{"id":32916,"date":"2011-08-11T12:28:42","date_gmt":"2011-08-11T04:28:42","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=32916"},"modified":"2021-10-06T16:24:54","modified_gmt":"2021-10-06T08:24:54","slug":"%e5%88%9d%e7%ad%89%e7%9a%84%e6%a9%9f%e7%8e%87%e8%ab%96%ef%bc%886%ef%bc%89%e6%a2%9d%e4%bb%b6%e6%a9%9f%e7%8e%87%e8%88%87bayes%e5%85%ac%e5%bc%8f","status":"publish","type":"post","link":"http:\/\/localhost\/%e5%88%9d%e7%ad%89%e7%9a%84%e6%a9%9f%e7%8e%87%e8%ab%96%ef%bc%886%ef%bc%89%e6%a2%9d%e4%bb%b6%e6%a9%9f%e7%8e%87%e8%88%87bayes%e5%85%ac%e5%bc%8f\/","title":{"rendered":"\u521d\u7b49\u7684\u6a5f\u7387\u8ad6\uff086\uff09\u689d\u4ef6\u6a5f\u7387\u8207Bayes\u516c\u5f0f"},"content":{"rendered":"

\u521d\u7b49\u7684\u6a5f\u7387\u8ad6\uff086\uff09\u689d\u4ef6\u6a5f\u7387\u8207Bayes\u516c\u5f0f
\n\uff08Elementary Probability\u00a0Theory-6. Conditional Probability and Bayes Formula\uff09
\n\u570b\u7acb\u81fa\u7063\u5927\u5b78\u6578\u5b78\u7cfb\u8521\u8070\u660e\u526f\u6559\u6388\/\u570b\u7acb\u81fa\u7063\u5927\u5b78\u6578\u5b78\u7cfb\u8521\u8070\u660e\u526f\u6559\u6388\u8cac\u4efb\u7de8\u8f2f<\/strong><\/span><\/span><\/span><\/strong><\/p>\n

\u9023\u7d50\uff1a\u521d\u7b49\u7684\u6a5f\u7387\u8ad6\uff085\uff09\u6709\u9650\u6a5f\u7387\u7a7a\u9593<\/a><\/p>\n

\u6458\u8981\uff1a<\/strong>\u672c\u7bc7\u4ecb\u7d39\u4f55\u8b02\u300c\u689d\u4ef6\u6a5f\u7387(conditional probability)\u300d\uff0c\u5f9e\u800c\u5c0e\u51fa\u300c\u6a5f\u7387\u7684\u4e58\u6cd5\u516c\u5f0f\u300d\u3001\u300c\u5168\u6a5f\u7387\u516c\u5f0f(the total probability formula)\u300d\u3001\u4ee5\u53ca\u300cBayes\u516c\u5f0f\u300d\uff0c\u4e26\u5206\u5225\u8209\u4f8b\u95e1\u8ff0\u5176\u5167\u6db5\u3002<\/strong><\/p>\n

\u5728\u6a5f\u7387\u8ad6\u4e2d\uff0c\u689d\u4ef6\u5f0f\u7684\u601d\u8003<\/strong>\u662f\u975e\u5e38\u91cd\u8981\u7684\u4e00\u7a2e\u601d\u8003\u65b9\u6cd5\u3002\u672c\u7bc0\u6211\u5011\u53ea\u4ecb\u7d39\u8f03\u7c21\u55ae\u7684\u689d\u4ef6\u6a5f\u7387\uff08conditional probability\uff09\u4e4b\u6982\u5ff5\u3002\u4e8b\u4ef6 $$A$$ \u7684\u6a5f\u7387 $$P(A)$$ \u662f\u5728 $$\\Omega$$ \u9435\u5b9a\u767c\u751f\u7684\u689d\u4ef6\u4e0b\uff0c\u63cf\u8ff0 $$A$$ \u767c\u751f\u7684\u6a5f\u7387\u3002\u73fe\u5728\u4f5c\u63a8\u5ee3\uff0c\u5047\u8a2d\u5df2\u7d93\u77e5\u9053\u4e8b\u4ef6 $$B$$ \u767c\u751f\u4e86\uff0c\u8981\u554f\u4e8b\u4ef6 $$A$$ \u767c\u751f\u7684\u6a5f\u7387\u3002\u7167\u7406\u8aaa\u6b64\u6642\u53ef\u80fd\u6703\u8ddf $$P(A)$$ \u4e0d\u4e00\u6a23\u3002\u6211\u5011\u5148\u89c0\u5bdf\u4e00\u500b\u4f8b\u5b50\u3002<\/p>\n

