{"id":5140,"date":"2020-12-07T15:15:13","date_gmt":"2020-12-07T14:15:13","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?page_id=5140"},"modified":"2023-04-16T16:21:17","modified_gmt":"2023-04-16T14:21:17","slug":"simulation-de-la-planche-de-galton-en-python","status":"publish","type":"page","link":"https:\/\/www.mathweb.fr\/euclide\/simulation-de-la-planche-de-galton-en-python\/","title":{"rendered":"Simulation de la planche de Galton en Python"},"content":{"rendered":"\n<p>Nous allons voir sur cette page comment r\u00e9aliser la simulation de la planche de Galton en Python, au programme de Terminale Math\u00e9matiques.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Simulation de la planche de Galton en Python: introduction<\/h2>\n\n\n\n<figure class=\"wp-block-embed-youtube wp-block-embed is-type-video is-provider-youtube wp-embed-aspect-4-3 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Planche-de-Galton.mp4\" width=\"750\" height=\"563\" src=\"https:\/\/www.youtube.com\/embed\/cpCo6mtdAnE?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<p>Cette vid\u00e9o illustre le principe de cette planche : on laisse tomber une bille tout en haut du &#8220;syst\u00e8me&#8221; (pyramide de clous). \u00c0 chaque clou que rencontre cette bille, elle a le choix entre aller \u00e0 droite ou \u00e0 gauche.<\/p>\n\n\n\n<p>On r\u00e9p\u00e8te ceci avec un grand nombre de billes et tout en bas, on les r\u00e9colte en observant leur distribution (la fa\u00e7on dont elles sont tomb\u00e9es).<\/p>\n\n\n\n<p>Il s&#8217;av\u00e8re qu&#8217;au final, toutes les billes forme une &#8220;courbe en cloche&#8221; appel\u00e9e &#8220;courbe de Gauss&#8221; (je vais vite car le but de cette page n&#8217;est pas tant d&#8217;expliquer ce qu&#8217;est l&#8217;exp\u00e9rience que de la r\u00e9aliser en Python). Pour plus d&#8217;informations, regarder la page de <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Planche_de_Galton\" target=\"_blank\" rel=\"noreferrer noopener\">wikipedia<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Simulation de la planche de Galton en Python: le programme<\/h2>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">from random import choice\n\nc = 7\nbase = c * [0]\n\nfor bille in range(1000):\n    position = c \/\/ 2\n    for clou in range(c-1):\n        position += choice([-1,1])\/2\n            \n    base[ int(position) ] += 1\n\nprint( base )<\/pre>\n\n\n\n<p>Le programme n&#8217;est pas si long que \u00e7a au final&#8230; Expliquons-le !<\/p>\n\n\n\n<p>Tout d&#8217;abord, j&#8217;importe la fonction <em>choice<\/em> du module <strong>random<\/strong>. Nous allons voir plus loin pourquoi elle est int\u00e9ressante&#8230;<\/p>\n\n\n\n<p>Je fixe le nombre de colonnes finales \u00e0 7 (c = 7). Les colonnes sont les endroits qui re\u00e7oivent les billes au final. Il faut que ce soit un nombre impair.<\/p>\n\n\n\n<p>Ensuite, je d\u00e9finis une liste nomm\u00e9e <em>base<\/em> qui vaut initialement [0,0,0,0,0,0,0] (<em>c<\/em> \u00e9l\u00e9ments nuls). Les \u00e9l\u00e9ments de cette liste repr\u00e9senteront le nombre de billes qu&#8217;il y aura \u00e0 la fin dans chacune des colonnes.<\/p>\n\n\n\n<p>En ligne 6, j&#8217;initialise une boucle it\u00e9rative pour simuler 1000 l\u00e2chers de billes.<\/p>\n\n\n\n<p>Pour chaque simulation de l\u00e2chers, je fixe la position de d\u00e9part de la bille: au milieu (donc c\/\/2).<\/p>\n\n\n\n<p>Ensuite, je pars du principe qu&#8217;il y a <em>c<\/em> &#8211; 1 lignes de clous (ce qui est toujours le cas), donc je simule la rencontre de <em>c<\/em> &#8211; 1 clous. Pour chacun d&#8217;eux, je choisis au hasard une direction. Ici, j&#8217;utilise <em>choice<\/em>([-1,1]) pour choisir entre &#8220;-1&#8221; et &#8220;1&#8221;. Si &#8220;-1&#8221; est obtenu, la position actuelle est diminu\u00e9e de 0,5; sinon, elle est augment\u00e9e de 0,5.<\/p>\n\n\n\n<p>\u00c0 l&#8217;issue de la boucle it\u00e9rative sur &#8220;clou&#8221;, j&#8217;obtiens la position de la bille (entre 0 et <em>c<\/em>-1). Je n&#8217;ai plus alors qu&#8217;\u00e0 incr\u00e9menter la valeur d\u00e9j\u00e0 contenue dans la liste <em>base<\/em> pour cette position (ligne 11 et 12).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Simulation de la planche de Galton en Python: r\u00e9sultats<\/h2>\n\n\n\n<p>Je vais effectuer 20 simulations de 1000 l\u00e2chers de billes :<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">from random import choice\n\nc = 7\n\nfor n in range(20):\n    base = c * [0]\n    for bille in range(1000):\n        position = c \/\/ 2\n        for clou in range(c-1):\n            position += choice([-1,1])\/2\n        \n        base[ int(position) ] += 1\n\n    print( base )<\/pre>\n\n\n\n<p>J&#8217;obtiens :<\/p>\n\n\n\n<pre class=\"wp-block-preformatted\">[17, 83, 232, 316, 242, 97, 13]\n[18, 96, 232, 334, 221, 82, 17]\n[17, 85, 243, 304, 240, 98, 13]\n[10, 90, 217, 320, 246, 106, 11]\n[12, 88, 230, 319, 