{"nbformat":4,"nbformat_minor":0,"metadata":{"colab":{"name":"Monte-Carlo.ipynb","version":"0.3.2","provenance":[],"collapsed_sections":[]},"kernelspec":{"name":"python3","display_name":"Python 3"}},"cells":[{"metadata":{"id":"7L6M6VVzoHNr","colab_type":"text"},"cell_type":"markdown","source":["# **MONTE-CARLO**"]},{"metadata":{"id":"_y1Zq1oCn2Ro","colab_type":"text"},"cell_type":"markdown","source":["![](http://eratosthene.com/images/montecarlo.png)"]},{"metadata":{"id":"Klfd03cNn1Ix","colab_type":"code","colab":{}},"cell_type":"code","source":["# Programme\n","\n","\n","\n","\n"],"execution_count":0,"outputs":[]},{"metadata":{"id":"sv7YMoO4omut","colab_type":"text"},"cell_type":"markdown","source":["# **Partie A**\n","\n","Questions 1, 2, 3 sur le document papier.\n","\n","4. Expliquer ce que fait la fonction  montecarlo()  dans le programme ci-dessous."]},{"metadata":{"id":"H-UDccMrpVGS","colab_type":"code","colab":{}},"cell_type":"code","source":["from random import*\n","\n","def f(x):\n","    return x/2\n","    \n","def montecarlo():          \n","    x=uniform(0,6)     # abscisse aléatoire dans l'intervalle [0;6]\n","    y=uniform(0,5)     # ordonnée aléatoire dans l'intervalle [0;5]\n","    if y<f(x):\n","        return True    \n","    else:\n","        return False"],"execution_count":0,"outputs":[]},{"metadata":{"id":"yHRFU6pbpkLE","colab_type":"text"},"cell_type":"markdown","source":["5. On souhaite appeler 1000 fois la fonction montecarlo() et compter le nombre de points inclus dans la surface étudiée. \n","\n","Compléter le programme ci-dessous:"]},{"metadata":{"id":"V7EoPW6EpswS","colab_type":"code","colab":{}},"cell_type":"code","source":["from random import*\n","\n","def f(x):\n","    return x/2\n","    \n","def montecarlo():          \n","    x=uniform(0,6)     # abscisse aléatoire dans l'intervalle [0;6]\n","    y=uniform(0,5)     # ordonnée aléatoire dans l'intervalle [0;5]\n","    if y<f(x):\n","        return True    \n","    else:\n","        return False\n","\n","compteur=0\n","\n","for i in range(1000):\n","    if montecarlo()== ..... :\n","        compteur= ..............\n","print(compteur/1000)"],"execution_count":0,"outputs":[]},{"metadata":{"id":"KJT4rf6Up8Jd","colab_type":"text"},"cell_type":"markdown","source":["6. A quoi correspond l'affichage obtenu en sortie de programme?\n"]},{"metadata":{"id":"B8xpYhezqeH2","colab_type":"text"},"cell_type":"markdown","source":["**Répondre ici : ...**"]},{"metadata":{"id":"J3jVknBQqiXr","colab_type":"text"},"cell_type":"markdown","source":["7. D'après l'introduction, une estimation de l'aire de $S_1$ est donnée par: \n","\n",">>>>$Aire~du~rectangle \\times f_{\\text{observée}}$.\n","\n","\n",">Modifier le programme pour qu'il affiche une estimation de l'aire de $S_1$."]},{"metadata":{"id":"JBQkvVtQqiGw","colab_type":"text"},"cell_type":"markdown","source":["8. Calculer $|\\text{valeur affichée}-9|$ et interpréter le résultat."]},{"metadata":{"id":"_fTkp0a1qfU8","colab_type":"text"},"cell_type":"markdown","source":["**Répondre ici : ...**"]},{"metadata":{"id":"zIWlHuDvrDhe","colab_type":"text"},"cell_type":"markdown","source":["# **Partie B   Application de la méthode dans un cadre général**\n","\n"," \n"," On souhaite à présent appliquer la méthode de Monte-Carlo pour estimer l'aire d'une surface que nous ne savons pas calculer.\n"," \n"," On considère la fonction $g$ définie sur $[1;5]$ par $g(x)=-x^2+6x-5$.\n"," \n","On note $S_2$ la surface délimitée par $C_g$ et l'axe des abscisses.\n","\n","1. Hachurer la surface $S_2$ sur votre document.\n","\n","2. Modifier le programme pour obtenir une estimation de l'aire de $S_2$.\n","\n","3. Le programme ci-dessous permet d'afficher un point rouge et un bleu dans un repère. \n","\n","\n","En vous inspirant de cet exemple, compléter le programme de la question 2 pour obtenir un graphique où les points inclus dans la surface $S_2$ sont affichés en rouge et les autres en bleu."]},{"metadata":{"id":"Xw-nNDlerzIY","colab_type":"code","outputId":"056061a1-979d-407f-bf12-bf7da570c199","executionInfo":{"status":"ok","timestamp":1547726461043,"user_tz":-540,"elapsed":906,"user":{"displayName":"Mohamed EL AHMADI","photoUrl":"","userId":"15958821790435656147"}},"colab":{"base_uri":"https://localhost:8080/","height":347}},"cell_type":"code","source":["from matplotlib.pyplot import*          # module graphique\n","\n","plot(1,2,'ro')       # point de coordonnées (1;2) motif 'o' de couleur rouge\n","plot(2,3,'bo')       # point de coordonnées (2;3) motif 'o' de couleur bleue\n","       \n","savefig(\"figure\")        # sauvegarde et affichage du graphique"],"execution_count":0,"outputs":[{"output_type":"display_data","data":{"image/png":"iVBORw0KGgoAAAANSUhEUgAAAeEAAAFKCAYAAAAqkecjAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nYII=\n","text/plain":["<Figure size 576x396 with 1 Axes>"]},"metadata":{"tags":[]}}]}]}