<\/p>\n

\u3010\u4f8b7\u3011<\/strong>\u8003\u616e\u67d0\u500b\u5bb6\u5ead\u6709 $$5$$ \u7537 $$7$$ \u5973\uff0c\u5176\u4e2d\u7537\u751f\u6709$$3$$\u4eba\u5c31\u696d\uff0c\u5973\u751f\u6709$$5$$\u4eba\u5c31\u696d\u3002\u4eca\u5f9e\u9019\u500b\u5bb6\u5ead\u4efb\u53d6\u4e00\u500b\u4eba\uff0c\u90a3\u9ebc\u6a23\u672c\u7a7a\u9593 $$\\Omega$$ \u70ba\u7531\u9019 $$12$$ \u4eba\u7d44\u6210\uff0c\u5047\u8a2d\u53d6\u5230\u6bcf\u500b\u4eba\u7684\u6a5f\u6703\u5747\u7b49\u3002\u4ee4 $$W$$ \u8868\u793a\u5973\u751f\u7684\u96c6\u5408\uff0c $$E$$ \u8868\u793a\u5c31\u696d\u8005\u7684\u96c6\u5408\u3002\u4efb\u53d6\u4e00\u500b\u4eba\u70ba\u5973\u751f\u7684\u6a5f\u7387\u662f $$P(W)=7\/12$$\u3002<\/p>\n

\u5047\u8a2d\u6211\u5011\u77e5\u9053\u6b64\u4eba\u662f\u5c31\u696d\u7684\uff0c\u90a3\u9ebc\u6a23\u672c\u7a7a\u9593\u5df2\u6539\u8b8a\u6210\u70ba $$E$$\uff0c\u6b64\u4eba\u70ba\u5973\u751f\u7684\u6a5f\u7387\u662f $$5\/8$$\u3002\u9019\u5c31\u662f\u689d\u4ef6\u6a5f\u7387<\/strong>\uff08conditional probability\uff09\u7684\u6982\u5ff5\u3002\u6211\u5011\u6ce8\u610f\u5230\uff0c\u82e5\u9084\u539f\u5230\u539f\u4f86\u7684\u6a23\u672c\u7a7a\u9593\uff0c\u8a08\u7b97\u65b9\u6cd5\u5c31\u662f<\/p>\n

$$\\displaystyle\\frac{P(W\\cap E)}{P(E)}=\\frac{5\/12}{8\/12}=\\frac{5}{8}$$<\/p>\n

\u9019\u5c31\u662f\u689d\u4ef6\u6a5f\u7387\uff0c\u662f\u5f88\u81ea\u7136\u800c\u91cd\u8981\u7684\u6a5f\u7387\u601d\u8003\u65b9\u6cd5\uff0c\u503c\u5f97\u6211\u5011\u53e6\u5275\u4e00\u500b\u65b0\u8a18\u865f $$P(W|E)$$ \u4f86\u8868\u9054\u5b83\uff0c\u4ea6\u5373\uff1a<\/p>\n

$$\\displaystyle P(W|E)=\\frac{P(W\\cap E)}{P(E)}=\\frac{5}{8}$$<\/p>\n

\u4e00\u822c\u800c\u8a00\uff0c\u6211\u5011\u6709\u5982\u4e0b\u7684\u5b9a\u7fa9\u3002<\/p>\n

