{"id":25495,"date":"2021-04-24T00:00:41","date_gmt":"2021-04-24T00:00:41","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25495"},"modified":"2024-06-05T04:24:20","modified_gmt":"2024-06-05T04:24:20","slug":"la-mesure-de-probabilite-et-ses-proprietes-cles","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/","title":{"rendered":"La mesure de probabilit\u00e9 et ses propri\u00e9t\u00e9s cl\u00e9s"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Propri\u00e9t\u00e9s cl\u00e9s de la mesure de probabilit\u00e9<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>R\u00e9sum\u00e9<\/strong><br \/><em>Dans cette classe, les d\u00e9finitions de base de la th\u00e9orie des probabilit\u00e9s sont pr\u00e9sent\u00e9es, y compris la mesure de probabilit\u00e9 et la sigma-alg\u00e8bre. De plus, la notion de probabilit\u00e9 en tant que limite des fr\u00e9quences relatives est explor\u00e9e, en utilisant l&#8217;exemple du lancer d&#8217;un d\u00e9 \u00e0 6 faces. \u00c0 travers la fonction de fr\u00e9quence relative, il est d\u00e9montr\u00e9 comment obtenir une mesure de probabilit\u00e9 \u00e0 travers la limite, ce qui est connu comme la \u00abprobabilit\u00e9 en tant que limite des fr\u00e9quences relatives\u00bb. Les propri\u00e9t\u00e9s cl\u00e9s de la mesure de probabilit\u00e9 sont pr\u00e9sent\u00e9es et il est montr\u00e9 comment on peut calculer la probabilit\u00e9 d&#8217;\u00e9v\u00e9nements \u00e0 travers des cas favorables sur des cas totaux. Cette connaissance est fondamentale pour comprendre la th\u00e9orie des probabilit\u00e9s et l&#8217;appliquer dans la prise de d\u00e9cision dans diff\u00e9rents contextes.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>OBJECTIFS D&#8217;APPRENTISSAGE :<\/strong><br \/>\nApr\u00e8s avoir termin\u00e9 cette classe, l&#8217;\u00e9tudiant sera capable de :\n<\/p>\n<ol>\n<li><strong>Conna\u00eetre<\/strong> la d\u00e9finition de la mesure de probabilit\u00e9 et ses propri\u00e9t\u00e9s fondamentales.<\/li>\n<li><strong>Comprendre<\/strong> la notion de probabilit\u00e9 en tant que limite des fr\u00e9quences relatives.<\/li>\n<li><strong>Calculer<\/strong> la probabilit\u00e9 d&#8217;\u00e9v\u00e9nements \u00e0 travers la \u00abprobabilit\u00e9 comme cas favorables sur cas totaux\u00bb.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX DES CONTENUS<\/strong><br \/>\n<a href=\"#1\">D\u00c9FINITIONS DE BASE<\/a><br \/>\n<a href=\"#2\">PROBABILIT\u00c9 COMME LIMITE DE FR\u00c9QUENCES RELATIVES<\/a><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/hJJpgf1jRvM\" title=\"Lecteur vid\u00e9o YouTube\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p><a name=\"1\"><\/a><\/p>\n<h2>D\u00e9finitions de base<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=hJJpgf1jRvM&amp;t=117s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">La partie qui condense l&#8217;essence d&#8217;un espace de probabilit\u00e9s<\/span><\/strong><\/a> est justement l&#8217;id\u00e9e de mesure de probabilit\u00e9. C&#8217;est la fonction qui quantifie, finalement, la possibilit\u00e9 qu&#8217;un certain \u00e9v\u00e9nement se produise.<\/p>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=hJJpgf1jRvM&amp;t=157s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Une mesure de probabilit\u00e9<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span> est une fonction telle que, donn\u00e9e une <strong><a href=\"http:\/\/toposuranos.com\/material\/es\/que-es-una-sigma-algebra-definicion-y-ejemplos\/\" rel=\"noopener\" target=\"_blank\">sigma-alg\u00e8bre<\/a><\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma=(\\Omega,\\mathcal{A})<\/span>, elle assigne une probabilit\u00e9 \u00e0 chaque \u00e9v\u00e9nement <span class=\"katex-eq\" data-katex-display=\"false\">E\\in\\mathcal{A}<\/span>. Cette mesure de probabilit\u00e9 satisfait les propri\u00e9t\u00e9s suivantes :<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">(E\\in \\mathcal{A}) \\rightarrow (0 \\leq P(E) \\leq 1)<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">P(\\Omega) = 1<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\left[E_1, E_2 \\in \\mathcal{A}\\right] \\rightarrow \\left[ (E_1 \\cap E_2 = \\emptyset) \\rightarrow \\left(P\\left(E_1 \\cup E_2 \\right) = P(E_1) + P(E_2) \\right) \\right]<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left[E_1 \\subseteq E_2 \\subseteq \\cdots \\subseteq E_n \\in \\mathcal{A}\\right] \\rightarrow \\left[P \\left(\\bigcup_{n=1}^\\infty