{"id":34054,"date":"2021-03-24T00:00:24","date_gmt":"2021-03-24T00:00:24","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34054"},"modified":"2025-08-15T08:46:03","modified_gmt":"2025-08-15T08:46:03","slug":"mensura-probabilitatis-eiusque-proprietates-claves","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/mensura-probabilitatis-eiusque-proprietates-claves\/","title":{"rendered":"Mensura probabilitatis eiusque proprietates clav\u0113s"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Proprietates clav\u0113s mensurae probabilitatis<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>Summarium<\/strong><br \/><em>In hac lectione exhibentur definitiones fundament\u0101l\u0113s theoriae probabilitatis, inter mensuram probabilitatis et sigma-algebram. Praeterea exploratur notio probabilitatis ut limes frequentiarum relativarum, exemplo iact\u016bs aleae sex facierum utens. Per functionem frequentiae relativae demonstratur quomodo mensura probabilitatis per limitem obtineri possit, quod appellatur \u00abprobabilitas ut limes frequentiarum relativarum\u00bb. Exhibentur proprietates clav\u0113s mensurae probabilitatis et ostenditur quomodo probabilitas eventuum computari possit per cas\u016bs faventes super cas\u016bs totales. Hic cognitio fundamentalis est ad intellegendam theoriam probabilitatis eamque applicandam in consiliis capiendis in diversis contextibus.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>PROPOSITA DISCENDI:<\/strong><br \/>\nHoc cursu absoluto discipulus poterit:\n<\/p>\n<ol>\n<li><strong>Cognoscere<\/strong> definitionem mensurae probabilitatis eiusque proprietates fundamentales.<\/li>\n<li><strong>Intellegere<\/strong> notionem probabilitatis ut limitem frequentiarum relativarum.<\/li>\n<li><strong>Computare<\/strong> probabilitatem eventuum per \u00abprobabilitatem ut cas\u016bs faventes super cas\u016bs totales\u00bb.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>INDEX CONTENTORUM<\/strong><br \/>\n<a href=\"#1\">DEFINITIONES FUNDAMENTALES<\/a><br \/>\n<a href=\"#2\">PROBABILITAS UT LIMES FREQUENTIARUM RELATIVARUM<\/a><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/hJJpgf1jRvM\" title=\"YouTube video player\" 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>Definitiones fundamentales<\/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;\">Pars quae essentiam spatii probabilitatum condensat<\/span><\/strong><\/a> est ipsa notio mensurae probabilitatis. Haec est functio quae, denique, quantificat possibilitatem ut certus eventus eveniat.<\/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;\">Mensura probabilitatis<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span> est functio talis ut, data <strong><a href=\"http:\/\/toposuranos.com\/material\/es\/que-es-una-sigma-algebra-definicion-y-ejemplos\/\" rel=\"noopener\" target=\"_blank\">sigma-algebra<\/a><\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma=(\\Omega,\\mathcal{A})<\/span>, probabilitatem cuique eventui <span class=\"katex-eq\" data-katex-display=\"false\">E\\in\\mathcal{A}<\/span> attribuat. Haec mensura probabilitatis proprietates sequentes satisfacit:<\/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,_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;\">(continuitas)<\/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=\"mensura probabilitatis, limes frequentiarum relativarum\" 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=\"mensura probabilitatis, limes frequentiarum relativarum\" 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>Probabilitas ut Limes Frequentiarum Relativarum<\/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;\">Haec notio probabilitatis intellegi potest<\/span><\/strong><\/a> per ideam intuitivam probabilitatis ut \u00ablimitem frequentiarum relativarum\u00bb. Sumamus, exempli gratia, experimentum in quo <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> vicibus alea sex facierum iactatur. <strong><a href=\"http:\/\/toposuranos.com\/material\/es\/conoce-el-espacio-muestral-de-la-teoria-de-las-probabilidades\/\" rel=\"noopener\" target=\"_blank\">Spatium exemplorum<\/a><\/strong> cuiusque iact\u016bs 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 eventus quivis ex <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1d6}<\/span>, tum possumus metiri frequentiam <span class=\"katex-eq\" data-katex-display=\"false\">f_N(A)<\/span> qua eventus <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> occurrit, factis <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> iactibus. Hoc facientes comprobabimus quod:<\/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 alterum eventum quivis <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> ex <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1d6}<\/span> consideremus, tunc fiet ut:<\/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;\">Ex hoc definire possumus novam \u00abfunctionem\u00bb <span class=\"katex-eq\" data-katex-display=\"false\">g_N<\/span>, quam <strong>frequentiam relativam appellamus,<\/strong> per expressionem<\/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;\">Animadvertimus functionem <span class=\"katex-eq\" data-katex-display=\"false\">g_N<\/span> primis tribus proprietatibus satisfacere, quartam vero (continuitatis) hoc modo obtineri non posse, sed ob rationes potius \u00abtechnicas\u00bb introduci, quoniam nonnulli event\u016bs sine hac proprietate consequi non possunt.