\u3010\u5b9a\u7fa9 1\u3011<\/span><\/p>\n

\u5728\u521d\u7b49\u6a5f\u7387\u7a7a\u9593 $$(\\Omega,\\mathfrak{A},P)$$ \u4e2d\uff0c\u5047\u8a2d $$A,B$$ \u70ba\u5169\u500b\u4e8b\u4ef6\uff0c\u4e26\u4e14 $$P(B)>0$$\uff0c\u5247\u5728\u7d66\u5b9a $$B$$ \u767c\u751f\u7684\u689d\u4ef6\u4e0b\uff0c$$A$$ \u7684\u689d\u4ef6\u6a5f\u7387 $$P(A|B)$$ \u5c31\u5b9a\u7fa9\u70ba<\/p>\n

$$\\displaystyle P(A|B)=\\frac{P(A\\cap B)}{P(B)}~~~~~~~~~(1)$$<\/p>\n

\u6ce8\u610f\u5230\uff0c$$P(A)$$ \u662f\u689d\u4ef6\u6a5f\u7387\u7684\u7279\u4f8b\uff1a$$P(A)=P(A|\\Omega)$$\u3002\u56e0\u6b64\uff0c\u76f8\u5c0d\u5730\uff0c\u6211\u5011\u7a31 $$P(A)$$ \u70ba\u7d55\u5c0d\u6a5f\u7387<\/strong>\uff08absolute probability\uff09\u3002\u53e6\u5916\uff0c\u6620\u5c04 $$P($$\u25cf$$|B):\\mathfrak{A}\\rightarrow [0,1]$$ \u4e5f\u662f\u4e00\u500b\u6a5f\u7387\u6e2c\u5ea6\u3002<\/p>\n

\u7531 $$(1)$$ \u5f0f\u7acb\u5f97\u6a5f\u7387\u7684\u4e58\u6cd5\u516c\u5f0f<\/strong>\uff1a<\/p>\n

$$P(A\\cap B)=P(B)\\cdot P(A|B)~~~~~~~~~(2)$$<\/p>\n

\u6b64\u5f0f\u8868\u793a\u4e8b\u4ef6 $$A$$ \u8207 $$B$$ \u540c\u6642\u90fd\u767c\u751f\u7684\u6a5f\u7387\uff0c\u7b49\u65bc $$B$$ \u767c\u751f\u7684\u6a5f\u7387\u4e58\u4ee5\u7d66 $$B$$ \u7684\u689d\u4ef6\u4e4b\u4e0b $$A$$ \u7684\u689d\u4ef6\u6a5f\u7387\u3002\u5982\u679c $$P(A)>0$$\uff0c$$(2)$$ \u5f0f\u53ef\u9032\u4e00\u6b65\u5beb\u6210<\/p>\n

$$P(A\\cap B)=P(B)\\cdot P(A|B)=P(A)\\cdot P(B|A)~~~~~~~~~(3)$$<\/p>\n

\u3010\u4f8b8\u3011<\/strong>\u4e00\u500b\u7515\u4e2d\u88dd\u6709 $$8$$ \u500b\u7d05\u7403\u8207 $$13$$ \u500b\u767d\u7403\uff0c\u6211\u5011\u63a5\u7e8c\u62bd\u51fa\u5169\u500b\u7403\uff0c\u62bd\u51fa\u7684\u7b2c\u4e00\u7403\u4e0d\u518d\u653e\u56de\u3002\u5047\u8a2d\u62bd\u5230\u6bcf\u4e00\u500b\u7403\u7684\u6a5f\u7387\u76f8\u7b49\u3002\u6c42\u5169\u7403\u90fd\u62bd\u5230\u7d05\u8272\u7684\u6a5f\u7387\u3002<\/p>\n

\u3010\u89e3\u7b54\u3011<\/strong>\u4ee4 $$R_1$$ \u8207 $$R_2$$ \u5206\u5225\u8868\u793a\u7b2c\u4e00\u6b21\u8207\u7b2c\u4e8c\u6b21\u62bd\u5230\u7d05\u8272\u7684\u4e8b\u4ef6\uff0c\u5247 $$P(R_1)=\\frac{8}{21}$$\u3002\u7576\u7b2c\u4e00\u6b21\u62bd\u5230\u7d05\u8272 $$R_1$$ \u7684\u689d\u4ef6\u4e0b\uff0c\u7515\u4e2d\u5269\u4e0b $$7$$ \u500b\u7d05\u7403\u8207 $$13$$ \u500b\u767d\u7403\uff0c\u6545\u7b2c\u4e8c\u6b21\u62bd\u5230\u7d05\u8272 $$R_2$$ \u7684\u689d\u4ef6\u6a5f\u7387\u70ba $$P(R_2|R_1)=\\frac{7}{20}$$\u3002\u56e0\u6b64\u5169\u7403\u90fd\u62bd\u5230\u7d05\u8272\u7684\u6a5f\u7387\u70ba<\/p>\n

$$\\displaystyle P(R_1\\cap R_2)=P(R_1)\\cdot P(R_2|R_1)=\\frac{8}{21}\\times\\frac{7}{20}=\\frac{2}{15}$$<\/p>\n

