Estimating the value of pi

Here's something I learned recently and found quite interesting. A neat way to calculate the value of pi using estimation is to take a unit circle inside a square of side 2 units and take random shots on the surface of the square.

Now, the ratio of the area between the circle and square can be given by,






Thus, we can estimate the value of pi as,






Now, I wrote a little script in Python which does just that. I also put in the value of pi to 10 decimal places and wrote out the error produced. Initially, I didn't take a circle of unit radius but that just produced too much error:



import math;
import random;
import sys;


#calculate the value of pi using probability
maxtries = float(sys.argv[1]);


radius = 1.0;
side = 2.0 * radius;
tries = 0.0;
hits = 0.0;
pi = 3.1415926536;


while (tries < maxtries):
tries = tries + 1;
#get a point within the square


rx = random.random()*side;
ry = random.random()*side;


#distance from center of circle
dist = math.sqrt(math.pow(rx - radius, 2.0) + math.pow(ry - radius, 2.0));
if (dist <= radius): hits = hits + 1;


#estimated value of pi
piest = (hits*4/tries);
print "Radius: %d, Tries: %d, Hits: %d, pi: %1.10f, Error:%1.10f\n" % (radius, tries, hits, piest, piest - pi);

My output looks something like:

~$ python pi.py 10.0
Radius: 1, Tries: 10, Hits: 6, pi: 2.4000000000, Error:-0.7415926536

~$ python pi.py 100.0
Radius: 1, Tries: 100, Hits: 83, pi: 3.3200000000, Error:0.1784073464

~$ python pi.py 1000.0
Radius: 1, Tries: 1000, Hits: 790, pi: 3.1600000000, Error:0.0184073464

~$ python pi.py 10000.0
Radius: 1, Tries: 10000, Hits: 7822, pi: 3.1288000000, Error:-0.0127926536

~$ python pi.py 100000.0
Radius: 1, Tries: 100000, Hits: 78591, pi: 3.1436400000, Error:0.0020473464

~$ python pi.py 1000000.0
Radius: 1, Tries: 1000000, Hits: 786368, pi: 3.1454720000, Error:0.0038793464

~$ python pi.py 10000000.0
Radius: 1, Tries: 10000000, Hits: 7853556, pi: 3.1414224000, Error:-0.0001702536

~$ python pi.py 100000000.0
Radius: 1, Tries: 100000000, Hits: 78539696, pi: 3.1415878400, Error:-0.0000048136

~$ python pi.py 1000000000.0
Radius: 1, Tries: 1000000000, Hits: 785382068, pi: 3.1415282720, Error:-0.0000643816

The last observation is the most interesting. It ran for a few hours on my machine but did absolutely nothing to improve the error. In fact, the error actually increased!