This is an exercise to estimate the values of π using Monte Carlo Algorithm

This method tries to find the value of π by estimating if a randomly chosen point lies inside a circle. We choose the radius of the circle to be 1000 units. The co-ordinates of the choosen points would range from (0, 0) to (1000, 1000).

Mathematically it boils down to

Area of quarter circle / area of square enclosing the quarter circle
   = number of points inside quarter circle / number of points inside square

⇒ [(3/4) * πr² / r² = k

where k = number of points inside quarter circle / number of points inside square (k is found by experiment)

⇒ (3/4) π = k

⇒ π = k * (4/3)

Now if we have n number of points, the value of k is found by the following logic

The distance of a point from origin is

d = sqrt(x'² + y'²)

x' and y' denote randomly generated numbers

The point lies inside the circle if

d < r

So k turns out to be the count of points for which

d < r

This algorithm can be used to test the randomness of the random number generator methods / function available in different languages.

Select a sample size and run the simulation for a thousand times. Increase the sample size and observe how the error in π reduces.

The sample size and the average, best and worst errors are tabulated.