\u3010\u5b9a\u7fa9 2\u3011<\/span><\/strong><\/p>\n

\u5982\u679c $$n$$ \u500b\u4e8b\u4ef6 $$B_1,B_2,\\cdots,B_n$$ \u5177\u6709\u5169\u5169\u4e92\u65a5\u4e14 $$\\Omega=B_1\\cup B_2\\cup\\cdots\\cup B_n$$<\/p>\n

\u5247\u6211\u5011\u7a31 $$\\{B_1,B_2,\\cdots,B_n\\}$$ \u70ba $$\\Omega$$ \u7684\u4e00\u500b\u53ef\u6e2c\u5206\u5272\uff08a partition\uff09\u3002<\/p>\n

\u3010\u5b9a\u7406 4\u3011\uff08\u5168\u6a5f\u7387\u516c\u5f0f\uff0cthe total probability formula\uff09<\/span><\/strong><\/p>\n

\u5047\u8a2d\u00a0$$\\{B_1,B_2,\\cdots,B_n\\}$$ \u70ba $$\\Omega$$ \u7684\u4e00\u500b\u53ef\u6e2c\u5206\u5272\uff0c$$P(B_k)>0,~k=1,2,\\cdots,n$$\uff0c<\/p>\n

\u4e26\u4e14 $$A\\in\\mathfrak{A}$$\uff0c\u5247<\/p>\n

$$P(A)=\\displaystyle\\sum^n_{k=1}P(A|B_k)P(B_k)~~~~~~~~~(4)$$<\/p>\n

\u7279\u5225\u5730\uff0c\u82e5 $$0<P(B)<1$$\uff0c\u5247\u6211\u5011\u6709 $$P(A)=P(A|B)P(B)+P(A|B^c)P(B^c)$$<\/p>\n

\u3010\u8a3b\u3011<\/strong>$$(4)$$ \u5f0f\u53eb\u505a\u300c\u5168\u6a5f\u7387\u516c\u5f0f<\/strong>\u300d\uff0c\u6709\u5316\u6574\u70ba\u96f6\u7684\u610f\u5473\u3002<\/p>\n

\u3010\u4f8b9\u3011<\/strong>\uff08\u6478\u5f69\u554f\u984c\uff09
\n\u5047\u8a2d\u7515\u4e2d\u6709 $$n$$ \u500b\u7403\uff0c\u5176\u4e2d\u6709 $$m$$ \u500b\u7403\u6709\u734e\uff08$$3<m<n$$\uff09\u3002\u4eca\u6709\u7532\u4e59\u4e19\u4e09\u4eba\u6309\u5e8f\u5404\u62bd\u51fa\u4e00\u500b\u7403\uff0c\u4e0d\u518d\u653e\u56de\uff0c\u554f\u4e09\u4eba\u4e2d\u734e\u7684\u6a5f\u7387\u5404\u70ba\u591a\u5c11\uff1f<\/p>\n

\u3010\u89e3\u7b54\u3011<\/strong>\u9019\u662f\u5168\u6a5f\u7387\u516c\u5f0f\u7684\u61c9\u7528\u3002\u9996\u5148\uff0c\u7532\u4e2d\u734e\u7684\u6a5f\u7387\u70ba $$m\/n$$\u3002\u4e59\u7684\u4e2d\u734e\u662f\u5728\u7532\u4e2d\u734e\u8207\u4e0d\u4e2d\u734e\u7684\u5206\u5272\u4e4b\u4e0b\u9032\u884c\u601d\u8003\uff0c\u5373\u7532\u4e2d\u734e\u5f8c\u4e59\u53c8\u4e2d\u734e\uff0c\u52a0\u4e0a\u7532\u4e0d\u4e2d\u734e\u5f8c\u4e59\u53c8\u4e2d\u734e\uff0c\u6545\u4e59\u4e2d\u734e\u7684\u6a5f\u7387\u70ba\uff1a<\/p>\n

$$\\displaystyle\\frac{m}{n}\\cdot\\frac{m-1}{n-1}+(1-\\frac{m}{n})\\cdot\\frac{m}{n-1}=\\frac{m}{n}$$<\/p>\n

