{"id":34093,"date":"2021-03-29T13:00:28","date_gmt":"2021-03-29T13:00:28","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34093"},"modified":"2025-08-16T06:49:48","modified_gmt":"2025-08-16T06:49:48","slug":"theoremata-utilia-ad-calculum-probabilitatum","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/theoremata-utilia-ad-calculum-probabilitatum\/","title":{"rendered":"Theoremata Utilia ad Calculum Probabilitatum"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Theoremata Utilia ad Calculum Probabilitatum<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>Summarium<\/strong><br \/><em>In hac lectione proponuntur exercitationes resolutae quibus demonstrantur nonnulla theoremata utilia ad calculum probabilitatum, cum demonstrationibus et deductionibus. Exercitationes tractant argumenta ut probabilitas complementaria, inclusio insectorum et convergentia eventuum. Has exercitationes perficere tibi praebebit firmam basim ad altius studium theoriae probabilitatum persequendum.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>OBJECTIVA DISCENDI:<\/strong><br \/>\nHis peractis lectionibus discipulus poterit:\n<\/p>\n<ol>\n<li><strong>Demonstrari<\/strong> proprietates fundamentales probabilitatum<\/li>\n<\/ol>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/SSkPFP5FpKM\" 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 style=\"text-align: justify;\">Quae infra videbimus est dux exercitationum (resolutarum) quarum propositum est demonstrare nonnulla theoremata utilia ad theoriam probabilitatum. Experire eas solvere et postea compara tua responsa &gt;:D<\/p>\n<ol style=\"text-align: justify;\">\n<li><strong><a href=\"https:\/\/www.youtube.com\/watch?v=SSkPFP5FpKM&amp;t=111s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">Demonstra <span class=\"katex-eq\" data-katex-display=\"false\">P(A^c) = 1 -P(A)<\/span> et,<\/span><\/a> ex hoc fac deductionem quae sinat arguere <span class=\"katex-eq\" data-katex-display=\"false\">P(\\emptyset) = 0<\/span><\/strong><br \/>\n<span class=\"collapseomatic \" id=\"id6a58b93db0223\"  tabindex=\"0\" title=\"MOSTRAR SOLUCI\u00d3N\"    >MOSTRAR SOLUCI\u00d3N<\/span><div id=\"target-id6a58b93db0223\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify;\">Ex definitione <a href=\"https:\/\/toposuranos.com\/el-espacio-de-probabilidades-medida-de-probabilidad\/\" target=\"_blank\" rel=\"noopener\">mensurae probabilitatis<\/a> habetur quod, si <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> sunt eventus mensurabiles quicumque, tum valebit<\/p>\n<table>\n<tbody>\n<tr>\n<td>[a]<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">0\\leq P(A) \\leq 1<\/span><\/td>\n<\/tr>\n<tr>\n<td>[b]<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">A\\cap B = \\emptyset \\rightarrow P(A\\cup B) = P(A) + P(B)<\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(\\Omega) = 1<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Nunc, cum <span class=\"katex-eq\" data-katex-display=\"false\">A\\cap A^c = \\emptyset,<\/span> ex parte [b], habebitur:<\/p>\n<table>\n<tbody>\n<tr>\n<td>[d]<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">A\\cap A^c = \\emptyset \\rightarrow P(A\\cup A^c) = P(A) + P(A^c)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Cum <span class=\"katex-eq\" data-katex-display=\"false\">A\\cap A^c = \\emptyset<\/span> semper verum sit et <span class=\"katex-eq\" data-katex-display=\"false\">A\\cup A^c = \\Omega<\/span>, tum habebitur<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">1=P(\\Omega) = P(A\\cup A^c) = P(A) + P(A^c)<\/span>\n<p style=\"text-align: justify;\">Et, proinde<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A^c) = 1-P(A)<\/span>\n<p style=\"text-align: justify;\">Quod erat demonstrandum.<\/p>\n<p style=\"text-align: justify;\">Ad demonstrandum <span class=\"katex-eq\" data-katex-display=\"false\">P(\\emptyset)=0<\/span>, sufficit sumere <span class=\"katex-eq\" data-katex-display=\"false\">A=\\Omega<\/span> in relatione nuper probata et habebitur<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(\\emptyset) = P(\\Omega^c) =1 - P(\\Omega) = 1-1 = 0<\/span>\n<\/div><\/li>\n<li><strong><a href=\"https:\/\/www.youtube.com\/watch?v=SSkPFP5FpKM&amp;t=534s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">a) Demonstra <span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B \\rightarrow P(B\\setminus A) = P(B) - P(A)<\/span><\/span><\/a>,num aequalitas in universum vera est?