Estimating pi using the MC technique - part 2

As is evident from the earlier post, the larger the sample size, the more is the number of digits in the estimate for pi. However, a question now arises as to whether or not this random sampling of marbles would hold good for every throw. The clichéd analogy is the well known problem that the probability of obtaining a head or a tail from a coin toss is 50%. While it is very well possible that the percentage may spike vastly from 50% for small sample sizes (aka beginner’s luck in gambling), the average gets closer to 50% as the number of tosses increases and stays pegged at 50% for very large sizes.

Getting back to estimating pi, the Monte Carlo analysis looks at analyzing the problem for several throws of the marbles, while each throw gets perturbed for its inputs (i.e., randomized probabilities) in order to obtain a contour map of the possibilities or regions for the occurrences of the solutions.

However, what’s to say that a random throw of marbles (no matter what the sample size is) that works for one throw will hold good for a 2^nd throw, or any subsequent one? A second level analysis utilizing the MC technique can be incorporated, with multiple throws of randomized sample sizes in order to determine a statistical parametric output for large number of throws. 

To be cont'd