Using Monte Carlo Simulation for the Estimation of Pi

A while ago on G+, +Sara Del Valle posted a link to this article The 12 Most Controversial Facts In Mathematics. Whether or not the title of that article is precise is a whole other question but one of the problems in there discusses the use of probability to estimate $\pi$.

At the time I wrote a small python script to randomly generate points (this approach of continuous random sampling is called Monte Carlo simulation) that lie in a square of length $2r$. If we let $S$ be the total number of points and $C$ to be the points that lie in an inscribed circle of radius $r$ then:

$$P(\text{point in circle})=\frac{\pi r^2}{4r^2}$$

Using this if we generate a high enough number of points we can estimate P(point in circle). We generate a point $(x,y)$ and check if that point is in the circle by checking if $x^2+y^2\leq r^2$. Using all this we get:

$\pi\approx \frac{4C}{S}$

This morning I've modified the script slightly so that it generates a few more plots:

• If the relevant option is set it will generate a new plot for every single plot;
• A plot of the estimate of $\pi$ as a function of the number of points.

Here's a plot of the estimate of $\pi$:

The main point of me changing this was to put together this screencast that describes the process:

If it's of interest the github repo with the code is available here.

On a related note, here's another video I put together a while back showing the basic process that can be used to simulate a queueing process: