{"id":7130,"date":"2011-11-01T19:40:45","date_gmt":"2011-11-01T19:40:45","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7130"},"modified":"2022-01-25T19:04:32","modified_gmt":"2022-01-25T19:04:32","slug":"um-cofre-o-oito-chaves","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7130","title":{"rendered":"Um cofre o oito chaves"},"content":{"rendered":"<p><ul id='GTTabs_ul_7130' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7130' class='GTTabs_curr'><a  id=\"7130_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7130' ><a  id=\"7130_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_7130'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/cofre.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7131\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7131\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/cofre.jpg\" data-orig-size=\"256,192\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Cofre\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/cofre.jpg\" class=\"alignright size-full wp-image-7131\" title=\"Cofre\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/cofre.jpg\" alt=\"\" width=\"154\" height=\"115\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/cofre.jpg 256w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/cofre-150x112.jpg 150w\" sizes=\"auto, (max-width: 154px) 100vw, 154px\" \/><\/a>Um homem tem 8 chaves, das quais apenas uma abre um cofre.<\/p>\n<p>Sabe-se que, ap\u00f3s cada tentativa, o homem separa a chave utilizada.<\/p>\n<ol>\n<li>Calcule a probabilidade dos acontecimentos:\n<p>A: &#8220;<em>Abriu o cofre na primeira tentativa<\/em>&#8220;;<\/p>\n<p>B: &#8220;<em>Abriu o cofre somente na segunda tentativa<\/em>&#8220;.<\/p>\n<\/li>\n<li>Considere a vari\u00e1vel aleat\u00f3ria X: &#8220;<em>n\u00famero de tentativas efetuadas at\u00e9 abrir o cofre<\/em>&#8221; e construa a respetiva distribui\u00e7\u00e3o de probabilidades.<\/li>\n<li>Determine a esperan\u00e7a matem\u00e1tica e o desvio padr\u00e3o da distribui\u00e7\u00e3o.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7130' onClick='GTTabs_show(1,7130)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7130'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Ora, $P(A)=\\frac{1}{8}$ e $P(B)=\\frac{7}{8}\\times \\frac{1}{7}=\\frac{1}{8}$.<br \/>\n\u00ad<\/li>\n<li>Vamos considerar que o enunciado da vari\u00e1vel aleat\u00f3ria tem sentido inclusivo.\n<p>Como,<\/p>\n<p>$P(X=1)=P(A)=\\frac{1}{8}$<br \/>\n$P(X=2)=P(B)=\\frac{7}{8}\\times \\frac{1}{7}=\\frac{1}{8}$<br \/>\n$P(X=3)=\\frac{7}{8}\\times \\frac{6}{7}\\times \\frac{1}{6}=\\frac{1}{8}$<br \/>\n$P(X=4)=\\frac{7}{8}\\times \\frac{6}{7}\\times \\frac{5}{6}\\times \\frac{1}{5}=\\frac{1}{8}$<br \/>\n$P(X=5)=\\frac{7}{8}\\times \\frac{6}{7}\\times \\frac{5}{6}\\times \\frac{4}{5}\\times \\frac{1}{4}=\\frac{1}{8}$<br \/>\n$P(X=6)=\\frac{7}{8}\\times \\frac{6}{7}\\times \\frac{5}{6}\\times \\frac{4}{5}\\times \\frac{3}{4}\\times \\frac{1}{3}=\\frac{1}{8}$<br \/>\n$P(X=7)=\\frac{7}{8}\\times \\frac{6}{7}\\times \\frac{5}{6}\\times \\frac{4}{5}\\times \\frac{3}{4}\\times \\frac{2}{3}\\times \\frac{1}{2}=\\frac{1}{8}$<br \/>\n$P(X=8)=\\frac{7}{8}\\times \\frac{6}{7}\\times \\frac{5}{6}\\times \\frac{4}{5}\\times \\frac{3}{4}\\times \\frac{2}{3}\\times \\frac{1}{2}\\times \\frac{1}{1}=\\frac{1}{8}$<\/p>\n<p>a distribui\u00e7\u00e3o de probabilidades \u00e9:<\/p>\n<table class=\" aligncenter\" style=\"width: 70%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">$X={{x}_{i}}$<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">1<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">2<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">3<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">4<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">5<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">6<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">7<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">8<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">$P(X={{x}_{i}})$<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">$\\frac{1}{8}$<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">$\\frac{1}{8}$<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">$\\frac{1}{8}$<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">$\\frac{1}{8}$<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">$\\frac{1}{8}$<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">$\\frac{1}{8}$<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">$\\frac{1}{8}$<\/td>\n<td style=\"text-align: center; border: #342ad4 1px solid;\" align=\"middle\">$\\frac{1}{8}$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u00ad<\/p>\n<\/li>\n<li>A esperan\u00e7a matem\u00e1tica \u00e9 \\[\\mu =\\frac{1}{8}\\times (1+2+3+4+5+6+7+8)=\\frac{1}{8}\\times \\frac{1+8}{2}\\times 8=\\frac{9}{2}=4,5\\] e o desvio padr\u00e3o \u00e9 \\[\\sigma =\\sqrt{\\frac{1}{8}\\times \\left[ {{\\left( 1-\\frac{9}{2} \\right)}^{2}}+{{\\left( 2-\\frac{9}{2} \\right)}^{2}}+&#8230;+{{\\left( 7-\\frac{9}{2} \\right)}^{2}}+{{\\left( 8-\\frac{9}{2} \\right)}^{2}} \\right]}=\\sqrt{\\frac{21}{4}}=\\frac{\\sqrt{21}}{2}\\approx 2,29\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7130' onClick='GTTabs_show(0,7130)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Um homem tem 8 chaves, das quais apenas uma abre um cofre. Sabe-se que, ap\u00f3s cada tentativa, o homem separa a chave utilizada. Calcule a probabilidade dos acontecimentos: A: &#8220;Abriu o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21020,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,245,244,246],"series":[],"class_list":["post-7130","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-probabilidades-e-combinatoria","tag-12-o-ano","tag-distribuicao-de-probabilidades","tag-variavel-aleatoria","tag-variavel-aleatoria-uniforme-discreta"],"views":1853,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/12V1Pag105-36_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7130","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7130"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7130\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21020"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7130"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7130"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7130"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7130"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}