{"id":25483,"date":"2021-03-24T00:00:04","date_gmt":"2021-03-24T00:00:04","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25483"},"modified":"2024-06-05T04:22:43","modified_gmt":"2024-06-05T04:22:43","slug":"%e6%a6%82%e7%8e%87%e6%b5%8b%e5%ba%a6%e5%8f%8a%e5%85%b6%e5%85%b3%e9%94%ae%e5%b1%9e%e6%80%a7","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/zh\/%e6%a6%82%e7%8e%87%e6%b5%8b%e5%ba%a6%e5%8f%8a%e5%85%b6%e5%85%b3%e9%94%ae%e5%b1%9e%e6%80%a7\/","title":{"rendered":"\u6982\u7387\u6d4b\u5ea6\u53ca\u5176\u5173\u952e\u5c5e\u6027"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>\u6982\u7387\u6d4b\u5ea6\u7684\u5173\u952e\u5c5e\u6027<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>\u6458\u8981<\/strong><br \/><em>\u5728\u8fd9\u5802\u8bfe\u4e2d\uff0c\u6211\u4eec\u4ecb\u7ecd\u4e86\u6982\u7387\u7406\u8bba\u7684\u57fa\u672c\u5b9a\u4e49\uff0c\u5305\u62ec\u6982\u7387\u6d4b\u5ea6\u548c\u03c3-\u4ee3\u6570\u3002\u6b64\u5916\uff0c\u8fd8\u63a2\u8ba8\u4e86\u4f5c\u4e3a\u76f8\u5bf9\u9891\u7387\u6781\u9650\u7684\u6982\u7387\u6982\u5ff5\uff0c\u4f7f\u7528\u63b7\u516d\u9762\u9ab0\u5b50\u7684\u4f8b\u5b50\u3002\u901a\u8fc7\u76f8\u5bf9\u9891\u7387\u51fd\u6570\uff0c\u6f14\u793a\u4e86\u5982\u4f55\u901a\u8fc7\u6781\u9650\u5f97\u5230\u6982\u7387\u6d4b\u5ea6\uff0c\u8fd9\u5c31\u662f\u6240\u8c13\u7684\u201c\u6982\u7387\u4f5c\u4e3a\u76f8\u5bf9\u9891\u7387\u7684\u6781\u9650\u201d\u3002\u4ecb\u7ecd\u4e86\u6982\u7387\u6d4b\u5ea6\u7684\u5173\u952e\u5c5e\u6027\uff0c\u5e76\u5c55\u793a\u4e86\u5982\u4f55\u901a\u8fc7\u6709\u5229\u6848\u4f8b\u4e0e\u603b\u6848\u4f8b\u6765\u8ba1\u7b97\u4e8b\u4ef6\u7684\u6982\u7387\u3002\u8fd9\u4e9b\u77e5\u8bc6\u5bf9\u4e8e\u7406\u89e3\u6982\u7387\u7406\u8bba\u5e76\u5728\u4e0d\u540c\u60c5\u5883\u4e2d\u8fdb\u884c\u51b3\u7b56\u81f3\u5173\u91cd\u8981\u3002<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>\u5b66\u4e60\u76ee\u6807\uff1a<\/strong><br \/>\n\u5b8c\u6210\u8fd9\u5802\u8bfe\u540e\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a\n<\/p>\n<ol>\n<li><strong>\u4e86\u89e3<\/strong>\u6982\u7387\u6d4b\u5ea6\u7684\u5b9a\u4e49\u53ca\u5176\u57fa\u672c\u5c5e\u6027\u3002<\/li>\n<li><strong>\u7406\u89e3<\/strong>\u4f5c\u4e3a\u76f8\u5bf9\u9891\u7387\u6781\u9650\u7684\u6982\u7387\u6982\u5ff5\u3002<\/li>\n<li><strong>\u8ba1\u7b97<\/strong>\u901a\u8fc7\u201c\u6709\u5229\u6848\u4f8b\u4e0e\u603b\u6848\u4f8b\u7684\u6982\u7387\u201d\u6765\u8ba1\u7b97\u4e8b\u4ef6\u7684\u6982\u7387\u3002<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>\u5185\u5bb9\u7d22\u5f15<\/strong><br \/>\n<a href=\"#1\">\u57fa\u672c\u5b9a\u4e49<\/a><br \/>\n<a href=\"#2\">\u6982\u7387\u4f5c\u4e3a\u76f8\u5bf9\u9891\u7387\u7684\u6781\u9650<\/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>\u57fa\u672c\u5b9a\u4e49<\/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;\">\u6d53\u7f29\u6982\u7387\u7a7a\u95f4\u672c\u8d28\u7684\u90e8\u5206<\/span><\/strong><\/a>\u6b63\u662f\u6982\u7387\u6d4b\u5ea6\u7684\u6982\u5ff5\u3002\u8fd9\u662f\u6700\u7ec8\u91cf\u5316\u67d0\u4e00\u4e8b\u4ef6\u53d1\u751f\u53ef\u80fd\u6027\u7684\u51fd\u6570\u3002<\/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;\">\u4e00\u4e2a\u6982\u7387\u6d4b\u5ea6<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span> \u662f\u8fd9\u6837\u4e00\u4e2a\u51fd\u6570\uff0c\u7ed9\u5b9a\u4e00\u4e2a <strong><a href=\"http:\/\/toposuranos.com\/material\/es\/que-es-una-sigma-algebra-definicion-y-ejemplos\/\" rel=\"noopener\" target=\"_blank\">\u03c3-\u4ee3\u6570<\/a><\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma=(\\Omega,\\mathcal{A})<\/span>\uff0c\u5b83\u4e3a\u6bcf\u4e2a\u4e8b\u4ef6 <span class=\"katex-eq\" data-katex-display=\"false\">E\\in\\mathcal{A}<\/span> \u5206\u914d\u4e00\u4e2a\u6982\u7387\u3002\u8fd9\u4e2a\u6982\u7387\u6d4b\u5ea6\u6ee1\u8db3\u4ee5\u4e0b\u5c5e\u6027\uff1a<\/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;\">(\u8fde\u7eed\u6027)<\/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=\"\u6982\u7387\u6d4b\u5ea6\uff0c\u76f8\u5bf9\u9891\u7387\u7684\u6781\u9650\" 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=\"\u6982\u7387\u6d4b\u5ea6\uff0c\u76f8\u5bf9\u9891\u7387\u7684\u6781\u9650\" 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>\u6982\u7387\u4f5c\u4e3a\u76f8\u5bf9\u9891\u7387\u6781\u9650<\/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;\">\u8fd9\u4e2a\u6982\u7387\u6982\u5ff5\u53ef\u4ee5\u901a\u8fc7<\/span><\/strong><\/a>\u201c\u76f8\u5bf9\u9891\u7387\u6781\u9650\u201d\u7684\u76f4\u89c2\u60f3\u6cd5\u6765\u7406\u89e3\u3002\u4f8b\u5982\uff0c\u8003\u8651\u4e00\u4e2a\u5b9e\u9a8c\uff0c\u628a\u4e00\u4e2a6\u9762\u7684\u9ab0\u5b50\u6295\u63b7<span class=\"katex-eq\" data-katex-display=\"false\">N<\/span>\u6b21\u3002\u6bcf\u6b21\u6295\u63b7\u7684<strong><a href=\"http:\/\/toposuranos.com\/material\/es\/conoce-el-espacio-muestral-de-la-teoria-de-las-probabilidades\/\" rel=\"noopener\" target=\"_blank\">\u6837\u672c\u7a7a\u95f4<\/a><\/strong>\u662f<span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1d6} = \\{1,2,3,4,5,6\\}<\/span>\u3002\u5982\u679c<span class=\"katex-eq\" data-katex-display=\"false\">A<\/span>\u662f<span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1d6}<\/span>\u4e2d\u7684\u4efb\u610f\u4e8b\u4ef6\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u6d4b\u91cf\u4e8b\u4ef6<span class=\"katex-eq\" data-katex-display=\"false\">A<\/span>\u5728<span class=\"katex-eq\" data-katex-display=\"false\">N<\/span>\u6b21\u6295\u63b7\u4e2d\u53d1\u751f\u7684\u9891\u7387<span class=\"katex-eq\" data-katex-display=\"false\">f_N(A)<\/span>\u3002\u8fd9\u6837\u505a\u6211\u4eec\u5c06\u9a8c\u8bc1\u5230\uff1a<\/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;\">\u5982\u679c\u6211\u4eec\u8003\u8651<span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1d6}<\/span>\u4e2d\u7684\u53e6\u4e00\u4e2a\u4efb\u610f\u4e8b\u4ef6<span class=\"katex-eq\" data-katex-display=\"false\">B<\/span>\uff0c\u90a3\u4e48\u5c06\u53d1\u751f\uff1a<\/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;\">\u57fa\u4e8e\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u4e00\u4e2a\u65b0\u7684\u201c\u51fd\u6570\u201d<span class=\"katex-eq\" data-katex-display=\"false\">g_N<\/span>\uff0c\u6211\u4eec<strong>\u79f0\u4e4b\u4e3a\u76f8\u5bf9\u9891\u7387\uff0c<\/strong>\u901a\u8fc7\u4ee5\u4e0b\u8868\u8fbe\u5f0f<\/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;\">\u6211\u4eec\u6ce8\u610f\u5230<span