E_n \\right) = \\lim_{n\\to\\infty}P(E_n)\\right]<\/span> <span style=\"color: #800000;\">(continuit\u00e9)<\/span><\/li>\n<\/ol>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/apuestas.jpg\" alt=\"mesure de probabilit\u00e9, limite des fr\u00e9quences relatives\" width=\"1280\" height=\"720\" class=\"aligncenter size-full wp-image-25465 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/apuestas.jpg\" alt=\"mesure de probabilit\u00e9, limite des fr\u00e9quences relatives\" width=\"1280\" height=\"720\" class=\"aligncenter size-full wp-image-25465 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/apuestas.jpg 1280w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/apuestas-300x169.jpg 300w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/apuestas-1024x576.jpg 1024w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/apuestas-768x432.jpg 768w\" sizes=\"(max-width: 1280px) 100vw, 1280px\" \/><\/noscript><\/center><br \/>\n<a name=\"2\"><\/a><\/p>\n<h2>Probabilit\u00e9 en tant que Limite des Fr\u00e9quences Relatives<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=hJJpgf1jRvM&amp;t=510s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Cette notion de probabilit\u00e9 peut \u00eatre comprise<\/span><\/strong><\/a> \u00e0 travers l&#8217;id\u00e9e intuitive de la probabilit\u00e9 en tant que \u00ablimite des fr\u00e9quences relatives\u00bb. Prenons, par exemple, l&#8217;exp\u00e9rience o\u00f9 l&#8217;on lance <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> fois un d\u00e9 \u00e0 6 faces. L&#8217;<strong><a href=\"http:\/\/toposuranos.com\/material\/es\/conoce-el-espacio-muestral-de-la-teoria-de-las-probabilidades\/\" rel=\"noopener\" target=\"_blank\">espace des \u00e9chantillons<\/a><\/strong> de chaque lancer est <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1d6} = \\{1,2,3,4,5,6\\}<\/span>. Si <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> est un \u00e9v\u00e9nement quelconque de <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1d6}<\/span>, nous pouvons alors mesurer la fr\u00e9quence <span class=\"katex-eq\" data-katex-display=\"false\">f_N(A)<\/span> \u00e0 laquelle l&#8217;\u00e9v\u00e9nement <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> se produit lors de <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> lancers. En faisant cela, nous v\u00e9rifierons que :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">0 \\leq f_N(A) \\leq f_N(\\Omega_{1d6}) = N<\/span>,<\/p>\n<p style=\"text-align: justify; color: #000000;\">et si nous consid\u00e9rons un second \u00e9v\u00e9nement quelconque <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> de <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1d6}<\/span>, il en r\u00e9sultera que :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">(A \\cap B = \\emptyset) \\rightarrow (f_N(A \\cup B) = f_N(A) + f_N(B))<\/span>\n<p style=\"text-align: justify; color: #000000;\">\u00c0 partir de cela, nous pouvons d\u00e9finir une nouvelle \u00abfonction\u00bb <span class=\"katex-eq\" data-katex-display=\"false\">g_N<\/span>, que <strong>nous appelons fr\u00e9quence relative,<\/strong> \u00e0 travers l&#8217;expression<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">g_N(A) = \\displaystyle \\frac{f_N(A)}{N}<\/span>\n<p style=\"text-align: justify; color: #000000;\">Nous remarquerons que la fonction <span class=\"katex-eq\" data-katex-display=\"false\">g_N<\/span> remplit les trois premi\u00e8res propri\u00e9t\u00e9s, seule la quatri\u00e8me (de continuit\u00e9) ne peut pas \u00eatre obtenue de cette mani\u00e8re, mais est introduite pour des raisons plut\u00f4t \u00abtechniques\u00bb car certains r\u00e9sultats ne peuvent pas \u00eatre obtenus sans cette propri\u00e9t\u00e9.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Malgr\u00e9 cela, <span class=\"katex-eq\" data-katex-display=\"false\">g_N<\/span> n&#8217;est pas encore une mesure de probabilit\u00e9, car elle n&#8217;est pas encore une fonction. Cela est d\u00fb au fait que, quelle que soit la valeur de <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">g_N<\/span> ne donne pas n\u00e9cessairement une valeur unique, comme le ferait une fonction. Pour r\u00e9soudre cela, nous ajoutons une \u00e9tape \u00e0 la limite pour finalement obtenir une mesure de probabilit\u00e9, obtenant :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(A) = \\lim_{N\\to\\infty} g_N(A)<\/span>\n<p style=\"text-align: justify; color: #000000;\">C&#8217;est ce que l&#8217;on appelle la Probabilit\u00e9 en tant que Limite des fr\u00e9quences relatives.