A simple Monte Carlo simulation in Python

Monte Carlo is a simulation method that can be useful in solving problems that are difficult to solve analytically.

Here’s an interesting application of the technique to estimate the value of pi.

Consider a circular dartboard placed against a square backing, with the sides of the square perfectly touching the circle tangentially at the top, bottom, left and right.

If you throw a dart blindly so it lands inside of the square, what are the chances the dart has landed inside the circle?

Given a circle radius of r, the chances are:

= area of circle / area of square
= (pi * r ^ 2) / (4 * r ^ 2)
= pi / 4

So the chances of hitting the circle are pi divided by 4.   In other words:

pi = 4 * the probability of hitting the circle.

So we can set up a Monte Carlo simulation that will give us the probability of getting the dart inside of the circle, and then we’ll be able to derive an approximation of the value of pi.

To do this, let’s think of the quarter circle below:

Note that the chances of hitting the quarter circle inside of its square is the same as hitting the larger circle inside of the bigger square.  

How can we easily simulate hitting the quarter circle?  We can obtain two random numbers between  0 and 1, representing the x and y coordinates of a point with the origin at the circle’s center.  If the distance between this point and the origin is less than 1, then we’re inside the quarter circle.  Otherwise we’re outside it.  

If we simulate this enough times, and multiply the percentage of times we’re inside the circle by 4, we should converge on the value of pi. Here’s python shell code that simulates this:

>>> from random import random
>>> len(filter(lambda x:(random()**2+random()**2)**.5<1,range(1000000)))/1000000.0*4
3.14064