{"id":380,"date":"2023-08-01T11:32:01","date_gmt":"2023-08-01T11:32:01","guid":{"rendered":"https:\/\/statorials.org\/tr\/toplam-olasilik-teoremi\/"},"modified":"2023-08-01T11:32:01","modified_gmt":"2023-08-01T11:32:01","slug":"toplam-olasilik-teoremi","status":"publish","type":"post","link":"https:\/\/statorials.org\/tr\/toplam-olasilik-teoremi\/","title":{"rendered":"Toplam olas\u0131l\u0131k teoremi"},"content":{"rendered":"<p>Bu makale toplam olas\u0131l\u0131k teoreminin ne oldu\u011funu ve olas\u0131l\u0131k ve istatistikte ne i\u00e7in kullan\u0131ld\u0131\u011f\u0131n\u0131 a\u00e7\u0131klamaktad\u0131r. B\u00f6ylece toplam olas\u0131l\u0131k teoreminin form\u00fcl\u00fcn\u00fc, \u00e7\u00f6z\u00fclm\u00fc\u015f al\u0131\u015ft\u0131rmalar\u0131 ve toplam olas\u0131l\u0131k teoreminin ne zaman kullan\u0131ld\u0131\u011f\u0131n\u0131 bulacaks\u0131n\u0131z. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-el-teorema-de-la-probabilidad-total\"><\/span> Toplam olas\u0131l\u0131k teoremi nedir?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Olas\u0131l\u0131k teorisinde <strong>toplam olas\u0131l\u0131k teoremi<\/strong> , bir \u00f6rnek uzay\u0131n par\u00e7as\u0131 olmayan bir olay\u0131n olas\u0131l\u0131\u011f\u0131n\u0131n, s\u00f6z konusu \u00f6rnek uzaydaki t\u00fcm olaylar\u0131n <a href=\"https:\/\/statorials.org\/tr\/kosullu-kosullu-olasilik\/\">ko\u015fullu olas\u0131l\u0131klar\u0131ndan<\/a> hesaplanmas\u0131n\u0131 m\u00fcmk\u00fcn k\u0131lan bir yasad\u0131r.<\/p>\n<p> Dolay\u0131s\u0131yla toplam olas\u0131l\u0131k teoremi, belirli bir olay\u0131n olas\u0131l\u0131\u011f\u0131n\u0131, o olayla ilgili k\u0131smi bilgilere dayanarak hesaplamak i\u00e7in kullan\u0131l\u0131r. Bazen gerekli t\u00fcm bilgilere sahip olmad\u0131\u011f\u0131m\u0131z i\u00e7in Laplace kural\u0131n\u0131 do\u011frudan uygulayarak bir olay\u0131n olas\u0131l\u0131\u011f\u0131n\u0131 belirleyemeyiz. Ancak bu olay hakk\u0131ndaki verileri di\u011fer olaylara g\u00f6re biliyorsak, toplam olas\u0131l\u0131k teoremi genellikle faydal\u0131d\u0131r.<\/p>\n<p> K\u0131saca toplam olas\u0131l\u0131k teoremi, bir olay\u0131n olas\u0131l\u0131\u011f\u0131n\u0131 hesaplamak istedi\u011fimizde ancak yaln\u0131zca belirli ko\u015fullar alt\u0131nda o olay hakk\u0131nda bilgi sahibi oldu\u011fumuzda kullan\u0131l\u0131r. \u00d6rne\u011fin, bu teoremin baz\u0131 uygulamalar\u0131 \u00e7oklu durum deneylerini, kuyruk teorisini ve hayatta kalma analizini i\u00e7erir. <\/p>\n<div style=\"background-color:#FFFDE7; padding-top: 10px; padding-bottom: 10px; padding-right: 20px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\"> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Bak\u0131n\u0131z:<\/strong> <a href=\"https:\/\/statorials.org\/tr\/laplace-kurali-veya-laplace-yasasi\/\">Laplace kural\u0131<\/a> <\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-del-teorema-de-la-probabilidad-total\"><\/span> Toplam olas\u0131l\u0131k teoremi form\u00fcl\u00fc<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Toplam olas\u0131l\u0131k teoremi, \u00f6rnek uzayda bir b\u00f6l\u00fcm olu\u015fturan bir dizi olay (A <sub>1<\/sub> , A <sub>2<\/sub> ,\u2026, A <sub>n<\/sub> ) verildi\u011finde, B olay\u0131n\u0131n olas\u0131l\u0131\u011f\u0131n\u0131n, her birinin olas\u0131l\u0131\u011f\u0131n\u0131n \u00e7arp\u0131mlar\u0131n\u0131n toplam\u0131na e\u015fit oldu\u011funu s\u00f6yler. P(A <sub>i<\/sub> ) olay\u0131 P(B|A <sub>i<\/sub> ) ko\u015fullu olas\u0131l\u0131\u011f\u0131na g\u00f6re hesaplan\u0131r.<\/p>\n<p> Bu nedenle <strong>toplam olas\u0131l\u0131k teoreminin form\u00fcl\u00fc<\/strong> \u015f\u00f6yledir: <\/p>\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/theoreme-de-probabilite-totale.png\" alt=\"toplam olas\u0131l\u0131k teoremi form\u00fcl\u00fc\" class=\"wp-image-8595\" width=\"291\" height=\"320\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p style=\"margin-bottom:7px\"> Alt\u0131n:<\/p>\n<ul style=\"color:#FF8A05; font-weight: bold;\">\n<li style=\"margin-bottom:8px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-3cf84b3bfcc955798a43fce91458ff42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(B)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"41\" style=\"vertical-align: -5px;\"><\/p>\n<p> B olay\u0131n\u0131n ger\u00e7ekle\u015fme olas\u0131l\u0131\u011f\u0131d\u0131r.<\/li>\n<li style=\"margin-bottom:8px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-676b61239ea67554603513360fbea72b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(B|A_i)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -5px;\"><\/p>\n<p> A <sub>i<\/sub> olay\u0131 g\u00f6z \u00f6n\u00fcne al\u0131nd\u0131\u011f\u0131nda B olay\u0131n\u0131n ko\u015fullu olas\u0131l\u0131\u011f\u0131d\u0131r.<\/li>\n<li style=\"margin-bottom:8px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-be31252343b496b792bee3e55a99979f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(A_i)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"46\" style=\"vertical-align: -5px;\"><\/p>\n<p> A <sub>i<\/sub> olay\u0131n\u0131n meydana gelme olas\u0131l\u0131\u011f\u0131d\u0131r.<\/li>\n<\/ul>\n<p> Olas\u0131l\u0131k durumunda, \u00f6rnek uzay\u0131n bir b\u00f6l\u00fcm\u00fcn\u00fcn, birle\u015fimi \u00f6rnek uzay\u0131 olu\u015fturan, kar\u015f\u0131l\u0131kl\u0131 olarak uyumsuz olaylar\u0131n bir k\u00fcmesi olarak tan\u0131mland\u0131\u011f\u0131n\u0131 unutmay\u0131n. <\/p>\n<div style=\"background-color:#FFFDE7; padding-top: 10px; padding-bottom: 10px; padding-right: 20px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\"> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Bak\u0131n\u0131z:<\/strong> <a href=\"https:\/\/statorials.org\/tr\/ornek-uzay-1\/\">\u00d6rnek uzay<\/a> <\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-resuelto-del-teorema-de-la-probabilidad-total\"><\/span> Toplam olas\u0131l\u0131k teoreminin somut \u00f6rne\u011fi<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Toplam olas\u0131l\u0131k teoreminin tan\u0131m\u0131n\u0131 ve form\u00fcl\u00fcn\u00fcn ne oldu\u011funu g\u00f6rd\u00fckten sonra, anlam\u0131n\u0131 daha iyi anlamak i\u00e7in bir olas\u0131l\u0131\u011f\u0131n toplam olas\u0131l\u0131k teoremi ile nas\u0131l hesapland\u0131\u011f\u0131 konusunda \u00e7\u00f6z\u00fcml\u00fc bir al\u0131\u015ft\u0131rma g\u00f6rece\u011fiz.