daly-gr/pi_monte_carlo

Simple python program to estimate the value of pi using a classic monte-carlo simulation.

pi_monte_carlo

Simple python program to estimate the value of pi using a classic monte-carlo simulation.

Monte-Carlo Methods
A monte-carlo method relies on random sampling to derive a numerical result.

Monte-Carlo Method for finding π
In this well-known approach, the value of π is estimated using a random number function which generates a number between 0 and 1. Consider a circle of radius r which is placed inside a square with length 2r; both the circle and square are centred at the origin.

If we were to pick a set of co-ordinates at random in the picture above, then the probability of our co-ordinates falling inside the circle is given by the area of the circle divided by the area of the square, i.e. P(inside circle) = πr^2/(2r)^2. The random number function affords a method for generating random co-ordiantes. Therefore, for a sample of n random co-ordinates, we can expect the ratio of co-ordinates falling inside the circle (m) to the total number of co-ordinates (n) to tend towards the probability P(inside circle).
n.b. A large value of n > 1000 is necessary to acquire a reasonably accurate estimation of π.

In fact, it is only necessary to consider a single quadrant of the picture above as πr^2/(2r)^2=(1/4)(πr^2)/r^2.
Calling the random number function twice generates an (x,y) co-ordinate for the upper-right quadrant of the square. If we do this n times, counting how many times the co-ordinates fall inside the circle m, we can compute the probability P(inside circle) = m/n. Given P(inside circle) = πr^2/(2r)^2, we can re-arrange the expression to find π like so π = 4(m/n).