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import random

class Point:

def __init__(self, x: float, y: float) -> None:

self.x = x

self.y = y

def is_in_unit_circle(self) -> bool:

"""

True, if the point lies in the unit circle

False, otherwise

"""

return (self.x ** 2 + self.y ** 2) <= 1

@classmethod

def random_unit_square(cls):

"""

Generates a point randomly drawn from the unit square [0, 1) x [0, 1).

"""

return cls(x=random.random(), y=random.random())

def estimate_pi(number_of_simulations: int) -> float:

"""

Generates an estimate of the mathematical constant PI.

See https://en.wikipedia.org/wiki/Monte_Carlo_method#Overview

The estimate is generated by Monte Carlo simulations. Let U be uniformly drawn from

the unit square [0, 1) x [0, 1). The probability that U lies in the unit circle is:

P[U in unit circle] = 1/4 PI

and therefore

PI = 4 * P[U in unit circle]

We can get an estimate of the probability P[U in unit circle].

See https://en.wikipedia.org/wiki/Empirical_probability by:

1. Draw a point uniformly from the unit square.

2. Repeat the first step n times and count the number of points in the unit

circle, which is called m.

3. An estimate of P[U in unit circle] is m/n

"""

if number_of_simulations < 1:

raise ValueError("At least one simulation is necessary to estimate PI.")

number_in_unit_circle = 0

for simulation_index in range(number_of_simulations):

random_point = Point.random_unit_square()

if random_point.is_in_unit_circle():

number_in_unit_circle += 1

return 4 * number_in_unit_circle / number_of_simulations

if __name__ == "__main__":

# import doctest

# doctest.testmod()

from math import pi

prompt = "Please enter the desired number of Monte Carlo simulations: "

my_pi = estimate_pi(int(input(prompt).strip()))

print(f"An estimate of PI is {my_pi} with an error of {abs(my_pi - pi)}")

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