Calculating pi using Monte Carlo methods¶

Relevant formulas¶

square area: $s = (2 r)^2$
circle area: $c = \pi r^2$
$c/s = (\pi r^2) / (4 r^2) = \pi / 4$
$\pi = 4 * c/s$

Image to visualize the concept¶

Darts

importing modules that we will need¶

import random import matplotlib.pyplot as plt

In [8]:

# importing modules that we will need

import random
import matplotlib.pyplot as plt

In [6]:

# initializing the number of "throws"

num_points = 1000

In [9]:

# here we "throw darts" and count the number of hits

points = []
hits = 0
for _ in range(num_points):
    x, y = random.random(), random.random()
    if x*x + y*y < 1.0:
        hits += 1
        points.append((x, y, "red"))
    else:
        points.append((x, y, "blue"))

In [10]:

# unzip points into 3 lists
x, y, colors = zip(*points)

# define figure dimensions
fig, ax = plt.subplots()
fig.set_size_inches(6.0, 6.0)

# plot results
ax.scatter(x, y, c=colors)

Out[10]:

<matplotlib.collections.PathCollection at 0x111e21f90>

In [11]:

# compute and print the estimate

fraction = hits / num_points
4 * fraction

Out[11]:

3.164

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