{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Import the numpy library\n",
    "import numpy as np\n",
    "# On the Computation of π\n",
    "# Buffon's Needle Method\n",
    "import numpy as np\n",
    "\n",
    "# Set a random seed for reproducibility\n",
    "np.random.seed(42)\n",
    "\n",
    "# Number of points to simulate\n",
    "n = 1000000  \n",
    "\n",
    "# Generate random points in the square [-1, 1] x [-1, 1]\n",
    "x = np.random.uniform(-1, 1, n)\n",
    "y = np.random.uniform(-1, 1, n)\n",
    "\n",
    "# Check if points are inside the circle\n",
    "inside = (x**2 + y**2) <= 1\n",
    "\n",
    "# Estimate π\n",
    "pi_estimate = 4 * np.sum(inside) / n\n",
    "print(\"Buffon's Needle π estimate:\", pi_estimate)\n",
    "# Monte Carlo Integration Approach\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Generate points and determine those inside the circle\n",
    "plt.figure(figsize=(6, 6))\n",
    "plt.scatter(x[inside], y[inside], color='blue', alpha=0.5, label='Inside Circle')\n",
    "plt.scatter(x[~inside], y[~inside], color='red', alpha=0.5, label='Outside Circle')\n",
    "plt.legend()\n",
    "plt.title(\"Monte Carlo Method - Points inside/outside the circle\")\n",
    "plt.show()\n",
    "\n",
    "# Final π Estimate\n",
    "pi_estimate_mc = 4 * np.sum(inside) / n\n",
    "print(\"Monte Carlo π estimate:\", pi_estimate_mc)\n",
    "## Conclusion\n",
    "Both Buffon’s Needle and Monte Carlo Integration provide useful approximations of π.\n",
    "In this case, the estimated value using Monte Carlo is closer to the known value of π ≈ 3.14159.\n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}