{"id":26672,"date":"2021-03-29T13:00:31","date_gmt":"2021-03-29T13:00:31","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=26672"},"modified":"2024-05-21T10:14:46","modified_gmt":"2024-05-21T10:14:46","slug":"%d0%bf%d0%be%d0%bb%d0%b5%d0%b7%d0%bd%d1%8b%d0%b5-%d1%82%d0%b5%d0%be%d1%80%d0%b5%d0%bc%d1%8b-%d0%b4%d0%bb%d1%8f-%d0%b2%d1%8b%d1%87%d0%b8%d1%81%d0%bb%d0%b5%d0%bd%d0%b8%d1%8f-%d0%b2%d0%b5%d1%80%d0%be","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/ru\/%d0%bf%d0%be%d0%bb%d0%b5%d0%b7%d0%bd%d1%8b%d0%b5-%d1%82%d0%b5%d0%be%d1%80%d0%b5%d0%bc%d1%8b-%d0%b4%d0%bb%d1%8f-%d0%b2%d1%8b%d1%87%d0%b8%d1%81%d0%bb%d0%b5%d0%bd%d0%b8%d1%8f-%d0%b2%d0%b5%d1%80%d0%be\/","title":{"rendered":"\u041f\u043e\u043b\u0435\u0437\u043d\u044b\u0435 \u0422\u0435\u043e\u0440\u0435\u043c\u044b \u0434\u043b\u044f \u0412\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u0412\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>\u041f\u043e\u043b\u0435\u0437\u043d\u044b\u0435 \u0422\u0435\u043e\u0440\u0435\u043c\u044b \u0434\u043b\u044f \u0412\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u0412\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>\u0420\u0435\u0437\u044e\u043c\u0435<\/strong><br \/><em>\u0412 \u044d\u0442\u043e\u043c \u0437\u0430\u043d\u044f\u0442\u0438\u0438 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u044b \u0440\u0435\u0448\u0435\u043d\u043d\u044b\u0435 \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f, \u0432 \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0434\u043e\u043a\u0430\u0437\u044b\u0432\u0430\u044e\u0442\u0441\u044f \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u043e\u043b\u0435\u0437\u043d\u044b\u0435 \u0442\u0435\u043e\u0440\u0435\u043c\u044b \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439, \u0432\u043a\u043b\u044e\u0447\u0430\u044f \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u0430 \u0438 \u0432\u044b\u0432\u043e\u0434\u044b. \u0423\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f \u043e\u0445\u0432\u0430\u0442\u044b\u0432\u0430\u044e\u0442 \u0442\u0430\u043a\u0438\u0435 \u0442\u0435\u043c\u044b, \u043a\u0430\u043a \u0434\u043e\u043f\u043e\u043b\u043d\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044c, \u0432\u043a\u043b\u044e\u0447\u0435\u043d\u0438\u0435 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432 \u0438 \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u044c \u0441\u043e\u0431\u044b\u0442\u0438\u0439. \u0417\u0430\u0432\u0435\u0440\u0448\u0435\u043d\u0438\u0435 \u044d\u0442\u0438\u0445 \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u0439 \u043f\u0440\u0435\u0434\u043e\u0441\u0442\u0430\u0432\u0438\u0442 \u0432\u0430\u043c \u043f\u0440\u043e\u0447\u043d\u0443\u044e \u043e\u0441\u043d\u043e\u0432\u0443 \u0434\u043b\u044f \u0443\u0433\u043b\u0443\u0431\u043b\u0435\u043d\u043d\u043e\u0433\u043e \u0438\u0437\u0443\u0447\u0435\u043d\u0438\u044f \u0442\u0435\u043e\u0440\u0438\u0438 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>\u0426\u0415\u041b\u0418 \u041e\u0411\u0423\u0427\u0415\u041d\u0418\u042f:<\/strong><br \/>\n\u0417\u0430\u0432\u0435\u0440\u0448\u0438\u0432 \u044d\u0442\u043e \u0437\u0430\u043d\u044f\u0442\u0438\u0435, \u0441\u0442\u0443\u0434\u0435\u043d\u0442 \u0441\u043c\u043e\u0436\u0435\u0442:\n<\/p>\n<ol>\n<li><strong>\u0414\u043e\u043a\u0430\u0437\u0430\u0442\u044c<\/strong> \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0435 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439<\/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; color: #000000;\">\u0427\u0442\u043e \u043c\u044b \u0443\u0432\u0438\u0434\u0438\u043c \u0434\u0430\u043b\u0435\u0435 &#8211; \u044d\u0442\u043e \u0440\u0443\u043a\u043e\u0432\u043e\u0434\u0441\u0442\u0432\u043e \u043f\u043e \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f\u043c (\u0441 \u0440\u0435\u0448\u0435\u043d\u0438\u044f\u043c\u0438), \u0446\u0435\u043b\u044c\u044e \u043a\u043e\u0442\u043e\u0440\u043e\u0433\u043e \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u043f\u043e\u043b\u0435\u0437\u043d\u044b\u0445 \u0442\u0435\u043e\u0440\u0435\u043c \u0434\u043b\u044f \u0442\u0435\u043e\u0440\u0438\u0438 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439. \u041f\u043e\u043f\u0440\u043e\u0431\u0443\u0439\u0442\u0435 \u0440\u0435\u0448\u0438\u0442\u044c \u0438\u0445 \u0441\u0430\u043c\u043e\u0441\u0442\u043e\u044f\u0442\u0435\u043b\u044c\u043d\u043e, \u0430 \u0437\u0430\u0442\u0435\u043c \u0441\u0440\u0430\u0432\u043d\u0438\u0442\u0435 \u0441\u0432\u043e\u0438 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b >:D<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><strong><a href=\"https:\/\/www.youtube.com\/watch?v=SSkPFP5FpKM&amp;t=111s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\">\u0414\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">P(A^c) = 1 -P(A)<\/span> \u0438,<\/span><\/a> \u043d\u0430 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0438 \u044d\u0442\u043e\u0433\u043e \u0441\u0434\u0435\u043b\u0430\u0439\u0442\u0435 \u0432\u044b\u0432\u043e\u0434, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u0435\u0442 \u0430\u0440\u0433\u0443\u043c\u0435\u043d\u0442\u0438\u0440\u043e\u0432\u0430\u0442\u044c, \u0447\u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">P(\\emptyset) = 0<\/span><\/strong><br \/>\n<span class=\"collapseomatic \" id=\"id69d4dbbe94ed8\"  tabindex=\"0\" title=\"\u041f\u041e\u041a\u0410\u0417\u0410\u0422\u042c \u0420\u0415\u0428\u0415\u041d\u0418\u0415\"    >\u041f\u041e\u041a\u0410\u0417\u0410\u0422\u042c \u0420\u0415\u0428\u0415\u041d\u0418\u0415<\/span><div id=\"target-id69d4dbbe94ed8\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify; color: #000000;\">\u0418\u0437 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u044f <a href=\"https:\/\/toposuranos.com\/el-espacio-de-probabilidades-medida-de-probabilidad\/\" target=\"_blank\" rel=\"noopener\">\u043c\u0435\u0440\u044b \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438<\/a> \u0441\u043b\u0435\u0434\u0443\u0435\u0442, \u0447\u0442\u043e, \u0435\u0441\u043b\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> \u044f\u0432\u043b\u044f\u044e\u0442\u0441\u044f \u043b\u044e\u0431\u044b\u043c\u0438 \u0438\u0437\u043c\u0435\u0440\u0438\u043c\u044b\u043c\u0438 \u0441\u043e\u0431\u044b\u0442\u0438\u044f\u043c\u0438, \u0442\u043e \u0431\u0443\u0434\u0435\u0442 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0442\u044c\u0441\u044f<\/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; color: #000000;\">\u0422\u0435\u043f\u0435\u0440\u044c, \u0442\u0430\u043a \u043a\u0430\u043a <span class=\"katex-eq\" data-katex-display=\"false\">A\\cap A^c = \\emptyset,<\/span> \u0438\u0437 \u0447\u0430\u0441\u0442\u0438 [b], \u043f\u043e\u043b\u0443\u0447\u0438\u043c:<\/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; color: #000000;\">\u041f\u043e\u0441\u043a\u043e\u043b\u044c\u043a\u0443 <span class=\"katex-eq\" data-katex-display=\"false\">A\\cap A^c = \\emptyset<\/span> \u0432\u0441\u0435\u0433\u0434\u0430 \u0438\u0441\u0442\u0438\u043d\u043d\u043e \u0438 <span class=\"katex-eq\" data-katex-display=\"false\">A\\cup A^c = \\Omega<\/span>, \u0442\u043e \u0438\u043c\u0435\u0435\u043c:<\/p>\n<p style=\"text-align: center; color: #000000;\"><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; color: #000000;\">\u0418, \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(A^c) = 1-P(A)<\/span>\n<p style=\"text-align: justify; color: #000000;\">\u0427\u0442\u043e \u0438 \u0442\u0440\u0435\u0431\u043e\u0432\u0430\u043b\u043e\u0441\u044c \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c.<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u0427\u0442\u043e\u0431\u044b \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c, \u0447\u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">P(\\emptyset)=0<\/span>, \u0434\u043e\u0441\u0442\u0430\u0442\u043e\u0447\u043d\u043e \u0432\u0437\u044f\u0442\u044c <span class=\"katex-eq\" data-katex-display=\"false\">A=\\Omega<\/span> \u0432 \u0442\u043e\u043b\u044c\u043a\u043e \u0447\u0442\u043e \u0434\u043e\u043a\u0430\u0437\u0430\u043d\u043d\u043e\u043c \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0438 \u0438 \u043f\u043e\u043b\u0443\u0447\u0438\u043c:<\/p>\n<p style=\"text-align: center; color: #000000;\"><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) \u0414\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B \\rightarrow P(B\\setminus A) = P(B) - P(A)<\/span><\/span><\/a>, \u0432\u0435\u0440\u043d\u043e \u043b\u0438 \u044d\u0442\u043e \u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e \u0432 \u043e\u0431\u0449\u0435\u043c \u0441\u043b\u0443\u0447\u0430\u0435?<\/strong><strong>b) \u0414\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B \\rightarrow P(B)\\leq P(A)<\/span><\/strong><span class=\"collapseomatic \" id=\"id69d4dbbe951b3\"  tabindex=\"0\" title=\"\u041f\u041e\u041a\u0410\u0417\u0410\u0422\u042c \u0420\u0415\u0428\u0415\u041d\u0418\u0415 a)\"    >\u041f\u041e\u041a\u0410\u0417\u0410\u0422\u042c \u0420\u0415\u0428\u0415\u041d\u0418\u0415 a)<\/span><div id=\"target-id69d4dbbe951b3\" 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>; \u041f\u0440\u0435\u043c\u0438\u0441\u0441\u0430<\/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>; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0442\u0435\u043e\u0440\u0438\u0438 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432<\/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>; \u0421\u0432\u043e\u0439\u0441\u0442\u0432\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432<\/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>; \u0421\u0432\u043e\u0439\u0441\u0442\u0432\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432<\/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>; \u0418\u0437 (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>; \u0418\u0437 (4) + \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u043c\u0435\u0440\u044b \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438<\/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>; \u0418\u0437 (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>; \u0418\u0437 (1)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center; color: #000000;\">\u0422\u0430\u043a\u0438\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u043c, <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; color: #000000;\">\u0417\u0430\u043c\u0435\u0442\u0438\u043c, \u0447\u0442\u043e \u044d\u0442\u043e \u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e \u043d\u0435 \u0432\u0435\u0440\u043d\u043e \u0432 \u043e\u0431\u0449\u0435\u043c \u0441\u043b\u0443\u0447\u0430\u0435, \u0442\u0430\u043a \u043a\u0430\u043a \u043e\u043d\u043e \u0437\u0430\u0432\u0438\u0441\u0438\u0442 \u043e\u0442 \u0432\u044b\u043f\u043e\u043b\u043d\u0435\u043d\u0438\u044f \u0443\u0441\u043b\u043e\u0432\u0438\u044f <span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B<\/span>. \u0418\u0437 \u044d\u0442\u0438\u0445 \u0440\u0430\u0441\u0441\u0443\u0436\u0434\u0435\u043d\u0438\u0439 \u0432\u0438\u0434\u043d\u043e, \u0447\u0442\u043e \u0435\u0441\u043b\u0438 \u044d\u0442\u043e \u0443\u0441\u043b\u043e\u0432\u0438\u0435 \u043d\u0435 \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f, \u0442\u043e \u0431\u0443\u0434\u0435\u0442 <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=\"id69d4dbbe953c7\"  tabindex=\"0\" title=\"\u041f\u041e\u041a\u0410\u0417\u0410\u0422\u042c \u0420\u0415\u0428\u0415\u041d\u0418\u0415 b)\"    >\u041f\u041e\u041a\u0410\u0417\u0410\u0422\u042c \u0420\u0415\u0428\u0415\u041d\u0418\u0415 b)<\/span><div id=\"target-id69d4dbbe953c7\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify; color: #000000;\">\u041d\u0430 \u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0438\u0438 \u0440\u0430\u0441\u0441\u0443\u0436\u0434\u0435\u043d\u0438\u0439 \u0432 \u0447\u0430\u0441\u0442\u0438 a) \u0438\u043c\u0435\u0435\u043c:<\/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; color: #000000;\">\u041d\u0430\u043a\u043e\u043d\u0435\u0446, \u0442\u0430\u043a \u043a\u0430\u043a <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span> \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043c\u0435\u0440\u043e\u0439 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438, \u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">\\forall X (P(X)\\geq 0),<\/span> \u0438, \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e:<\/p>\n<p style=\"text-align: center; color: #000000;\"><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) \u0414\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cup B) = P(A) + P(B) - P(A\\cup B).<\/span><\/span><\/a> \u0421\u043e\u043f\u0440\u043e\u0432\u043e\u0434\u0438\u0442\u0435 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u043e \u0434\u0438\u0430\u0433\u0440\u0430\u043c\u043c\u043e\u0439.<\/strong><strong>b) \u0418\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u044f \u043f\u0440\u0435\u0434\u044b\u0434\u0443\u0449\u0438\u0439 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442, \u0434\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e <span class=\"katex-eq\" data-katex-display=\"false\">P(A\\cup B) \\leq P(A) + P(B)<\/span><\/strong><span class=\"collapseomatic \" id=\"id69d4dbbe954db\"  tabindex=\"0\" title=\"\u041f\u041e\u041a\u0410\u0417\u0410\u0422\u042c \u0420\u0415\u0428\u0415\u041d\u0418\u0415 \u0427\u0410\u0421\u0422\u042c a)\"    >\u041f\u041e\u041a\u0410\u0417\u0410\u0422\u042c \u0420\u0415\u0428\u0415\u041d\u0418\u0415 \u0427\u0410\u0421\u0422\u042c a)<\/span><div id=\"target-id69d4dbbe954db\" 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>; \u0421\u0432\u043e\u0439\u0441\u0442\u0432\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432<\/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>; \u041e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438\u0435 \u0441\u0438\u043c\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u043e\u0439 \u0440\u0430\u0437\u043d\u043e\u0441\u0442\u0438<\/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>; \u0421\u0432\u043e\u0439\u0441\u0442\u0432\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432<\/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>; \u0421\u0432\u043e\u0439\u0441\u0442\u0432\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432<\/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>; \u0418\u0437 (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>; \u0418\u0437 (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>; \u0418\u0437 (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>; \u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u0430 \u0438\u0437 \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f 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>; \u0422\u043e \u0436\u0435 \u0441\u0430\u043c\u043e\u0435, \u0447\u0442\u043e (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>; \u0418\u0437 (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=\"id69d4dbbe9566c\"  tabindex=\"0\" title=\"\u041f\u041e\u041a\u0410\u0417\u0410\u0422\u042c \u0420\u0415\u0428\u0415\u041d\u0418\u0415 \u0427\u0410\u0421\u0422\u042c b)\"    >\u041f\u041e\u041a\u0410\u0417\u0410\u0422\u042c \u0420\u0415\u0428\u0415\u041d\u0418\u0415 \u0427\u0410\u0421\u0422\u042c b)<\/span><div id=\"target-id69d4dbbe9566c\" 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>; \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442 \u0447\u0430\u0441\u0442\u0438 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;\">\u0415\u0441\u043b\u0438 <span class=\"katex-eq\" data-katex-display=\"false\">\\{E_n\\}<\/span> &#8211; \u044d\u0442\u043e \u0431\u0435\u0441\u043a\u043e\u043d\u0435\u0447\u043d\u043e\u0435 \u0441\u0435\u043c\u0435\u0439\u0441\u0442\u0432\u043e \u0441\u043e\u0431\u044b\u0442\u0438\u0439<\/span> <\/a>\u0442\u0430\u043a\u0438\u0445, \u0447\u0442\u043e <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> \u0414\u043e\u043a\u0430\u0436\u0438\u0442\u0435, \u0447\u0442\u043e \u0432\u044b\u043f\u043e\u043b\u043d\u044f\u0435\u0442\u0441\u044f:<\/strong>\n<p style=\"text-align: center; color: #000000;\"><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=\"id69d4dbbe95764\"  tabindex=\"0\" title=\"\u041f\u041e\u041a\u0410\u0417\u0410\u0422\u042c \u0420\u0415\u0428\u0415\u041d\u0418\u0415\"    >\u041f\u041e\u041a\u0410\u0417\u0410\u0422\u042c \u0420\u0415\u0428\u0415\u041d\u0418\u0415<\/span><div id=\"target-id69d4dbbe95764\" 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>; \u0413\u0438\u043f\u043e\u0442\u0435\u0437\u0430<\/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>; \u041f\u043e \u0434\u043e\u043f\u043e\u043b\u043d\u0435\u043d\u0438\u044e \u0438\u0437 (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>; \u0421\u0432\u043e\u0439\u0441\u0442\u0432\u043e \u043d\u0435\u043f\u0440\u0435\u0440\u044b\u0432\u043d\u043e\u0441\u0442\u0438<\/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>; \u0417\u0430\u043a\u043e\u043d\u044b \u0414\u0435\u041c\u043e\u0440\u0433\u0430\u043d\u0430<\/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>; \u0414\u043e\u043a\u0430\u0437\u0430\u043d\u043e \u0432 \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u0438 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>; \u0438\u0437 (2,4,5) \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u043e \u043a (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; color: #000000;\"><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; color: #000000;\">\u0420\u0435\u0448\u0430\u044f \u044d\u0442\u0438 \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f, \u0432\u044b \u0437\u0430\u043b\u043e\u0436\u0438\u0442\u0435 \u043f\u0435\u0440\u0432\u044b\u0439 \u0441\u0442\u043e\u043b\u043f, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u043f\u043e\u0441\u043b\u0443\u0436\u0438\u0442 \u043e\u043f\u043e\u0440\u043e\u0439 \u0434\u043b\u044f \u0434\u0430\u043b\u044c\u043d\u0435\u0439\u0448\u0435\u0433\u043e \u0438\u0437\u0443\u0447\u0435\u043d\u0438\u044f \u0442\u0435\u043e\u0440\u0438\u0438 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u041f\u043e\u043b\u0435\u0437\u043d\u044b\u0435 \u0422\u0435\u043e\u0440\u0435\u043c\u044b \u0434\u043b\u044f \u0412\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u0412\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439 \u0420\u0435\u0437\u044e\u043c\u0435\u0412 \u044d\u0442\u043e\u043c \u0437\u0430\u043d\u044f\u0442\u0438\u0438 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u044b \u0440\u0435\u0448\u0435\u043d\u043d\u044b\u0435 \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f, \u0432 \u043a\u043e\u0442\u043e\u0440\u044b\u0445 \u0434\u043e\u043a\u0430\u0437\u044b\u0432\u0430\u044e\u0442\u0441\u044f \u043d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u043e\u043b\u0435\u0437\u043d\u044b\u0435 \u0442\u0435\u043e\u0440\u0435\u043c\u044b \u0434\u043b\u044f \u0432\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439, \u0432\u043a\u043b\u044e\u0447\u0430\u044f \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u0430 \u0438 \u0432\u044b\u0432\u043e\u0434\u044b. \u0423\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u044f \u043e\u0445\u0432\u0430\u0442\u044b\u0432\u0430\u044e\u0442 \u0442\u0430\u043a\u0438\u0435 \u0442\u0435\u043c\u044b, \u043a\u0430\u043a \u0434\u043e\u043f\u043e\u043b\u043d\u0438\u0442\u0435\u043b\u044c\u043d\u0430\u044f \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u044c, \u0432\u043a\u043b\u044e\u0447\u0435\u043d\u0438\u0435 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432 \u0438 \u0441\u0445\u043e\u0434\u0438\u043c\u043e\u0441\u0442\u044c \u0441\u043e\u0431\u044b\u0442\u0438\u0439. \u0417\u0430\u0432\u0435\u0440\u0448\u0435\u043d\u0438\u0435 \u044d\u0442\u0438\u0445 \u0443\u043f\u0440\u0430\u0436\u043d\u0435\u043d\u0438\u0439 \u043f\u0440\u0435\u0434\u043e\u0441\u0442\u0430\u0432\u0438\u0442 \u0432\u0430\u043c \u043f\u0440\u043e\u0447\u043d\u0443\u044e \u043e\u0441\u043d\u043e\u0432\u0443 \u0434\u043b\u044f \u0443\u0433\u043b\u0443\u0431\u043b\u0435\u043d\u043d\u043e\u0433\u043e \u0438\u0437\u0443\u0447\u0435\u043d\u0438\u044f \u0442\u0435\u043e\u0440\u0438\u0438 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439. \u0426\u0415\u041b\u0418 \u041e\u0411\u0423\u0427\u0415\u041d\u0418\u042f: \u0417\u0430\u0432\u0435\u0440\u0448\u0438\u0432 \u044d\u0442\u043e \u0437\u0430\u043d\u044f\u0442\u0438\u0435, \u0441\u0442\u0443\u0434\u0435\u043d\u0442 \u0441\u043c\u043e\u0436\u0435\u0442: [&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":0,"footnotes":""},"categories":[680,573],"tags":[],"class_list":["post-26672","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-680","category-573"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u041f\u043e\u043b\u0435\u0437\u043d\u044b\u0435 \u0422\u0435\u043e\u0440\u0435\u043c\u044b \u0434\u043b\u044f \u0412\u044b\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f \u0412\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0435\u0439 - toposuranos.com\/material<\/title>\n<meta name=\"description\" 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