{"id":7265,"date":"2011-12-07T00:06:49","date_gmt":"2011-12-07T00:06:49","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7265"},"modified":"2022-01-26T01:40:27","modified_gmt":"2022-01-26T01:40:27","slug":"uma-sala-de-espetaculos","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7265","title":{"rendered":"Uma sala de espet\u00e1culos"},"content":{"rendered":"<p><ul id='GTTabs_ul_7265' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7265' class='GTTabs_curr'><a  id=\"7265_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7265' ><a  id=\"7265_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_7265'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<div id=\"attachment_7266\" style=\"width: 310px\" class=\"wp-caption alignright\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Teatro+Ribeiro+Concei\u00e7\u00e3o2.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-7266\" data-attachment-id=\"7266\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7266\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Teatro+Ribeiro+Concei\u00e7\u00e3o2.jpg\" data-orig-size=\"500,310\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;Picasa 3.0&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=\"Teatro Ribeiro Concei\u00e7\u00e3o\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Teatro+Ribeiro+Concei\u00e7\u00e3o2.jpg\" class=\"wp-image-7266 size-medium\" title=\"Teatro Ribeiro Concei\u00e7\u00e3o\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Teatro+Ribeiro+Concei\u00e7\u00e3o2-300x186.jpg\" alt=\"\" width=\"300\" height=\"186\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Teatro+Ribeiro+Concei\u00e7\u00e3o2-300x186.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Teatro+Ribeiro+Concei\u00e7\u00e3o2-150x93.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Teatro+Ribeiro+Concei\u00e7\u00e3o2-400x248.jpg 400w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/Teatro+Ribeiro+Concei\u00e7\u00e3o2.jpg 500w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><p id=\"caption-attachment-7266\" class=\"wp-caption-text\">Teatro Ribeiro Concei\u00e7\u00e3o &#8211; Lamego<\/p><\/div>\n<p>Uma sala de espet\u00e1culos prop\u00f5e para a \u00e9poca a compra de bilhetes para 4, 5 ou 6 espet\u00e1culos, a pre\u00e7os especiais.<\/p>\n<p>De entre o conjunto de pessoas que compraram os bilhetes especiais, a reparti\u00e7\u00e3o foi a seguinte:<\/p>\n<ul>\n<li>43,5% escolheram a compra de 4 espet\u00e1culos;<\/li>\n<li>33% escolheram a compra de 5 espet\u00e1culos;<\/li>\n<li>os restantes escolheram a compra de 6 espet\u00e1culos.<\/li>\n<\/ul>\n<p>Por outro lado, 65% dos compradores s\u00e3o jovens com menos de 25 anos e nessa popula\u00e7\u00e3o a reparti\u00e7\u00e3o \u00e9 a seguinte:<\/p>\n<ul>\n<li>40% compraram uma s\u00e9rie de 4 espet\u00e1culos;<\/li>\n<li>40% compraram uma s\u00e9rie de 5 espet\u00e1culos;<\/li>\n<li>os restantes compraram uma s\u00e9rie de 6 espet\u00e1culos.<\/li>\n<\/ul>\n<p>Interroga-se um qualquer dos compradores, ao acaso. Considere os acontecimentos:<\/p>\n<ul>\n<li>A: &#8220;<em>o comprador inquirido tem menos de 25 anos<\/em>&#8220;;<\/li>\n<li>B: &#8220;<em>o comprador inquirido comprou uma s\u00e9rie de 5 espet\u00e1culos<\/em>&#8220;.<\/li>\n<\/ul>\n<ol>\n<li>a) Qual a probabilidade do comprador inquirido ter 25 anos ou mais?\n<p>b) Sabendo que o comprador inquirido tem menos de 25 anos, qual \u00e9 a probabilidade de que ele tenha escolhido a s\u00e9rie de 5 espet\u00e1culos?<\/p>\n<p>c) Descreva o acontecimento $A\\cap B$ e demonstre que a probabilidade deste acontecimento \u00e9 igual a $0,26$.<\/p>\n<\/li>\n<li>a) Demonstre que $P(\\overline{A}\\cap B)=0,07$.\n<p>b) Indique a probabilidade condicionada de B, sabendo que $\\overline{A}$ se realizou.<\/p>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7265' onClick='GTTabs_show(1,7265)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7265'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Comecemos por tabelar os dados:<\/p>\n<table class=\" aligncenter\" style=\"width: 70%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #696969 1px solid;\"><strong>N.