{"id":2820,"date":"2023-07-20T13:32:55","date_gmt":"2023-07-20T13:32:55","guid":{"rendered":"https:\/\/statorials.org\/tr\/en-az-iki-olasilik\/"},"modified":"2023-07-20T13:32:55","modified_gmt":"2023-07-20T13:32:55","slug":"en-az-iki-olasilik","status":"publish","type":"post","link":"https:\/\/statorials.org\/tr\/en-az-iki-olasilik\/","title":{"rendered":"\u201cen az iki\u201d ba\u015far\u0131 olas\u0131l\u0131\u011f\u0131 nas\u0131l bulunur?"},"content":{"rendered":"<p><\/p>\n<hr>\n<p><span style=\"color: #000000;\">Bir dizi denemede <strong>en az iki ba\u015far\u0131 olas\u0131l\u0131\u011f\u0131n\u0131<\/strong> bulmak i\u00e7in a\u015fa\u011f\u0131daki genel form\u00fcl\u00fc kullanabiliriz:<\/span><\/p>\n<pre style=\"background-color: #ececec; font-size: 17px;\"> <strong>P(at least two successes) = 1 - P(zero successes) - P(one success)<\/strong><\/pre>\n<p> <span style=\"color: #000000;\">Yukar\u0131daki form\u00fclde, <a href=\"https:\/\/statorials.org\/tr\/binom-dagilimi-1\/\" target=\"_blank\" rel=\"noopener\">binom da\u011f\u0131l\u0131m\u0131<\/a> i\u00e7in a\u015fa\u011f\u0131daki form\u00fcl\u00fc kullanarak her olas\u0131l\u0131\u011f\u0131 hesaplayabiliriz:<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>P(X=k) = <sub>n<\/sub> C <sub>k<\/sub> * p <sup>k<\/sup> * (1-p) <sup>nk<\/sup><\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\">Alt\u0131n:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\"><strong>n:<\/strong> deneme say\u0131s\u0131<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>k:<\/strong> ba\u015far\u0131 say\u0131s\u0131<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong>p:<\/strong> belirli bir denemede ba\u015far\u0131 olas\u0131l\u0131\u011f\u0131<\/span><\/li>\n<li> <span style=\"color: #000000;\"><strong><sub>n<\/sub> C <sub>k<\/sub> :<\/strong> <em>n<\/em> denemede <em>k<\/em> ba\u015far\u0131 elde etmenin yollar\u0131n\u0131n say\u0131s\u0131<\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">A\u015fa\u011f\u0131daki \u00f6rnekler, farkl\u0131 senaryolarda &#8220;en az iki&#8221; ba\u015far\u0131 olas\u0131l\u0131\u011f\u0131n\u0131 bulmak i\u00e7in bu form\u00fcl\u00fcn nas\u0131l kullan\u0131laca\u011f\u0131n\u0131 g\u00f6stermektedir.<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>\u00d6rnek 1: Serbest at\u0131\u015f denemeleri<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">Ty serbest at\u0131\u015f denemelerinin %25&#8217;ini yap\u0131yor. E\u011fer 5 serbest at\u0131\u015f atarsa en az iki at\u0131\u015f yapma olas\u0131l\u0131\u011f\u0131n\u0131 bulun.<\/span><\/p>\n<p> <span style=\"color: #000000;\">\u00d6ncelikle tam olarak s\u0131f\u0131r serbest at\u0131\u015f veya tam olarak bir serbest at\u0131\u015f yapma olas\u0131l\u0131\u011f\u0131n\u0131 hesaplayal\u0131m:<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>P(X=0)<\/strong> = <sub>5<\/sub> C <sub>0<\/sub> * 0,25 <sup>0<\/sup> * (1-0,25) <sup>5-0<\/sup> = 1 * 1 * 0,75 <sup>5<\/sup> = <strong>0,2373<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>P(X=1)<\/strong> = <sub>5<\/sub> C <sub>1<\/sub> * 0,25 <sup>1<\/sup> * (1-0,25) <sup>5-1<\/sup> = 5 * 0,25 * 0,75 <sup>4<\/sup> = <strong>0,3955<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\">Daha sonra Ty&#8217;\u0131n en az iki serbest at\u0131\u015f yapma olas\u0131l\u0131\u011f\u0131n\u0131 bulmak i\u00e7in bu de\u011ferleri a\u015fa\u011f\u0131daki form\u00fcle yerle\u015ftirelim:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">P(X\u22652) = 1 \u2013 P(X=0) \u2013 P(X=1)<\/span><\/li>\n<li> <span style=\"color: #000000;\">P(X\u22652) = 1 \u2013 0,2372 \u2013 0,3955<\/span><\/li>\n<li> <span style=\"color: #000000;\">P(X\u22652) = <strong>0,3673<\/strong><\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Ty&#8217;\u0131n be\u015f denemede en az iki serbest at\u0131\u015f yapma olas\u0131l\u0131\u011f\u0131 <strong>0,3673&#8217;t\u00fcr<\/strong> .<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>\u00d6rnek 2: Widget&#8217;lar<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">Belirli bir fabrikada t\u00fcm aletlerin %2&#8217;si kusurludur. 10 aletten olu\u015fan rastgele bir \u00f6rnekte en az ikisinin kusurlu olma olas\u0131l\u0131\u011f\u0131n\u0131 belirleyin.