Quote:
Originally Posted by yaser
Calculating these quantities analytically is not tractable. Estimating them using Monte Carlo methods, i.e., by running many random instances of the problem and averaging, is what we are after here. As you point out, there are many sources of randomness and some will result in significant variation. However, repeating the experiment a large number of times will overcome that variance. The numbers given for this problem were chosen to achieve that.

It should be possible to give an order of magnitude estimate, though if that was easy then you wouldn't have to write the program to answer the problem.
It looks a little similar to the relaxation algorithms for solving partial differential equations. In that case, the theory is well studied.
Even without knowing that, you could find the scaling law. How the number of iterations changes, on the average, with the number of points. But once you have the MC program, it is easy enough to change the number of points.