# Wiki

# monte_carlo_pi / Home

## Monte Carlo Pi

Did you know you can estimate the value of pi by throwing random points on a square?

Now draw a square around that circle, like this:

The area of the square is 4 = (1+1)*(1+1), and the area of the circle is = 3.1459...

If you throw darts at the square randomly, the number that land in the circle divided by the total darts thrown will approximate pi! Neat, huh? We use a quarter of the square and then multiply the estimate by 4. This is important, because the fraction of (darts in circle/darts thrown) will always be less than or equal to 1. To get pi, I start by using 1/4 of the square, and so multiplying by 4 brings us "full circle".

*[Note: This is also one simple way to perform rejection sampling over the distribution defined by the area under pi/2 radians on the unit circle.]*

The approximation approaches the true value of pi as the number of darts approaches infinity (i.e. you throw enough darts to cover the entire circle), and assuming truly random throws. A successful run should produce something like:

## Pulling the Repo

You can pull this repo with:

$ git clone https://github.com/entangledloops/monte_carlo_pi.git

See the readme for further details on running this script on your machine.

## Python Code

# Author: Stephen Dunn # Date: 8/16/13 # File: pi.py # Description: Monte-Carlo Pi Estimator # Notes: 1) exceeding ~pi to 4 decimal places with anything better than luck has low probability # 2) yes, this could make better use of pygame functions (e.g. not hand-computing the semicircle) import sys import pygame import math import random window = "" width = 800 height = 600 resolution = 2000 # precision of line segments points = 10000 # of points def graph(): x = y = ex = ey = 0 dp = (math.pi/2)/resolution # draw a quarter-circle for i in range(0,resolution): x = ex y = ey ex = ex+dp ey = ey+dp pygame.draw.aaline(window, (128,255,128), \ ( math.cos(x)*width, math.sin(y)*height ), \ ( math.cos(ex)*width, math.sin(ey)*height ), 3) # estimate pi from random points hits = 0 for i in range(0,points): px = random.random() py = random.random() c = math.sqrt( (px*px) + (py*py) ) if (c <= 1.0): hits += 1 px *= width py *= height pygame.draw.circle( window, (148,0,211), (int(px),int(py)), 1, 1 ) api = 4.0*(float(hits)/float(points)) # approximate pi font = pygame.font.SysFont('Arial', 18) pi_str = u'\u03c0' + ' ' + u'\u2248' + " " + "{0:.5f}".format(api) pi_str.encode('utf-8') print pi_str text = font.render(pi_str, True, (72,209,204)) window.blit( text, (width/4,height/4) ) #---------------------------------- random.seed() pygame.init() window = pygame.display.set_mode( (width, height) ) print "graphing..." graph() print "done" pygame.display.flip() while True: for event in pygame.event.get(): if event.type == pygame.QUIT: sys.exit(0)

Updated