Statistical Physics: Determine an approximate value of by the following method,

Question

Statistical Physics:

Determine an approximate value of by the following method, called Monte Carlo. Generate by a computer pairs of numbers. {x, y}. each uniformly distributed between -1/2 and 1/2: equivalently. this means you generate random points in a unit square. Count the fraction of those which happen to be inside the circle inscribed in this square. This should allow you to estimate pi. How accurate is your result? As you should have learned from other examples, the quality of a computational method is determined by how rapidly the result improves with the increasing computational load. Try performing your Monte Carlo determination of pi with varying number of Monte Carlo trials, n, and look how Delta_n = | pi_estimated - pi |/pi depends on n.

Explanation / Answer

You can try the following it works well. For part 2 try changing the trials

Chegg answer

#!/usr/bin/env python

from random import random

from math import hypot

from PIL import Image

from PIL import ImageDraw

TRIALS=10**6

XMAX=500

YMAX=500

def main():

    inside = 0

    img = Image.new('RGB', (XMAX, YMAX), "white")

    draw = ImageDraw.Draw(img)

    for i in xrange(TRIALS):

        x = random()

        y = random()

        if hypot(x, y) < 1:

            inside +=1

            draw.point((int(x*XMAX), int(y*YMAX)), fill="red")

    img.save('pi.png')

    print 'pi is: ', 4.0 * inside / TRIALS

if __name__ == '__main__':

    main()

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