262, 75, 14]\n[13, 84, 245, 302, 245, 100, 11]\n[18, 94, 224, 332, 219, 101, 12]\n[16, 101, 241, 285, 254, 88, 15]\n[23, 92, 239, 302, 220, 103, 21]\n[21, 100, 218, 311, 224, 100, 26]\n[15, 113, 230, 288, 249, 89, 16]\n[15, 92, 231, 310, 252, 85, 15]\n[19, 100, 211, 308, 253, 86, 23]\n[12, 83, 237, 310, 248, 93, 17]\n[16, 97, 239, 302, 228, 103, 15]\n[20, 96, 210, 322, 236, 96, 20]\n[17, 97, 223, 313, 241, 97, 12]\n[14, 89, 246, 330, 216, 91, 14]\n[17, 87, 222, 333, 239, 86, 16]\n[15, 90, 246, 307, 222, 102, 18]<\/pre>\n\n\n\n<h2 class=\"wp-block-heading\">Repr\u00e9sentation graphique<\/h2>\n\n\n\n<p>On va d&#8217;abord repr\u00e9senter en mode texte la liste obtenue avec une simulation. Bien entendu, je ne vais pas simuler le l\u00e2cher de 1000 billes (car je risque de manquer de place&#8230; comme disait Fermat !).<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">from random import choice\n\ndef simulation_galton(c = 7 , n = 20):\n    base = c * [0]\n    for bille in range(n):\n        position = c \/\/ 2\n        for clou in range(c-1):\n            position += choice([-1,1])\/2\n        \n        base[ int(position) ] += 1\n\n    return base\n\ndef graphique( L ):\n    M = max(L)\n    for l in range( max(L) ):\n        ligne = ''\n        for e in L:\n            if e == M:\n                ligne += ' * '\n                L[L.index(e)] -= 1\n            else:\n                ligne += ' ' * 3\n        print(ligne)\n        M -= 1\n\ngraphique( simulation_galton(n=50) )<\/pre>\n\n\n\n<p>qui donne par exemple :<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"186\" height=\"290\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/12\/simulation-planche-galton-python.jpg\" alt=\"simulation planche de Galton en Python, mode texte\" class=\"wp-image-5142\"\/><\/figure><\/div>\n\n\n\n<p>Si l&#8217;on souhaite une repr\u00e9sentation plus \u00e9l\u00e9gante, on pourra par exemple ajouter la fonction Python suivante :<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">def graph_plot( L ):\n    from numpy import arange\n    from matplotlib.pyplot import bar, show\n    c = len( L )\n    x = arange(0,c,1)\n    bar(x,L,width=0.8,color='red',alpha=0.5)\n    show()<\/pre>\n\n\n\n<p>qui donne, en lan\u00e7ant :<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">graph_plot( simulation_galton(n=1000) )<\/pre>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"640\" height=\"480\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/12\/simulation-planche-galton-python-mode-graphique.jpg\" alt=\"simulation planche de Galton en Python, mode graphique\" class=\"wp-image-5143\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/12\/simulation-planche-galton-python-mode-graphique.jpg 640w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/12\/simulation-planche-galton-python-mode-graphique-300x225.jpg 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/12\/simulation-planche-galton-python-mode-graphique-600x450.jpg 600w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/figure><\/div>\n\n\n\n<p>Bien s\u00fbr, cette derni\u00e8re solution est meilleure car s&#8217;adaptant \u00e0 de grandes valeurs de <em>c<\/em> et <em>n<\/em>.<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">graph_plot( simulation_galton(c = 50, n=1000) )<\/pre>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"640\" height=\"480\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/12\/simulation-planche-galton-python-mode-graphique-2.jpg\" alt=\"simulation planche de Galton en Python, mode graphique grandes valeurs\" class=\"wp-image-5144\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/12\/simulation-planche-galton-python-mode-graphique-2.jpg 640w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/12\/simulation-planche-galton-python-mode-graphique-2-300x225.jpg 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/12\/simulation-planche-galton-python-mode-graphique-2-600x450.jpg 600w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Nous allons voir sur cette page comment r\u00e9aliser la simulation de la planche de Galton en Python, au programme de Terminale Math\u00e9matiques. Simulation de la planche de Galton en Python: introduction Cette vid\u00e9o illustre le principe de cette planche : on laisse tomber une bille tout en haut du &#8220;syst\u00e8me&#8221; [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-5140","page","type-page","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Simulation de la planche de Galton en Python - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Nous allons voir sur cette page comment r\u00e9aliser la simulation de la planche de Galton en Python, au programme de Terminale Math\u00e9matiques.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/simulation-de-la-planche-de-galton-en-python\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Simulation de la planche de Galton en Python - 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