\u540c\u7406\uff0c\u4e19\u7684\u4e2d\u734e\u662f\u5728\uff1a\u7532\u4e2d\u734e\u4e14\u4e59\u4e2d\u734e\uff0c\u7532\u4e2d\u734e\u4e14\u4e59\u4e0d\u4e2d\u734e\uff0c\u7532\u4e0d\u4e2d\u734e\u4e14\u4e59\u4e2d\u734e\uff0c\u7532\u4e0d\u4e2d\u734e\u4e14\u4e59\u4e0d\u4e2d\u734e\uff0c\u9019\u500b\u5206\u5272\u4e4b\u4e0b\u9032\u884c\u601d\u8003\u3002\u5f9e\u800c\uff0c\u4e19\u4e2d\u734e\u7684\u6a5f\u7387\u70ba\uff1a<\/p>\n

$$\\frac{m}{n}\\frac{m-1}{n-1}\\frac{m-2}{n-2}+\\frac{m}{n}\\frac{n-m}{n-1}\\frac{m-1}{n-2}+\\frac{n-m}{n}\\frac{m}{n-1}\\frac{m-1}{n-2}+\\frac{n-m}{n}\\frac{n-m-1}{n-1}\\frac{m}{n-2}=\\frac{m}{n}$$<\/p>\n

\u56e0\u6b64\uff0c\u7406\u6027\u8ad6\u8ff0\u544a\u8a34\u6211\u5011\uff0c\u4e09\u4eba\u4e2d\u734e\u7684\u6a5f\u7387\u90fd\u76f8\u7b49\uff0c\u8ddf\u62bd\u7403\u7684\u9806\u5e8f\u7121\u95dc\uff0c\u4f46\u662f\u4e00\u822c\u5e38\u8b58\u537b\u8b93\u6211\u5011\u932f\u89ba\u70ba\u5148\u62bd\u7403\u8005\u6bd4\u8f03\u6709\u5229\u3002<\/p>\n

\u5047\u8a2d $$H$$ \u8207 $$A$$ \u70ba\u5169\u500b\u4e8b\u4ef6\uff0c\u4e26\u4e14 $$P(H)> 0$$\uff0c$$P(A)>0$$\u3002\u7531\u689d\u4ef6\u6a5f\u7387\u7684\u5b9a\u7fa9\u8207\u4e58\u6cd5\u516c\u5f0f\u5f97\u5230\uff1a<\/p>\n

$$\\displaystyle P(A|H)=\\frac{P(A\\cap H)}{P(H)}=\\frac{P(A)P(H|A)}{P(H)}~~~~~~~~~(5)$$<\/p>\n

$$P(H)P(A|H)=P(A)P(H|A)$$<\/p>\n

\u56e0\u6b64\u00a0$$\\displaystyle P(H|A)=\\frac{P(H)P(A|H)}{P(A)}~~~~~~~~~(6)$$<\/p>\n

\u7531 $$(5)$$ \u5f0f\u7684 $$P(A|H)$$ \u5230 $$(6)$$ \u5f0f\u7684 $$P(H|A)$$\uff0c\u5728\u79d1\u5b78\u65b9\u6cd5\u8ad6\u4e0a\u6709\u91cd\u5927\u610f\u7fa9\u3002<\/p>\n

\u5728\u79d1\u5b78\u7814\u7a76\u4e2d\uff0c\u63a2\u6c42\u4e8b\u7269\u7684\u56e0\u679c\u95dc\u4fc2\u662f\u4e00\u500b\u7684\u6838\u5fc3\u8ab2\u984c\u3002\u6709\u56e0\u5fc5\u6709\u679c\uff0c\u6709\u679c\u5fc5\u6709\u56e0\uff0c\u6211\u5011\u8981\u300c\u7531\u56e0\u63a8\u679c\u300d\uff0c\u4e26\u4e14\u300c\u7531\u679c\u8ffd\u56e0\u300d\u3002<\/p>\n

