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    "# À propos de π\n",
    "\n",
    "Dans ce mini-notebook, nous rappelons deux représentations classiques de π :\n",
    "\n",
    "$$ \\pi = \\frac{C}{D} $$  \n",
    "$$ \\displaystyle \\pi = 4 \\sum_{k=0}^{\\infty} \\frac{(-1)^k}{2k+1} . $$\n",
    "\n",
    "La célèbre expérience des [aiguilles de Buffon](https://fr.wikipedia.org/wiki/Aiguille_de_Buffon) permet aussi d’estimer π de manière probabiliste."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "math.pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random, math\n",
    "\n",
    "random.seed(42)        # pour reproduire la même estimation\n",
    "N = 100_000            # nombre de lancers\n",
    "l, d = 1.0, 1.0        # longueur de l’aiguille et espacement des lignes\n",
    "\n",
    "cross = 0\n",
    "for _ in range(N):\n",
    "    x      = random.uniform(0, d/2)          # distance centre-ligne la plus proche\n",
    "    theta  = random.uniform(0, math.pi/2)    # angle avec la verticale\n",
    "    if x <= (l/2) * math.sin(theta):\n",
    "        cross += 1\n",
    "\n",
    "pi_est = (2 * l * N) / (cross * d)\n",
    "pi_est"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def buffon_once(n):\n",
    "    \"\"\"Une estimation de π par Buffon sur n lancers\"\"\"\n",
    "    crosses = sum(\n",
    "        random.uniform(0, d/2) <= (l/2) * math.sin(random.uniform(0, math.pi/2))\n",
    "        for _ in range(n)\n",
    "    )\n",
    "    return (2 * l * n) / (crosses * d)\n",
    "\n",
    "random.seed(0)\n",
    "ests = [buffon_once(2_000) for _ in range(1_000)]\n",
    "\n",
    "plt.hist(ests, bins=30)\n",
    "plt.axvline(math.pi, linestyle='--')\n",
    "plt.title(\"Distribution des estimations de π (méthode de Buffon)\")\n",
    "plt.xlabel(\"π estimé\")\n",
    "plt.ylabel(\"Fréquence\")\n",
    "plt.show()"
   ]
  }
 ],
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