As we’ve seen in class and in our textbook, the estimation of Pi through Monte Carlo simulations is a pretty casual and standard example.  It’s a good one too since it allows us to visualize the norms of setting up a Monte Carlo simulation, and see what the method actually achieves.  The Monte Carlo method though obviously has some reliances on a computer to randomly (or technically pseudo-randomly) generate numbers to serve as a probabilistic test for a certain problem.  In the example of calculating Pi, the locations of the dropped points were what was random.  (Aside, this is a good part of the reason why the Manhattan Project really took off when the computer came into existence is the ability to use the Monte Carlo method for their required simulations).  Nonetheless, an interesting precursor that sets up the same sort of prediction of the value of Pi is a probabilistic problem called Buffon’s needle.

The problem is quite simple and stated as follows in conjunction with the following picture:

What is the probability that the needle will cross one of the green lines in the picture (or both).  This assumes that A < .

It isn’t quite obvious how this problem solves the value of pi, but this is what the web site gets into and I won’t go too much into it here except for the fact that when calculating the probability of this happening and equating this to a theoretical performance of this experiment, we’re left with the following equation:

which means that pi can be estimated with n number of drops and the outcome of how many lines were crossed with M replacing E(M) to statistically estimate.  This is a really interesting outcome and not so necessarily straight forward (or really practical), but it serves to show another way to calculate pi in a Monte Carlo like way.

The article that I’ve linked to as the source for this information also goes into more details about the statistics behind the Monte Carlo method, but I did want to mention a few more points that were interesting and pertinent to the class.  For one, while it may not serve as a confusion in our class since we haven’t really delved into the Monte Carlo outside an academic standpoint, the uses of this method our numerous and actually confusing when seeing that many applications are probabilistic guesses of problems like weather forecasting or price changes, but this method is also very good at calculating certain values that have no probabilistic tendencies associated with them at all (like pi!).  The reason for this leads into my second point is that Monte Carlo methods are good for integrating (which is why in class, we’ve also referred to it and studied it as Monte Carlo integration).  This seems obvious, but because of this, it makes sense that Monte Carlo is used for a lot of probability distributions.  One can even see this through Buffon’s needle as the first estimation for Buffon’s needle is the probability density of the needle landing across the line (which requires an integration to solve.  Refer to the source page for more details).  In any case, because of these two points, it is plainly easy to see why Monte Carlo methods are important in our scientific computing age.  It’s just interesting to see that examples of this started more than 200 years earlier.

Sources:

http://www.riskglossary.com/link/monte_carlo_method.htm

http://www.csm.ornl.gov/ssi-expo/MChist.html