\u6211\u5011\u5c0d $$(6)$$ \u5f0f\u4f5c\u5982\u4e0b\u7684\u89e3\u91cb\uff1a\u5728\u4e00\u500b\u56e0 $$H$$ \u4e4b\u4e0b\uff0c\u53ef\u80fd\u7522\u751f\u5404\u7a2e\u7684\u679c\uff0c\u800c\u7522\u751f\u679c $$A$$ \u7684\u689d\u4ef6\u6a5f\u7387\u70ba $$P(A|H)$$\u3002\u4eca\u5df2\u77e5 $$A$$ \u767c\u751f\u4e86\uff0c\u6211\u5011\u8981\u53cd\u904e\u4f86\u8ffd\u7a76\u5b83\u662f\u7531\u56e0 $$H$$ \u7522\u751f\u7684\u689d\u4ef6\u6a5f\u7387 $$P(H|A)=?$$ \u7b54\u6848\u5c31\u662f $$(6)$$ \u5f0f\u3002<\/p>\n

\u63a8\u800c\u5ee3\u4e4b\uff0c\u5047\u8a2d $$\\mathcal{H}=\\{H_1,H_2,\\dots,H_n\\}$$ \u70ba $$\\Omega$$ \u7684\u4e00\u500b\u5206\u5272\uff0c$$A\\in\\mathfrak{A}$$ \uff0c\u6211\u5011\u5c07 $$\\{H_1,H_2,\\dots,H_n\\}$$\u00a0\u60f3\u50cf\u6210\u662f\u7522\u751f $$A$$ \u7684 $$n$$ \u7a2e\u56e0\uff0c\u4e26\u4e14\u5047\u8a2d\u77e5\u9053 $$P(A|H_k)$$ \u8207 $$P(H_k)$$\u3002\u73fe\u5728\u5df2\u77e5\u4e8b\u4ef6 $$A$$ \u767c\u751f\u4e86\uff0c\u6211\u5011\u8981\u53cd\u904e\u4f86\u8ffd\u7a76 $$A$$ \u662f\u7531\u56e0 $$H_k$$ \u4fc3\u6210\u7684\u689d\u4ef6\u6a5f\u7387 $$P(H_k|A)$$\u3002<\/p>\n

\u7b54\u6848\u5c31\u662f\u4e0b\u5217\u7684Bayes\u516c\u5f0f\uff1a<\/p>\n

\u3010\u5b9a\u7406 6\u3011\uff08Bayes \u516c\u5f0f\uff09<\/span><\/strong><\/p>\n

$$\\displaystyle P(H_k|A)=\\frac{P(H_k)P(A|H_k)}{\\sum\\limits^n_{i=1}P(H_i)P(A|H_i)},~~k=1,2,\\cdots,n~~~~~~~~~(7)$$<\/p>\n

\u3010\u8a3b\u3011<\/strong>\u6b64\u5f0f\u662fBayes\u7d71\u8a08\u7684\u767c\u6e90\u5730\u3002$$P(H_i)$$ \u53eb\u505a $$H_i$$ \u7684\u5148\u9a57\u6a5f\u7387\uff08a priori probability of $$H_i$$\uff09\uff1b$$P(H_i|A)$$ \u53eb\u505a $$A$$ \u767c\u751f\u5f8c\uff0c$$H_i$$ \u7684\u5f8c\u9a57\u6a5f\u7387\uff08a posteriori probability of $$H_i$$\u00a0after the occurrence of event $$A$$\uff09\u3002\u5148\u9a57\u6a5f\u7387 $$P(H_i)$$\uff0c\u5728 $$A$$ \u767c\u751f\u7684\u8cc7\u8a0a\u4e0b\uff0c\u4fee\u6b63\u70ba\u5f8c\u9a57\u6a5f\u7387 $$P(H_i|A)$$\u3002\u56e0\u6b64Bayes\u516c\u5f0f\u751a\u6709\u5f9e\u7d93\u9a57\u4e2d\u5b78\u7fd2\u3001\u4e0d\u65b7\u4fee\u6b63\u7684\u610f\u5473\u3002<\/p>\n

