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Monte Carlo Estimation of \(\pi\)ΒΆ

OverviewΒΆ

Monte Carlo methods provide a stochastic way to solve problems that might be deterministic in principle. Estimating the value of \(\pi\) is a classic problem in this domain, often demonstrated using simulations like:

  1. Circle and Square Method: Simulating random points in a square and checking if they fall inside an inscribed circle.
  2. Buffon's Needle Problem: Dropping needles on a floor with parallel lines and calculating the probability of crossing a line.

Both methods highlight the power of randomness in approximating mathematical constants.

Python Code Example: Estimating \(\pi\) with Monte CarloΒΆ

1. Circle and Square MethodΒΆ

import numpy as np
import matplotlib.pyplot as plt

# Parameters
total_points = 10000

# Generate random points in a unit square
x = np.random.uniform(-1, 1, total_points)
y = np.random.uniform(-1, 1, total_points)

# Check if points are inside the unit circle
inside_circle = x**2 + y**2 <= 1

# Estimate Ο€
pi_estimate = 4 * np.sum(inside_circle) / total_points

# Visualization
plt.figure(figsize=(8, 8))
plt.scatter(x[inside_circle], y[inside_circle], s=1, color='blue', label="Inside Circle")
plt.scatter(x[~inside_circle], y[~inside_circle], s=1, color='red', label="Outside Circle")
circle = plt.Circle((0, 0), 1, color='black', fill=False)
plt.gca().add_artist(circle)
plt.title(f"Monte Carlo Estimation of Ο€: {pi_estimate:.4f}")
plt.xlabel("x")
plt.ylabel("y")
plt.legend()
plt.axis("equal")
plt.grid()
plt.savefig("docs/Mechanics/pic/monte_carlo_circle.png")
plt.show()

2. Buffon's Needle ProblemΒΆ

import numpy as np

# Parameters
needle_length = 1
line_spacing = 2
total_needles = 10000

# Simulate needle drops
theta = np.random.uniform(0, np.pi / 2, total_needles)  # Angle with the horizontal
x_center = np.random.uniform(0, line_spacing / 2, total_needles)  # Distance to nearest line

# Check if needle crosses a line
crosses_line = x_center <= (needle_length / 2) * np.sin(theta)

# Estimate Ο€
pi_estimate = (2 * needle_length * total_needles) / (np.sum(crosses_line) * line_spacing)

# Output
print(f"Monte Carlo Estimation of Ο€ using Buffon's Needle: {pi_estimate:.4f}")

Key InsightsΒΆ

  1. Randomness and Approximation: Monte Carlo methods rely on randomness to approximate deterministic quantities. Increasing the number of simulations improves the accuracy of the result.

  2. Circle and Square: This method illustrates the geometric relationship between a circle and its circumscribing square.

  3. Buffon's Needle: A classic probabilistic approach linking geometry and probability theory.

Suggested ProjectsΒΆ

  1. Accuracy vs. Sample Size: Analyze how the estimation of \(\pi\) improves as the number of points or needles increases.

  2. Higher Dimensions: Extend the circle and square method to estimate the volume of a hypersphere in higher dimensions.

  3. Visualization Enhancements: Animate the needle drops or point generation for better understanding.

  4. Statistical Analysis: Perform multiple runs and compute confidence intervals for the estimated \(\pi\).

These examples showcase how randomness can uncover mathematical truths. Experiment with these methods and variations to appreciate the elegance of Monte Carlo techniques in estimating \(\pi\).