<\/p>\n<h3>Exemple : le lancer d&#8217;un d\u00e9<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=hJJpgf1jRvM&amp;t=932s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Poursuivant l&#8217;analyse du lancer d&#8217;un d\u00e9 \u00e0 6 faces,<\/span> <\/strong><\/a>il s&#8217;av\u00e9rera que :<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\left(\\forall x\\in\\{1,2,3,4,5,6\\}\\right) \\left( g_N(\\{x\\}) = \\frac{f_N(\\{x\\})}{N}\\right)<\/span>\n<p style=\"text-align: justify; color: #000000;\">et si l&#8217;exp\u00e9rience est r\u00e9p\u00e9t\u00e9e un grand nombre de fois, on pourra v\u00e9rifier que<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\forall x\\in\\{1,2,3,4,5,6\\} \\right) \\left( \\lim_{N \\to \\infty} g_N(\\{x\\}) = \\displaystyle \\frac{1}{6} \\right)<\/span>\n<p style=\"text-align: justify; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{N \\to \\infty} g_N(\\Omega_{1d6}) = 1<\/span>\n<p style=\"text-align: justify; color: #000000;\">Par cons\u00e9quent, si nous d\u00e9finissons <span class=\"katex-eq\" data-katex-display=\"false\">P(\\{x\\}) = \\lim_{N\\to\\infty} g_N(\\{x\\})<\/span>, il en r\u00e9sultera que :<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(x=\\{1,3,5\\}) = P(\\{1\\}\\cup\\{3\\}\\cup\\{5\\}) = P(\\{1\\})+P(\\{3\\})+P(\\{5\\}) = \\frac{1}{6}+\\frac{1}{6}+\\frac{1}{6}= \\frac{1}{2} <\/span>\n<p style=\"text-align: justify; color: #000000;\">\u00c0 partir de cet exemple, il est \u00e9galement simple d&#8217;intuire que, pour un \u00e9v\u00e9nement <span class=\"katex-eq\" data-katex-display=\"false\">E<\/span> d&#8217;un espace d&#8217;\u00e9chantillonnage <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(E) = \\frac{\\# E}{\\# \\Omega}<\/span>\n<p style=\"text-align: justify; color: #000000;\">C&#8217;est ce que l&#8217;on appelle la \u00abprobabilit\u00e9 comme cas favorables sur cas totaux\u00bb et est fondamental pour comprendre de nombreux raisonnements concernant l&#8217;espace de probabilit\u00e9s et la mesure de probabilit\u00e9.<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/OAzCnbW3pSw?si=pNyBUqlZEuhRJO4i\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Propri\u00e9t\u00e9s cl\u00e9s de la mesure de probabilit\u00e9 R\u00e9sum\u00e9Dans cette classe, les d\u00e9finitions de base de la th\u00e9orie des probabilit\u00e9s sont pr\u00e9sent\u00e9es, y compris la mesure de probabilit\u00e9 et la sigma-alg\u00e8bre. De plus, la notion de probabilit\u00e9 en tant que limite des fr\u00e9quences relatives est explor\u00e9e, en utilisant l&#8217;exemple du lancer d&#8217;un d\u00e9 \u00e0 6 faces. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":25477,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":15,"footnotes":""},"categories":[569,682],"tags":[],"class_list":["post-25495","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","category-probabilites-et-statistiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>La mesure de probabilit\u00e9 et ses propri\u00e9t\u00e9s cl\u00e9s - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Apprenez la mesure de probabilit\u00e9 et l&#039;analyse de probabilit\u00e9 comme limite des fr\u00e9quences relatives avec des exemples pratiques\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"La mesure de probabilit\u00e9 et ses propri\u00e9t\u00e9s cl\u00e9s\" \/>\n<meta property=\"og:description\" content=\"Apprenez la mesure de probabilit\u00e9 et l&#039;analyse de probabilit\u00e9 comme limite des fr\u00e9quences relatives avec des exemples pratiques\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-04-24T00:00:41+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-06-05T04:24:20+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/03\/MEDIDADEPROBABILIDAD-1-1024x576.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"La mesure de probabilit\u00e9 et ses propri\u00e9t\u00e9s cl\u00e9s\" \/>\n<meta name=\"twitter:description\" content=\"Apprenez la mesure de probabilit\u00e9 et l&#039;analyse de probabilit\u00e9 comme limite des fr\u00e9quences relatives avec des exemples pratiques\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/03\/MEDIDADEPROBABILIDAD-1.