\u3010\u4f8b10\u3011<\/strong>\u6709\u4e00\u500b\u7515\u88dd\u6709\u5169\u500b\u786c\u5e63\uff1a $$A_1$$ \u70ba\u516c\u6b63\u786c\u5e63\uff0c\u64f2\u51fa $$H$$ \u7684\u6a5f\u7387\u70ba $$1\/2$$\uff1b\u53e6\u4e00\u500b $$A_2$$ \u70ba\u4e0d\u516c\u6b63\u786c\u5e63\uff0c\u64f2\u51fa $$H$$ \u7684\u6a5f\u7387\u70ba $$1\/3$$\u3002\u4eca\u6211\u5011\u5f9e\u7515\u4e2d\u4efb\u610f\u53d6\u51fa\u4e00\u500b\u786c\u5e63\uff08\u6a5f\u6703\u5747\u7b49\uff09\uff0c\u518d\u64f2\u6b64\u786c\u5e63\uff0c\u5f97\u5230\u6b63\u9762 $$H$$\u3002\u6c42\u6b64\u786c\u5e63\u70ba\u516c\u6b63\u786c\u5e63 $$A_1$$ \u7684\u6a5f\u7387\u3002<\/p>\n

\u3010\u89e3\u7b54\u3011<\/strong>\u8b93\u6211\u5011\u5148\u5efa\u69cb\u76f8\u61c9\u7684\u6a5f\u7387\u6a21\u578b\u3002\u53d6\u6a23\u672c\u7a7a\u9593\u70ba $$\\Omega=\\{A_{1}H, A_{1}T,A_{2}H, A_{2}T\\}$$<\/p>\n

\u9019\u63cf\u8ff0\u8457\u5f9e\u7515\u4e2d\u4efb\u610f\u53d6\u51fa\u4e00\u500b\u786c\u5e63\uff0c\u518d\u64f2\u6b64\u786c\u5e63\u7684\u96a8\u6a5f\u5be6\u9a57\u4e4b\u6240\u6709\u7d50\u679c\uff08$$A_{1}H$$\u8868\u793a\u53d6\u5230\u516c\u6b63\u786c\u5e63$$A_1$$\uff0c\u518d\u64f2\u51fa\u6b63\u9762$$H$$\uff09\u3002<\/p>\n

\u6839\u64da\u5047\u8a2d\u689d\u4ef6\u5f97\u77e5 $$P(A_1)=P(A_2)=\\frac{1}{2}$$ \u4e26\u4e14 $$P(H|A_1)=\\frac{1}{2}$$ \uff0c$$ P(H|A_2)=\\frac{1}{3}$$<\/p>\n

\u7531\u6b64\u53ef\u552f\u4e00\u5f97\u5230\u6a23\u672c\u9ede\u7684\u6a5f\u7387\u6e2c\u5ea6 $$\\displaystyle P(A_{1}H)=P(A_1)P(H|A_1)=\\frac{1}{2}\\cdot\\frac{1}{2}=\\frac{1}{4}$$<\/p>\n

\u540c\u7406 $$\\displaystyle P(A_{1}T)=\\frac{1}{4}$$\uff0c$$\\displaystyle P(A_{2}H)=\\frac{1}{6}$$\uff0c$$\\displaystyle P(A_{2}T)=\\frac{1}{3}$$<\/p>\n

\u73fe\u5728\u7531Bayes\u516c\u5f0f\u5f97\u5230\u6240\u8981\u7684\u7b54\u6848<\/p>\n

$$\\displaystyle P(A_1|H)=\\frac{P(A_1)P(H|A_1)}{P(A_1)P(H|A_1)+P(A_2)P(H|A_2)}=\\frac{3}{5}$$<\/p>\n

\u5f9e\u800c\u53c8\u6709 $$\\displaystyle P(A_2|H)=\\frac{2}{5}$$<\/p>\n

\u9023\u7d50\uff1a\u521d\u7b49\u7684\u6a5f\u7387\u8ad6\uff087\uff09\u7368\u7acb\u4e8b\u4ef6\u7684\u6982\u5ff5<\/a><\/p>\n

