Monte Carlo approximation of $\pi$

P2.5.10Monte Carlo approximation of

\pi

The value of $\pi$ may be approximated by a Monte Carlo method. Consider the region of the $$xy$$ -plane bounded by $0 \le x \le 1$ and $0 \le y \le 1$ . By selecting a large number of random points within this region and counting the proportion of them lying beneath the function $y=\sqrt{1-x^2}$ describing a quarter-circle, one can estimate $\pi/4$ , this being the area bounded by the axes and $$y(x)$$ . Write a program to estimate the value of $\pi$ by this method.

Hint: Use Python's random module: The method random.random() generates a (pseudo-)random number between 0. and 1. See Section 4.5.1 of the book for more information.

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Note that this Monte Carlo method converges extremely slowly (to get each new decimal place we have to increase the number of random points by a factor of 10):

import math
import random

N = 10000

def f(x):
    return math.sqrt(1.-x**2)

pi = 0.
for i in range(N):
    xi, yi = random.random(), random.random()
    if yi <= f(xi):
        pi += 1.
pi *= 4 / N
print('pi is approximately {:.4f}'.format(pi))

The output is:

pi is approximately 3.1432

P2.5.10Monte Carlo approximation of \(\pi\)

P2.5.10Monte Carlo approximation of $\pi$