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module2/exo1/module2_exo1_toy_notebook_fr.ipynb

+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "toy_notebook_fr\n",
+    "\n",
+    "March 28,2019"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# 1 A propos du calcul de $\\pi$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## 1.1 En demandant à la lib maths"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Mon ordinateur m'indique que $\\pi$ vaut _approximativement_"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "3.141592653589793\n"
+     ]
+    }
+   ],
+   "source": [
+    "from math import *\n",
+    "print(pi)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## 1.2 En utilisant la méthode des aiguilles de Buffon"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Mais calculé avec la **méthode** des [aiguilles de Buffon](https://fr.wikipedia.org/wiki/Aiguille_de_Buffon), on obtiendrait comme approximation :"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "3.144654088050314"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "import numpy as np\n",
+    "np.random.seed(seed=42)\n",
+    "N = 1000\n",
+    "x = np.random.uniform(size=N, low=0, high=1)\n",
+    "theta = np.random.uniform(size=N, low=0, high=pi/2)\n",
+    "2/(sum((x+np.sin(theta))>1)/N)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## 1.3 Avec un argument \"fréquentiel\" de surface"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Sinon, une méthode plus simple à comprendre et ne faisant pas intervenir d'appel à la fonction sinus se base sur le fait que si X et Y alors $\\X^2$ + $\\Y^2$ <=1} = pi/4$ ([voir méthode Monte Carlo sur Wikipedia](https://fr.wikipedia.org/wiki/M%C3%A9thode_de_Monte-Carlo#D%C3%A9termination_de_la_valeur_de_%CF%80)\n",
+    "). Le code suivant illustre ce fait :"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "%matplotlib inline\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "np.random.seed(seed=42)\n",
+    "N = 100\n",
+    "x = np.random.uniform(size=N, low=0, high=1)\n",
+    "y = np.random.uniform(size=N, low=0, high=1)\n",
+    "accept = (x*x+y*y) <= 1\n",
+    "reject = np.logical_not(accept)\n",
+    "\n",
+    "fig, ax = plt.subplots(1)\n",
+    "ax.scatter(x[accept], y[accept], c='b', alpha=0.2, edgecolor=None)\n",
+    "ax.scatter(x[reject], y[reject], c='r', alpha=0.2, edgecolor=None)\n",
+    "ax.set_aspect('equal')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Il est alors aisé d'obtenir une approximation (pas terrible) de $\\pi$ en comptant combien de fois, en moyenne, $X^2$ + $Y^2$ est inférieur à 1:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "3.2"
+      ]
+     },
+     "execution_count": 11,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "4*np.mean(accept)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "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": "3.7.13"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}