in performance gains caused by a statistical effect akin to the so-called "Birthday Paradox". This latter, in turn, is the term given to the surprisingly high observed probability of at least one coincidence of two different random variables in the same range (eg two or more people's birthdays)
The following Wikipedia link talks about Monte Carlo effect in general:
http://en.wikipedia.org/wiki/Monte_Carlo_method
and here is a link to the Birthday Paradox:
http://en.wikipedia.org/wiki/Birthday_Paradox