R or Python Use Monte Carlo simulation to estimate pi Draw a

R or Python

Use Monte Carlo simulation to estimate pi. Draw a one by one square and draw a circle in it. Choose 54 points with a quasi random uniform sequence on the 1x1 sqaure and measure if they are in the circle or not.

How do we do that? Divide 1x1 square into 9 sub squares (1/3 x 1/3 each) and randomly pick 6 points for each sub square and measure if they are in the circle or not

Solution

Following is the required python code

import random;

def estimatePiForMe():
   #circle inside square is of radius 1/2
   #assuming circle and square are centered at origin
   #any point (x,y) is in circle, if it satisfies x^x + y*y < 1/4
   limits = [ [-1.0/2,-1.0/2 + 1.0/3], [-1.0/2+1.0/3, -1.0/2+ 2.0/3], [-1.0/2 + 2.0/3 , 1.0/2 ] ];
   xlimitsForSubsquares= [];
   ylimitsForSubsquares= [];  
   for i in range(3):
       xlimitsForSubsquares.extend(limits);
   for i in range(len(limits)):
       for j in range(3):
           add = [ limits[ len(limits)-1-i ][1] , limits[ len(limits)-1-i ][0] ];
           ylimitsForSubsquares.append( add );
  
   #got 9 subsquares
   incircle= 0;
   for ss in range(9):
       for rpoint in range(6):
           xval = random.random()* 1.0/3 + xlimitsForSubsquares[ss][0];
           yval = random.random()* 1.0/3 + xlimitsForSubsquares[ss][1];
           if (xval*xval + yval*yval) < 1.0/4:
               incircle= incircle + 1;

   print (4*incircle*1.0)/54;

estimatePiForMe();