class=\"katex-eq\" data-katex-display=\"false\">g_N<\/span>\u51fd\u6570\u7b26\u5408\u524d\u4e09\u4e2a\u5c5e\u6027\uff0c\u53ea\u6709\u7b2c\u56db\u4e2a\uff08\u8fde\u7eed\u6027\uff09\u5c5e\u6027\u4e0d\u80fd\u4ee5\u8fd9\u79cd\u65b9\u5f0f\u83b7\u5f97\uff0c\u4f46\u7531\u4e8e\u67d0\u4e9b\u7ed3\u679c\u4e0d\u80fd\u5728\u6ca1\u6709\u8fd9\u4e2a\u5c5e\u6027\u7684\u60c5\u51b5\u4e0b\u83b7\u5f97\uff0c\u6240\u4ee5\u51fa\u4e8e\u201c\u6280\u672f\u201d\u539f\u56e0\u5f15\u5165\u4e86\u8fd9\u4e2a\u5c5e\u6027\u3002<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u5373\u4fbf\u5982\u6b64\uff0c<span class=\"katex-eq\" data-katex-display=\"false\">g_N<\/span>\u8fd8\u4e0d\u662f\u4e00\u4e2a\u6982\u7387\u6d4b\u5ea6\uff0c\u56e0\u4e3a\u5b83\u8fd8\u4e0d\u662f\u4e00\u4e2a\u51fd\u6570\u3002\u8fd9\u662f\u56e0\u4e3a\u65e0\u8bba<span class=\"katex-eq\" data-katex-display=\"false\">N<\/span>\u7684\u503c\u5982\u4f55\uff0c<span class=\"katex-eq\" data-katex-display=\"false\">g_N<\/span>\u5e76\u4e0d\u603b\u662f\u5fc5\u7136\u7ed9\u51fa\u4e00\u4e2a\u552f\u4e00\u7684\u503c\uff0c\u5c31\u50cf\u4e00\u4e2a\u51fd\u6570\u6240\u505a\u7684\u90a3\u6837\u3002\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u6211\u4eec\u6dfb\u52a0\u4e86\u4e00\u4e2a\u6781\u9650\u6b65\u9aa4\uff0c\u6700\u7ec8\u5f97\u5230\u4e00\u4e2a\u6982\u7387\u6d4b\u5ea6\uff0c\u5f97\u5230\uff1a<\/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;\">\u8fd9\u5c31\u662f\u6240\u8c13\u7684\u6982\u7387\u4f5c\u4e3a\u76f8\u5bf9\u9891\u7387\u7684\u6781\u9650\u3002<\/p>\n<h3>\u4f8b\u5b50\uff1a\u63b7\u9ab0\u5b50<\/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;\">\u7ee7\u7eed\u5206\u6790\u63b7\u4e00\u4e2a6\u9762\u9ab0\u5b50\u7684\u60c5\u51b5\uff0c<\/span> <\/strong><\/a>\u6211\u4eec\u4f1a\u53d1\u73b0\uff1a<\/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;\">\u5982\u679c\u5b9e\u9a8c\u91cd\u590d\u591a\u6b21\uff0c\u6211\u4eec\u53ef\u4ee5\u9a8c\u8bc1<\/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;\">\u56e0\u6b64\uff0c\u5982\u679c\u6211\u4eec\u5b9a\u4e49<span class=\"katex-eq\" data-katex-display=\"false\">P(\\{x\\}) = \\lim_{N\\to\\infty} g_N(\\{x\\})<\/span>\uff0c\u90a3\u4e48\u5c06\u53d1\u751f\uff1a<\/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;\">\u4ece\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\u6211\u4eec\u4e5f\u53ef\u4ee5\u7b80\u5355\u5730\u63a8\u65ad\u51fa\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u6837\u672c\u7a7a\u95f4<span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span>\u4e2d\u7684\u4e8b\u4ef6<span class=\"katex-eq\" data-katex-display=\"false\">E<\/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;\">\u8fd9\u5c31\u662f\u6240\u8c13\u7684\u201c\u6982\u7387\u4f5c\u4e3a\u6709\u5229\u60c5\u51b5\u4e0e\u603b\u60c5\u51b5\u4e4b\u6bd4\u201d\uff0c\u8fd9\u5bf9\u4e8e\u7406\u89e3\u6982\u7387\u7a7a\u95f4\u548c\u6982\u7387\u6d4b\u5ea6\u7684\u5927\u91cf\u63a8\u7406\u81f3\u5173\u91cd\u8981\u3002<\/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>\u6982\u7387\u6d4b\u5ea6\u7684\u5173\u952e\u5c5e\u6027 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