About

Title: Monte Carlo Simulation for π Estimation

Overview:
I built a Python-based Monte Carlo simulation that estimates the value of π using random sampling. The system generates 1 million random points in a 2D unit square, identifies which points fall inside the unit circle, and calculates π using the formula:

𝜋
≈
4
×
number of points inside circle
total points
π≈4×
total points
number of points inside circle
​


The solution also tracks the cumulative estimate of π as more points are sampled and visualizes the convergence of the estimate through a matplotlib plot.

Problem it Solves:
Estimating π is a classical problem in numerical methods and simulations. This project demonstrates how random sampling can be used to approximate mathematical constants efficiently. It also provides insight into the concept of convergence in Monte Carlo simulations, which is widely applicable in fields like finance, physics, engineering, and machine learning where analytical solutions are difficult to compute.

Validation Approach:
To validate the solution:

Analytical Comparison: The estimated π value is compared with the actual value of π (3.14159…), and the error is tracked.

Convergence Analysis: A plot shows how the π estimate stabilizes as the number of points increases, demonstrating that the simulation converges towards the true value.

Large Sample Size: Using 1 million points ensures the estimate is statistically significant and reduces random fluctuations.

Results:

The final estimated value of π: ~3.1416

Convergence plot shows the estimate stabilizing close to the true value of π.

Demonstrates the effectiveness of Monte Carlo methods for numerical estimation and probabilistic simulations.