The Monte Carlo method has been used a lot to estimate integrals, the value of pi, the value of the Euler's constant and so on. Having an estimate for the n! is also interesting.
You may find the following article helpful:
I. Dimov, R. Georgieva, Monte Carlo method for numerical integration based on Sovel's
sequences:
http://www.google.ca/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0CCMQFjAB&url=http%3A%2F%2Fwww.researchgate.net%2Fpublication%2F225246114_Monte_Carlo_Method_for_Numerical_Integration_Based_on_Sobols_Sequences%2Ffile%2F9fcfd50913ce7f0cba.pdf&ei=jqGrU_u3HcOQqAbZpYCQDg&usg=AFQjCNEvZrG0EtTFFnb877SFUogAOpxXIQ&bvm=bv.69837884,d.b2k
More to the point is the following article:
J.F. Traub, H. Wozniakowski, The Monte Carlo algorithm with a pseudorandom generator, Mathematica of Computation 58 (197), 1992, 323-339:
http://www.ams.org/journals/mcom/1992-58-197/S0025-5718-1992-1106984-4/S0025-5718-1992-1106984-4.pdf
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