<\/span><\/p>\n<p> <span style=\"color: #000000;\">\u00d6ncelikle tam olarak s\u0131f\u0131r veya tam olarak bir tanesinin kusurlu olma olas\u0131l\u0131\u011f\u0131n\u0131 hesaplayal\u0131m:<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>P(X=0)<\/strong> = <sub>10<\/sub> C <sub>0<\/sub> * 0,02 <sup>0<\/sup> * (1-0,02) <sup>10-0<\/sup> = 1 * 1 * 0,98 <sup>10<\/sup> = <strong>0,8171<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>P(X=1)<\/strong> = <sub>10<\/sub> C <sub>1<\/sub> * 0,02 <sup>1<\/sup> * (1-0,02) <sup>10-1<\/sup> = 10 * 0,02 * 0,98 <sup>9<\/sup> = <strong>0,1667<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\">Daha sonra, en az iki widget&#8217;\u0131n hatal\u0131 olma olas\u0131l\u0131\u011f\u0131n\u0131 bulmak i\u00e7in bu de\u011ferleri a\u015fa\u011f\u0131daki form\u00fcle yerle\u015ftirelim:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">P(X\u22652) = 1 \u2013 P(X=0) \u2013 P(X=1)<\/span><\/li>\n<li> <span style=\"color: #000000;\">P(X\u22652) = 1 \u2013 0,8171 \u2013 0,1667<\/span><\/li>\n<li> <span style=\"color: #000000;\">P(X\u22652) = <strong>0,0162<\/strong><\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Bu rastgele 10 \u00f6rneklemde en az iki widget&#8217;\u0131n kusurlu olma olas\u0131l\u0131\u011f\u0131 <strong>0,0162&#8217;dir<\/strong> .<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>\u00d6rnek 3: Trivia sorular\u0131<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">Bob, \u00f6nemsiz sorular\u0131n %60&#8217;\u0131n\u0131 do\u011fru yan\u0131tl\u0131yor. Ona 5 \u00f6nemsiz soru sorarsak en az ikisini do\u011fru cevaplama olas\u0131l\u0131\u011f\u0131n\u0131 bulun.<\/span><\/p>\n<p> <span style=\"color: #000000;\">\u00d6ncelikle tam olarak s\u0131f\u0131r veya tam olarak bir yan\u0131t verme olas\u0131l\u0131\u011f\u0131n\u0131 hesaplayal\u0131m:<\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>P(X=0)<\/strong> = <sub>5<\/sub> C <sub>0<\/sub> * 0,60 <sup>0<\/sup> * (1-0,60) <sup>5-0<\/sup> = 1 * 1 * 0,40 <sup>5<\/sup> = <strong>0,01024<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\"><strong>P(X=1)<\/strong> = <sub>5<\/sub> C <sub>1<\/sub> * 0,60 <sup>1<\/sup> * (1-0,60) <sup>5-1<\/sup> = 5 * 0,60 * 0,40 <sup>4<\/sup> = <strong>0,0768<\/strong><\/span><\/p>\n<p> <span style=\"color: #000000;\">Daha sonra, en az iki soruyu do\u011fru cevaplama olas\u0131l\u0131\u011f\u0131n\u0131 bulmak i\u00e7in bu de\u011ferleri a\u015fa\u011f\u0131daki form\u00fcle yerle\u015ftirelim:<\/span><\/p>\n<ul>\n<li> <span style=\"color: #000000;\">P(X\u22652) = 1 \u2013 P(X=0) \u2013 P(X=1)<\/span><\/li>\n<li> <span style=\"color: #000000;\">P(X\u22652) = 1 \u2013 0,01024 \u2013 0,0768<\/span><\/li>\n<li> <span style=\"color: #000000;\">P(X\u22652) = <strong>0,91296<\/strong><\/span><\/li>\n<\/ul>\n<p> <span style=\"color: #000000;\">Be\u015f sorudan en az ikisini do\u011fru cevaplama olas\u0131l\u0131\u011f\u0131 <strong>0,91296&#8217;d\u0131r<\/strong> .<\/span><\/p>\n<h3> <span style=\"color: #000000;\"><strong>Bonus: \u201cEn az iki\u201d olas\u0131l\u0131k hesaplay\u0131c\u0131s\u0131<\/strong><\/span><\/h3>\n<p> <span style=\"color: #000000;\">Belirli bir denemedeki ba\u015far\u0131 olas\u0131l\u0131\u011f\u0131na ve toplam deneme say\u0131s\u0131na ba\u011fl\u0131 olarak &#8220;en az iki&#8221; ba\u015far\u0131 olas\u0131l\u0131\u011f\u0131n\u0131 otomatik olarak bulmak i\u00e7in bu hesaplay\u0131c\u0131y\u0131 kullan\u0131n.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Bir dizi denemede en az iki ba\u015far\u0131 olas\u0131l\u0131\u011f\u0131n\u0131 bulmak i\u00e7in a\u015fa\u011f\u0131daki genel form\u00fcl\u00fc kullanabiliriz: P(at least two successes) = 1 &#8211; P(zero successes) &#8211; P(one success) Yukar\u0131daki form\u00fclde, binom da\u011f\u0131l\u0131m\u0131 i\u00e7in a\u015fa\u011f\u0131daki form\u00fcl\u00fc kullanarak her olas\u0131l\u0131\u011f\u0131 hesaplayabiliriz: P(X=k) = n C k * p k * (1-p) nk Alt\u0131n: n: deneme say\u0131s\u0131 k: ba\u015far\u0131 say\u0131s\u0131 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[],"class_list":["post-2820","post","type-post","status-publish","format-standard","hentry","category-rehber"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.3 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>&quot;En az iki&quot; 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