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/la-mesure-de-probabilite-et-ses-proprietes-cles\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/la-mesure-de-probabilite-et-ses-proprietes-cles\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"La mesure de probabilit\u00e9 et ses propri\u00e9t\u00e9s cl\u00e9s\",\"datePublished\":\"2021-04-24T00:00:41+00:00\",\"dateModified\":\"2024-06-05T04:24:20+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/la-mesure-de-probabilite-et-ses-proprietes-cles\\\/\"},\"wordCount\":958,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/la-mesure-de-probabilite-et-ses-proprietes-cles\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/03\\\/MEDIDADEPROBABILIDAD-1.jpg\",\"articleSection\":[\"Math\u00e9matiques\",\"Probabilit\u00e9s et Statistiques\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/la-mesure-de-probabilite-et-ses-proprietes-cles\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/la-mesure-de-probabilite-et-ses-proprietes-cles\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/la-mesure-de-probabilite-et-ses-proprietes-cles\\\/\",\"name\":\"La mesure de probabilit\u00e9 et ses propri\u00e9t\u00e9s cl\u00e9s - toposuranos.com\\\/material\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/la-mesure-de-probabilite-et-ses-proprietes-cles\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/la-mesure-de-probabilite-et-ses-proprietes-cles\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/03\\\/MEDIDADEPROBABILIDAD-1.jpg\",\"datePublished\":\"2021-04-24T00:00:41+00:00\",\"dateModified\":\"2024-06-05T04:24:20+00:00\",\"description\":\"Apprenez la mesure de probabilit\u00e9 et l'analyse de probabilit\u00e9 comme limite des fr\u00e9quences relatives avec des exemples pratiques\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/la-mesure-de-probabilite-et-ses-proprietes-cles\\\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/la-mesure-de-probabilite-et-ses-proprietes-cles\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/la-mesure-de-probabilite-et-ses-proprietes-cles\\\/#primaryimage\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/03\\\/MEDIDADEPROBABILIDAD-1.jpg\",\"contentUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/03\\\/MEDIDADEPROBABILIDAD-1.jpg\",\"width\":1280,\"height\":720},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/la-mesure-de-probabilite-et-ses-proprietes-cles\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/cursos-de-matematica-y-fisica\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"La mesure de probabilit\u00e9 et ses propri\u00e9t\u00e9s cl\u00e9s\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#website\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/\",\"name\":\"toposuranos.com\\\/material\",\"description\":\"\",\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"es\"},{\"@type\":\"Organization\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\",\"name\":\"toposuranos.com\\\/material\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"width\":2400,\"height\":2059,\"caption\":\"toposuranos.com\\\/material\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\"},\"sameAs\":[\"https:\\\/\\\/www.facebook.com\\\/groups\\\/toposuranos\",\"https:\\\/\\\/x.com\\\/topuranos\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UC16yDm12cPcrwsE0fAM7X1g\",\"https:\\\/\\\/www.linkedin.com\\\/company\\\/69429190\"]},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\",\"name\":\"giorgio.reveco\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"caption\":\"giorgio.reveco\"},\"description\":\"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.\",\"sameAs\":[\"http:\\\/\\\/toposuranos.com\\\/material\"],\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/author\\\/giorgio-reveco\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"La mesure de probabilit\u00e9 et ses propri\u00e9t\u00e9s cl\u00e9s - toposuranos.com\/material","description":"Apprenez la mesure de probabilit\u00e9 et l'analyse de probabilit\u00e9 comme limite des fr\u00e9quences relatives avec des exemples pratiques","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/","og_locale":"es_ES","og_type":"article","og_title":"La mesure de probabilit\u00e9 et ses propri\u00e9t\u00e9s cl\u00e9s","og_description":"Apprenez la mesure de probabilit\u00e9 et l'analyse de probabilit\u00e9 comme limite des fr\u00e9quences relatives avec des exemples pratiques","og_url":"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2021-04-24T00:00:41+00:00","article_modified_time":"2024-06-05T04:24:20+00:00","og_image":[{"url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/03\/MEDIDADEPROBABILIDAD-1-1024x576.jpg","type":"","width":"","height":""}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"La