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Etiam sic, <span class=\"katex-eq\" data-katex-display=\"false\">g_N<\/span> nondum est mensura probabilitatis, quia nondum est functio. Hoc fit quia, cuiuslibet valoris <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">g_N<\/span> non necessario unum tantum valorem reddit, ut functio faceret. Ad hoc solvendi gradum limitem addimus, ut demum mensuram probabilitatis habeamus, obtinentes:<\/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;\">Hoc est quod appellatur Probabilitas ut Limes frequentiarum relativarum.<\/p>\n<h3>Exemplum: iactus aleae<\/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;\">Pergendo in analysi iact\u016bs aleae sex facierum,<\/span> <\/strong><\/a>erit ut:<\/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 experimentum magna vice repetatur, comprobari poterit quod<\/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;\">Quare si definiamus <span class=\"katex-eq\" data-katex-display=\"false\">P(\\{x\\}) = \\lim_{N\\to\\infty} g_N(\\{x\\})<\/span>, tum fiet ut:<\/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;\">Ex hoc exemplo etiam facile est intellegere quod, dato eventu <span class=\"katex-eq\" data-katex-display=\"false\">E<\/span> ex spatio exemplorum <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;\">Hoc est quod appellatur \u00abprobabilitas ut cas\u016bs faventes super cas\u016bs totales\u00bb et fundamentale est ad intellegendum magnam partem ratiocinationum de spatio probabilitatum et mensura probabilitatis.<\/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>Proprietates clav\u0113s mensurae probabilitatis SummariumIn hac lectione exhibentur definitiones fundament\u0101l\u0113s theoriae probabilitatis, inter mensuram probabilitatis et sigma-algebram. Praeterea exploratur notio probabilitatis ut limes frequentiarum relativarum, exemplo iact\u016bs aleae sex facierum utens. Per functionem frequentiae relativae demonstratur quomodo mensura probabilitatis per limitem obtineri possit, quod appellatur \u00abprobabilitas ut limes frequentiarum relativarum\u00bb. Exhibentur proprietates clav\u0113s mensurae probabilitatis [&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":3,"footnotes":""},"categories":[1298,1364],"tags":[],"class_list":["post-34054","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematica","category-probabilitates-et-statistica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Mensura probabilitatis eiusque proprietates clav\u0113s - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce de mensura probabilitatis et analysi probabilitatis ut limite frequentiarum relativarum cum exemplis practicis\" \/>\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\/la\/mensura-probabilitatis-eiusque-proprietates-claves\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Mensura probabilitatis eiusque proprietates clav\u0113s\" \/>\n<meta property=\"og:description\" content=\"Disce de mensura probabilitatis et analysi probabilitatis ut limite frequentiarum relativarum cum exemplis practicis\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/mensura-probabilitatis-eiusque-proprietates-claves\/\" \/>\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=\"2021-03-24T00:00:24+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-08-15T08:46:03+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=\"Mensura probabilitatis eiusque proprietates clav\u0113s\" \/>\n<meta name=\"twitter:description\" content=\"Disce de mensura probabilitatis et analysi probabilitatis ut limite frequentiarum relativarum cum exemplis practicis\" \/>\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=\"3 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/mensura-probabilitatis-eiusque-proprietates-claves\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/mensura-probabilitatis-eiusque-proprietates-claves\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Mensura probabilitatis eiusque proprietates clav\u0113s\",\"datePublished\":\"2021-03-24T00:00:24+00:00\",\"dateModified\":\"2025-08-15T08:46:03+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/mensura-probabilitatis-eiusque-proprietates-claves\\\/\"},\"wordCount\":686,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/mensura-probabilitatis-eiusque-proprietates-claves\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/03\\\/MEDIDADEPROBABILIDAD-1.jpg\",\"articleSection\":[\"Mathematica\",\"Probabilitates et Statistica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/mensura-probabilitatis-eiusque-proprietates-claves\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/mensura-probabilitatis-eiusque-proprietates-claves\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/mensura-probabilitatis-eiusque-proprietates-claves\\\/\",\"name\":\"Mensura probabilitatis eiusque proprietates clav\u0113s - 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