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   "source": [
    "# À propos de π\n",
    "\n",
    "Ce notebook présente différentes méthodes pour estimer la valeur de π.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Formules mathématiques\n",
    "\n",
    "La valeur de π peut être estimée par différentes méthodes.  \n",
    "Par exemple, avec la formule suivante :\n",
    "\n",
    "$$\\pi = \\frac{C}{D}$$\n",
    "\n",
    "où *C* est la circonférence et *D* le diamètre d’un cercle.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "print(f\"La valeur de π selon la bibliothèque math est : {math.pi}\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Méthode des aiguilles de Buffon\n",
    "\n",
    "Pour en savoir plus :  \n",
    "[Expérience des aiguilles de Buffon (Wikipedia)](https://fr.wikipedia.org/wiki/Aiguille_de_Buffon)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random\n",
    "import math\n",
    "\n",
    "def buffon_needle(n_drops=100000):\n",
    "    hits = 0\n",
    "    for _ in range(n_drops):\n",
    "        y = random.random()\n",
    "        theta = random.uniform(0, math.pi / 2)\n",
    "        if y <= (math.sin(theta) / 2):\n",
    "            hits += 1\n",
    "    return (2 * n_drops) / hits\n",
    "\n",
    "pi_est = buffon_needle()\n",
    "print(f\"Estimation de π par la méthode de Buffon : {pi_est}\")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Exemple : afficher la convergence de l'estimation de π\n",
    "import numpy as np\n",
    "\n",
    "n_values = np.arange(1000, 20000, 1000)\n",
    "pi_estimations = [buffon_needle(n) for n in n_values]\n",
    "\n",
    "plt.plot(n_values, pi_estimations)\n",
    "plt.axhline(y=math.pi, color='r', linestyle='--', label='π réel')\n",
    "plt.xlabel(\"Nombre d'expériences\")\n",
    "plt.ylabel(\"Estimation de π\")\n",
    "plt.legend()\n",
    "plt.title(\"Convergence de la méthode de Buffon\")\n",
    "plt.show()\n"
   ]
  }
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