"Completed Buffon’s Needle and Monte Carlo estimation of π"

module2/exo1/toy_notebook_en.ipynb

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+   "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"
+   ]
+  }
+ ],
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@@ -16,10 +64,9 @@
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