\u00ba de espet\u00e1culos comprados<\/strong><\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\"><strong>4<\/strong><\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\"><strong>5<\/strong><\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\"><strong>6<\/strong><\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">Total<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #696969 1px solid;\">Percentagem de compradores<\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">43,5%<\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">33,0%<\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">23,5%<\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">100%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\" aligncenter\" style=\"width: 70%;\" border=\"0\" align=\"center\">\n<caption><strong>65% dos compradores s\u00e3o jovens com menos de 25 anos<\/strong><\/caption>\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #696969 1px solid;\"><strong>N.\u00ba de espet\u00e1culos comprados<\/strong><\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\"><strong>4<\/strong><\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\"><strong>5<\/strong><\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\"><strong>6<\/strong><\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">Total<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #696969 1px solid;\">Percentagem de compradores<\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">40%<\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">40%<\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">20%<\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">100%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ul>\n<li>A: &#8220;<em>o comprador inquirido tem menos de 25 anos<\/em>&#8220;;<\/li>\n<li>B: &#8220;<em>o comprador inquirido comprou uma s\u00e9rie de 5 espet\u00e1culos<\/em>&#8220;.<\/li>\n<\/ul>\n<ol>\n<li>a)<br \/>\nComo $P(A)=0,65$, ent\u00e3o $P(\\overline{A})=1-0,65=0,35$.<\/p>\n<p>b)<br \/>\nPretende-se calcular $P(B|A)$.<br \/>\nPor interpreta\u00e7\u00e3o da probabilidade condicionada, resulta $P(B|A)=0,4$.<\/p>\n<p>c)<br \/>\n$A\\cap B$: &#8220;<em>o comprador inquirido tem menos de 25 anos e comprou uma s\u00e9rie de 5 espet\u00e1culos<\/em>&#8220;.<\/p>\n<p>Ora, $P(A\\cap B)=P(A)\\times P(B|A)=0,65\\times 0,4=0,26$.<\/p>\n<\/li>\n<li>a)<br \/>\nRegistemos, numa tabela de dupla entrada, algumas das probabilidades j\u00e1 conhecidas:<\/p>\n<table class=\" aligncenter\" style=\"width: 70%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #696969 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">$B$<\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">$\\overline{B}$<\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\"><strong>Total\u00a0<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #696969 1px solid;\">$A$<\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">$0,26$<\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">$0,65$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #696969 1px solid;\">$\\overline{A}$<\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">$0,35$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #696969 1px solid;\"><strong>Total<\/strong><\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\">$0,33$<\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #696969 1px solid;\"><strong>$1$<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Ora, $P(\\overline{A}\\cap B)=P(B)-P(A\\cap B)=0,33-0,26=0,07$.<\/p>\n<p>c)<br \/>\nA probabilidade pedida \u00e9 $$P(B|\\overline{A})=\\frac{P(\\overline{A}\\cap B)}{P(\\overline{A})}=\\frac{0,07}{0,35}=\\frac{7}{35}=\\frac{1}{5}$$<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7265' onClick='GTTabs_show(0,7265)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Uma sala de espet\u00e1culos prop\u00f5e para a \u00e9poca a compra de bilhetes para 4, 5 ou 6 espet\u00e1culos, a pre\u00e7os especiais. De entre o conjunto de pessoas que compraram os bilhetes&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21074,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,234,215,235],"series":[],"class_list":["post-7265","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-axiomatica","tag-probabilidade","tag-probabilidade-condicionada"],"views":3177,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12V1Pag182-76_520x245-1.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7265","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=7265"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7265\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21074"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7265"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7265"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7265"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}