<\/p>\n<ul>\n<li> Bir elektronik ma\u011fazas\u0131nda X, Y, Z olmak \u00fczere \u00fc\u00e7 marka televizyon sat\u0131lmaktad\u0131r. Sat\u0131\u015flar\u0131n %20&#8217;sinin marka televizyon, %4&#8217;\u00fcn\u00fcn Z marka televizyon oldu\u011fu tahmin edilmektedir. televizyonlar ar\u0131zal\u0131d\u0131r. Ar\u0131zal\u0131 bir TV sat\u0131n alma olas\u0131l\u0131\u011f\u0131 nedir?<\/li>\n<\/ul>\n<p> Sorun ifadesi bize bir m\u00fc\u015fterinin her marka TV&#8217;yi sat\u0131n alma olas\u0131l\u0131\u011f\u0131n\u0131 verir:<\/p>\n<ul>\n<li> Olay A <sub>1<\/sub> : Bir m\u00fc\u015fteri <sub>bir<\/sub> televizyon markas\u0131 sat\u0131n al\u0131r<\/li>\n<li> Olay A <sub>2<\/sub> : Bir m\u00fc\u015fteri Y markal\u0131 bir televizyon al\u0131yor \u2192 P(A <sub>2<\/sub> )=0,50<\/li>\n<li> Olay A <sub>3<\/sub> : Bir m\u00fc\u015fteri bir televizyon markas\u0131 sat\u0131n al\u0131yor Z \u2192 P(A <sub>3<\/sub> )=0,30<\/li>\n<\/ul>\n<p> Ek olarak, uygulama ifadesi bize her markan\u0131n televizyonunun ar\u0131zal\u0131 olma olas\u0131l\u0131\u011f\u0131n\u0131 da veriyor:<\/p>\n<p class=\"has-text-align-center\"> Olay B: TV ar\u0131zal\u0131<\/p>\n<ul>\n<li> B|A <sub>1<\/sub> : X marka televizyon verildi\u011finde televizyon ar\u0131zal\u0131d\u0131r \u2192 P(B|A <sub>1<\/sub> )=0,05<\/li>\n<li> B|A <sub>2<\/sub> : Y marka televizyon verildi\u011finde televizyon ar\u0131zal\u0131d\u0131r \u2192 P(B|A <sub>2<\/sub> )=0,03<\/li>\n<li> B|A <sub>3<\/sub> : Z marka televizyon verildi\u011finde televizyon ar\u0131zal\u0131d\u0131r \u2192 P(B|A <sub>3<\/sub> )=0,04<\/li>\n<\/ul>\n<p> Buna g\u00f6re problemin <a href=\"https:\/\/statorials.org\/tr\/agac-diyagrami\/\">olas\u0131l\u0131k a\u011fac\u0131<\/a> \u015fu \u015fekildedir: <\/p>\n<figure class=\"wp-block-image aligncenter size-full\"><img decoding=\"async\" loading=\"lazy\" width=\"370\" height=\"257\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/theoreme-de-probabilite-de-bayes.png\" alt=\"\" class=\"wp-image-8525\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Bu nedenle, ar\u0131zal\u0131 bir TV sat\u0131n alma olas\u0131l\u0131\u011f\u0131n\u0131 hesaplamak i\u00e7in toplam olas\u0131l\u0131k kural\u0131 form\u00fcl\u00fcn\u00fc kullanmam\u0131z gerekir:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-4bd2e4c30e05d7abab89dabb7b85cd6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle P(B)=\\sum_{i=1}^n P(B|A_i)\\cdot P(A_i)\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"218\" style=\"vertical-align: -21px;\"><\/p>\n<\/p>\n<p> Bizim durumumuzda \u00f6rnek uzay \u00fc\u00e7 olaydan (A <sub>1<\/sub> , A <sub>2<\/sub> ve A <sub>3<\/sub> ) olu\u015fur, dolay\u0131s\u0131yla toplam olas\u0131l\u0131k teoreminin form\u00fcl\u00fc a\u015fa\u011f\u0131daki gibidir:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-0fe9d6f5c6241bf124144269ba498188_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle P(B)=P(B|A_1)\\cdot P(A_1)+P(B|A_2)\\cdot P(A_2)+P(B|A_3)\\cdot P(A_3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"496\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Dolay\u0131s\u0131yla ar\u0131zal\u0131 bir televizyon sat\u0131n alma olas\u0131l\u0131\u011f\u0131n\u0131 bulmak i\u00e7in \u00f6nceki ifadedeki olas\u0131l\u0131klar\u0131 yerine koymak yeterlidir:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-78411084ca76e5e0de6a2b1794e61b28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(B)&amp;=P(B|A_1)\\cdot P(A_1)+P(B|A_2)\\cdot P(A_2)+P(B|A_3)\\cdot P(A_3)\\\\[2ex]&amp;=0,05\\cdot 0,20+0,03\\cdot 0,50+0,04\\cdot 0,30\\\\[2ex]&amp;=0,037\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"102\" width=\"496\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Sonu\u00e7 olarak televizyon ald\u0131\u011f\u0131m\u0131zda ar\u0131zal\u0131 olma ihtimalimiz %3,7&#8217;dir. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"teorema-de-la-probabilidad-total-y-teorema-de-bayes\"><\/span> Toplam olas\u0131l\u0131k teoremi ve Bayes teoremi<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Toplam olas\u0131l\u0131k teoremi ve Bayes teoremi olas\u0131l\u0131k teorisindeki iki \u00f6nemli teoremdir, \u00e7\u00fcnk\u00fc \u00f6zellikle ko\u015fullu olas\u0131l\u0131k de\u011ferlerinden olas\u0131l\u0131klar\u0131 hesaplamam\u0131za izin verirler.<\/p>\n<p> Bayes teoremi, bir olay hakk\u0131nda \u00f6nceden bilgi bilindi\u011finde, olay\u0131n olas\u0131l\u0131\u011f\u0131n\u0131 hesaplamak i\u00e7in kullan\u0131lan bir olas\u0131l\u0131k teorisi yasas\u0131d\u0131r.<\/p>\n<p> Spesifik olarak, <strong>toplam olas\u0131l\u0131k teoremi ve Bayes teoremi<\/strong> birbiriyle ili\u015fkilidir, asl\u0131nda Bayes teoremi form\u00fcl\u00fcn\u00fcn paydas\u0131, toplam olas\u0131l\u0131k teoremi form\u00fcl\u00fcne e\u015fde\u011ferdir.<\/p>\n<p> Bayes teoreminin ne oldu\u011funu ve uygulama \u00f6rneklerini g\u00f6rmek i\u00e7in a\u015fa\u011f\u0131daki ba\u011flant\u0131ya t\u0131klay\u0131n: <\/p>\n<div style=\"background-color:#FFFDE7; padding-top: 10px; padding-bottom: 10px; padding-right: 20px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\"> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Bak\u0131n\u0131z:<\/strong> <a href=\"https:\/\/statorials.org\/tr\/bayes-teoremi\/\">Bayes teoremi<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Bu makale toplam olas\u0131l\u0131k teoreminin ne oldu\u011funu ve olas\u0131l\u0131k ve istatistikte ne i\u00e7in kullan\u0131ld\u0131\u011f\u0131n\u0131 a\u00e7\u0131klamaktad\u0131r. B\u00f6ylece toplam olas\u0131l\u0131k teoreminin form\u00fcl\u00fcn\u00fc, \u00e7\u00f6z\u00fclm\u00fc\u015f al\u0131\u015ft\u0131rmalar\u0131 ve toplam olas\u0131l\u0131k teoreminin ne zaman kullan\u0131ld\u0131\u011f\u0131n\u0131 bulacaks\u0131n\u0131z. Toplam olas\u0131l\u0131k teoremi nedir? Olas\u0131l\u0131k teorisinde toplam olas\u0131l\u0131k teoremi , bir \u00f6rnek uzay\u0131n par\u00e7as\u0131 olmayan bir olay\u0131n olas\u0131l\u0131\u011f\u0131n\u0131n, s\u00f6z konusu \u00f6rnek uzaydaki t\u00fcm olaylar\u0131n ko\u015fullu [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[],"class_list":["post-380","post","type-post","status-publish","format-standard","hentry","category-olasilik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Toplam olas\u0131l\u0131k teoremi<\/title>\n<meta name=\"description\" content=\"Burada toplam olas\u0131l\u0131k teoreminin ne oldu\u011funu, ne i\u00e7in kullan\u0131ld\u0131\u011f\u0131n\u0131, form\u00fcl\u00fcn\u00fc ve toplam olas\u0131l\u0131k teoreminin 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