\u53c3\u8003\u66f8\u76ee\uff1a<\/strong><\/p>\n

    \n
  1. William Feller: An Introduction to Probability Theory and its Applications. Vol.1 John-Wiley & Sons, INC. Third Edition, 1967.<\/li>\n
  2. Sheldon M. Ross: A First Course in Probability. 8th Edition, Prentice Hall, 2009.
    \n\uff08\u9019\u5169\u672c\u662f\u516c\u8a8d\u7684\u6a5f\u7387\u8ad6\u5165\u9580\u7d55\u4f73\u7684\u66f8\u3002\u7b2c\u4e00\u672c\u662f\u7d93\u5178\uff1b\u7b2c\u4e8c\u672c\u662f\u6bd4\u8f03\u665a\u8fd1\u5beb\u6210\u7684\u66f8\uff0c\u7d93\u5e38\u88ab\u62ff\u4f86\u7576\u4f5c\u5927\u5b78\u90e8\u300c\u521d\u7b49\u6a5f\u7387\u8ad6\u300d\u9019\u9580\u8ab2\u7684\u6559\u79d1\u66f8\u3002\uff09<\/li>\n
  3. Kai Lai Chung: Elementary Probability Theory. Springer, 2004.<\/li>\n
  4. Hugh Gordon: Discrete Probability. Springer, 1997.<\/li>\n
  5. Eugene Lukacs: Probability and Mathematical Statistics. Academic Press, 1972.<\/li>\n
  6. David Stirzaker: Elementary Probability. Cambridge University Press, 1994.<\/li>\n
  7. Jim Pitman: Probability. Springer-Verlag, 1993.<\/li>\n
  8. Janos Galambos: Introductory Probability Theory. Marcel Dekker, INC. 1984.<\/li>\n<\/ol>\n

    \u8a3b\uff1a\u901a\u5e38\u8981\u8b1b\u8ff0\u6a5f\u7387\u8ad6\u5fc5\u9808\u7528\u5230\u300c\u6e2c\u5ea6\u7a4d\u5206\u8ad6\u300d\u7684\u6578\u5b78\u5de5\u5177\uff0c\u6216\u81f3\u5c11\u8981\u7528\u5230\u5fae\u7a4d\u5206\u3002\u56e0\u6b64\u8981\u70ba\u4e00\u822c\u8b80\u8005\u4ecb\u7d39\u6a5f\u7387\u8ad6\u7684\u8b80\u7269\u8aa0\u5c6c\u4e0d\u5bb9\u6613\u3002\u4e0a\u8ff0\u516b\u672c\u66f8\u76e1\u91cf\u58d3\u4f4e\u8981\u7528\u5230\u7684\u6578\u5b78\u5de5\u5177\uff0c\u5927\u90e8\u5206\u53ea\u9700\u6392\u5217\u8207\u7d44\u5408\uff0c\u53ea\u6709\u5c11\u90e8\u4efd\u8981\u7528\u5230\u4e00\u9ede\u5152\u5fae\u7a4d\u5206\u3002<\/p>\n

    \u5f9e\u79d1\u5b78\u65b9\u6cd5\u8ad6\u7684\u89c0\u9ede\u4f86\u770b\uff0c\u6a5f\u7387\u8ad6\u8207\u7d71\u8a08\u5b78\u662f\u4e00\u9ad4\u7684\u5169\u9762\uff0c\u6a5f\u7387\u8ad6\u662f\u300c\u6f14\u7e79\u6cd5\u300d\uff0c\u7d71\u8a08\u5b78\u662f\u300c\u6b78\u7d0d\u6cd5\u300d\u3002\u56e0\u6b64\uff0c\u672c\u6587\u7684\u4e3b\u984c\u96d6\u7136\u662f\u6a5f\u7387\u8ad6\uff0c\u4f46\u662f\u4e5f\u9806\u4fbf\u4ecb\u7d39\u4e00\u9ede\u9ede\u7d71\u8a08\u5b78\u7684\u6982\u5ff5\u3002<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

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