<\/strong><strong>b) Demonstra <span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B \\rightarrow P(B)\\leq P(A)<\/span><\/strong><span class=\"collapseomatic \" id=\"id6a58b93db04fc\"  tabindex=\"0\" title=\"MOSTRAR SOLUCI\u00d3N a)\"    >MOSTRAR SOLUCI\u00d3N a)<\/span><div id=\"target-id6a58b93db04fc\" class=\"collapseomatic_content \">\n<table>\n<tbody>\n<tr>\n<td>(1)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B<\/span>; Praemissa<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv A\\cap B = A<\/span><\/td>\n<\/tr>\n<tr>\n<td>(2)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">B\\setminus A = B\\cap A^c<\/span>; Definitio Theor. Coniunctorum<\/td>\n<\/tr>\n<tr>\n<td>(3)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">B= (B\\cap A) \\cup (B\\cap A^c)<\/span>; Proprietas coniunctorum<\/td>\n<\/tr>\n<tr>\n<td>(4)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(B\\cap A)\\cap (B\\cap A^c)=\\emptyset<\/span>; Proprietas coniunctorum<\/td>\n<\/tr>\n<tr>\n<td>(5)<\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(B)= P[(B\\cap A) \\cup (B\\cap A^c)]<\/span>; Ex (3)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(B)= P (B\\cap A) + P(B\\cap A^c)<\/span>; Ex (4) + Def. Mensurae Probabilitatis<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(B)= P (B\\cap A) + P(B\\setminus A)<\/span>; Ex (2)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> P(B\\setminus A) =P(B) - P (B\\cap A) <\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">{P(B\\setminus A) =P(B) - P (A) }<\/span>; Ex (1)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\">Itaque <span class=\"katex-eq\" data-katex-display=\"false\">{\\{A\\subseteq B\\}\\vdash P(B\\setminus A) = P(B) - P(A)}.<\/span>\n<p style=\"text-align: justify;\">Animadvertamus hanc aequalitatem non universaliter veram esse, quoniam pendet ex eo quod <span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B<\/span> sit ad obtinendum. Ex his deductionibus videri potest quod, si hoc non impletur, tum habebitur <span class=\"katex-eq\" data-katex-display=\"false\">P(B\\setminus A) = P(B) - P(A\\cap B).<\/span>\n<\/div>\n<span class=\"collapseomatic \" id=\"id6a58b93db0709\"  tabindex=\"0\" title=\"MOSTRAR SOLUCI\u00d3N b)\"    >MOSTRAR SOLUCI\u00d3N b)<\/span><div id=\"target-id6a58b93db0709\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify;\">Ex iis quae in a) rationata sunt habetur:<\/p>\n<table>\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\{A\\subseteq B\\}\\vdash P(B\\setminus A) = P(B) - P(A)<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv \\{A\\subseteq B\\}\\vdash P(A) + P(B\\setminus A) = P(B)<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Denique, cum <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span> sit mensura probabilitatis, habetur <span class=\"katex-eq\" data-katex-display=\"false\">\\forall X (P(X)\\geq 0),<\/span> ita ut, proinde:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">{\\{A\\subseteq B\\}\\vdash P(A) \\leq P(B)}.<\/span>\n<\/div>\n<p>&nbsp;<\/li>\n<li><strong><a href=\"https:\/\/www.youtube.com\/watch?v=SSkPFP5FpKM&amp;t=892s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">a) Demonstra <span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cup B) = P(A) + P(B) - P(A\\cup B).<\/span><\/span><\/a> Comitetur demonstrationem cum diagrammate.<\/strong><strong>b) Utendo priori resultado demonstra <span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cup B) \\leq P(A) + P(B)<\/span><\/strong><span class=\"collapseomatic \" id=\"id6a58b93db0828\"  tabindex=\"0\" title=\"MOSTRAR SOLUCI\u00d3N PARTE a)\"    >MOSTRAR SOLUCI\u00d3N PARTE a)<\/span><div id=\"target-id6a58b93db0828\" class=\"collapseomatic_content \">\n<table>\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">A\\cup B = (A\\triangle B) \\cup (A\\cap B)<\/span>; Proprietas coniunctorum<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">A\\triangle B := (A\\setminus B) \\cup (B\\setminus A)<\/span>; Definitio differentiae symmetricae<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(A\\triangle B)\\cap ( A \\cap B) = \\emptyset<\/span>; proprietas coniunctorum<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(A\\setminus B)\\cap (B\\setminus A) = \\emptyset<\/span>; proprietas coniunctorum<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cup B) = P[(A\\triangle B) \\cup (A\\cap B)]<\/span>; Ex (1)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cup B) = P(A\\triangle B) + P(A\\cap B)]<\/span>; Ex (3)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cup B) = P(A\\setminus B) + P(B\\setminus A) + P(A\\cap B)]<\/span>; Ex (2,4)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(B\\setminus A) = P(B) - P(A\\cap B)<\/span>; Applicando eventus exercitationis 