mesure de probabilit\u00e9 et ses propri\u00e9t\u00e9s cl\u00e9s","twitter_description":"Apprenez la mesure de probabilit\u00e9 et l'analyse de probabilit\u00e9 comme limite des fr\u00e9quences relatives avec des exemples pratiques","twitter_image":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/03\/MEDIDADEPROBABILIDAD-1.jpg","twitter_creator":"@topuranos","twitter_site":"@topuranos","twitter_misc":{"Escrito por":"giorgio.reveco","Tiempo de lectura":"4 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/#article","isPartOf":{"@id":"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/"},"author":{"name":"giorgio.reveco","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"La mesure de probabilit\u00e9 et ses propri\u00e9t\u00e9s cl\u00e9s","datePublished":"2021-04-24T00:00:41+00:00","dateModified":"2024-06-05T04:24:20+00:00","mainEntityOfPage":{"@id":"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/"},"wordCount":958,"commentCount":0,"publisher":{"@id":"https:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/#primaryimage"},"thumbnailUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/03\/MEDIDADEPROBABILIDAD-1.jpg","articleSection":["Math\u00e9matiques","Probabilit\u00e9s et Statistiques"],"inLanguage":"es","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/","url":"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/","name":"La mesure de probabilit\u00e9 et ses propri\u00e9t\u00e9s cl\u00e9s - toposuranos.com\/material","isPartOf":{"@id":"https:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/#primaryimage"},"image":{"@id":"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/#primaryimage"},"thumbnailUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/03\/MEDIDADEPROBABILIDAD-1.jpg","datePublished":"2021-04-24T00:00:41+00:00","dateModified":"2024-06-05T04:24:20+00:00","description":"Apprenez la mesure de probabilit\u00e9 et l'analyse de probabilit\u00e9 comme limite des fr\u00e9quences relatives avec des exemples pratiques","breadcrumb":{"@id":"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/#primaryimage","url":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/03\/MEDIDADEPROBABILIDAD-1.jpg","contentUrl":"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/03\/MEDIDADEPROBABILIDAD-1.jpg","width":1280,"height":720},{"@type":"BreadcrumbList","@id":"https:\/\/toposuranos.com\/material\/fr\/la-mesure-de-probabilite-et-ses-proprietes-cles\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"http:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"La mesure de probabilit\u00e9 et ses propri\u00e9t\u00e9s cl\u00e9s"}]},{"@type":"WebSite","@id":"https:\/\/toposuranos.com\/material\/#website","url":"https:\/\/toposuranos.com\/material\/","name":"toposuranos.com\/material","description":"","publisher":{"@id":"https:\/\/toposuranos.com\/material\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/toposuranos.com\/material\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"es"},{"@type":"Organization","@id":"https:\/\/toposuranos.com\/material\/#organization","name":"toposuranos.com\/material","url":"https:\/\/toposuranos.com\/material\/","logo":{"@type":"ImageObject","inLanguage":"es","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","width":2400,"height":2059,"caption":"toposuranos.com\/material"},"image":{"@id":"https:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/groups\/toposuranos","https:\/\/x.com\/topuranos","https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g","https:\/\/www.linkedin.com\/company\/69429190"]},{"@type":"Person","@id":"https:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1","name":"giorgio.reveco","image":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","caption":"giorgio.reveco"},"description":"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"https:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/25495","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=25495"}],"version-history":[{"count":0,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/25495\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/25477"}],"wp:attachment":[{"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=25495"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=25495"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=25495"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}