2<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(7)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(A\\setminus B) = P(A) - P(A\\cap B)<\/span>; Idem quod (6)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(8)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cup B) = P(A) - P(A\\cap B) + P(B) - P(A\\cap B) + P(A\\cap B)<\/span>; ex (5,6,7)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">{P(A\\cup B) = P(A) + P(B) - P(A\\cap B)}. <\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<span class=\"collapseomatic \" id=\"id6a58b93db09b3\"  tabindex=\"0\" title=\"MOSTRAR SOLUCI\u00d3N PARTE b)\"    >MOSTRAR SOLUCI\u00d3N PARTE b)<\/span><div id=\"target-id6a58b93db09b3\" class=\"collapseomatic_content \">\n<table>\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash P(A\\cup B) = P(A) + P(B) - P(A\\cap B)<\/span>; eventus partis a)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv\\; \\vdash P(A\\cup B) + P(A\\cap B) = P(A) + P(B) <\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv\\; \\vdash {P(A\\cup B) \\leq P(A) + P(B)} <\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/li>\n<li><strong><a href=\"https:\/\/www.youtube.com\/watch?v=SSkPFP5FpKM&amp;t=1625s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">Si <span class=\"katex-eq\" data-katex-display=\"false\">\\{E_n\\}<\/span> est familia infinita eventuum<\/span> <\/a>talis ut <span class=\"katex-eq\" data-katex-display=\"false\">E_1\\supseteq E_2 \\supseteq E_2 \\supseteq \\cdots \\supseteq E_n \\supseteq E_{n+1}\\supseteq \\cdots.<\/span> Demonstra verum esse<\/strong>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P\\left( \\bigcap_{n=1}^\\infty E_n \\right) = \\lim_{n\\to \\infty}P(E_n)<\/span>\n<span class=\"collapseomatic \" id=\"id6a58b93db0aa6\"  tabindex=\"0\" title=\"MOSTRAR SOLUCI\u00d3N\"    >MOSTRAR SOLUCI\u00d3N<\/span><div id=\"target-id6a58b93db0aa6\" class=\"collapseomatic_content \">\n<table>\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">E_n \\supseteq E_{n+1}<\/span>; Hypothesis<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">E_n^c \\subseteq E_{n+1}^c<\/span>; Per complementum ex (1)<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle(E_n^c \\subseteq E_{n+1}^c) \\rightarrow P\\left( \\bigcup_{n=1}^\\infty E_n^c \\right)= \\lim_{n\\to\\infty}P(E_n^c)<\/span>; Proprietas Continuitatis<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcup_{n=1}^\\infty E_n^c = \\left( \\bigcap_{n=1}^\\infty E_n \\right)^c<\/span>; Leges DeMorgan<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">P\\left(E_n^c\\right) = 1 - P(E_n)<\/span>; Demonstratum in exercitio 1<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">(6)<\/span><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P\\left(\\left[ \\bigcap_{n=1}^\\infty E_n\\right]^c \\right) = \\lim_{n\\to\\infty}[1-P(E_n)] = 1 - \\lim_{n\\to\\infty}P(E_n)<\/span>; ex (2,4,5) applicatum super (3)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 1 - P\\left( \\bigcap_{n=1}^\\infty E_n \\right) = 1 - \\lim_{n\\to\\infty}P(E_n) <\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">{\\displaystyle P\\left( \\bigcap_{n=1}^\\infty E_n \\right) = \\lim_{n\\to\\infty}P(E_n)} <\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\therefore\\{E_n \\supseteq E_{n+1}\\}\\vdash P\\left( \\bigcap_{n=1}^\\infty E_n \\right) = \\lim_{n\\to\\infty}P(E_n).<\/span>\n<p>&nbsp;<\/li>\n<\/ol>\n<\/div>\n<p style=\"text-align: justify;\">His exercitationibus resolutis primum fundamentum perficies quod tibi erit sustentaculum ad prosequendum studium theoriae probabilitatum.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Theoremata Utilia ad Calculum Probabilitatum SummariumIn hac lectione proponuntur exercitationes resolutae quibus demonstrantur nonnulla theoremata utilia ad calculum probabilitatum, cum demonstrationibus et deductionibus. Exercitationes tractant argumenta ut probabilitas complementaria, inclusio insectorum et convergentia eventuum. Has exercitationes perficere tibi praebebit firmam basim ad altius studium theoriae probabilitatum persequendum. OBJECTIVA DISCENDI: His peractis lectionibus discipulus poterit: Demonstrari [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":26394,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":7,"footnotes":""},"categories":[1298,1364],"tags":[],"class_list":["post-34093","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>Theoremata Utilia ad Calculum Probabilitatum - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explora theoremata praecipua probabilitatis per exercitationes resolutas proposita. 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