{"id":7057,"date":"2011-10-15T21:32:37","date_gmt":"2011-10-15T20:32:37","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7057"},"modified":"2022-01-25T17:29:56","modified_gmt":"2022-01-25T17:29:56","slug":"numa-escola-secundaria","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7057","title":{"rendered":"Numa escola secund\u00e1ria"},"content":{"rendered":"<p><ul id='GTTabs_ul_7057' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7057' class='GTTabs_curr'><a  id=\"7057_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7057' ><a  id=\"7057_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<li id='GTTabs_li_2_7057' ><a  id=\"7057_2\" onMouseOver=\"GTTabsShowLinks('ES2,3 da S\u00e9 &#8211; 3D'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>ES2,3 da S\u00e9 &#8211; 3D<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_7057'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/escola2.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7059\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7059\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/escola2.jpg\" data-orig-size=\"238,127\" 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=\"Escola\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/escola2.jpg\" class=\"alignright wp-image-7059 size-full\" title=\"Escola\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/escola2.jpg\" alt=\"\" width=\"238\" height=\"127\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/escola2.jpg 238w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/escola2-150x80.jpg 150w\" sizes=\"auto, (max-width: 238px) 100vw, 238px\" \/><\/a>Numa Escola Secund\u00e1ria fez-se um estudo sobe o n\u00famero de alunos do 12.\u00ba ano que se matricularam nas disciplinas de F\u00edsica e de Qu\u00edmica, tendo-se conclu\u00eddo que:<\/p>\n<ul>\n<li>30% dos alunos matricularam-se em ambas;<\/li>\n<li>20% dos alunos matricularam-se apenas em F\u00edsica;<\/li>\n<li>40% dos alunos matricularam-se apenas em Qu\u00edmica.<\/li>\n<\/ul>\n<ol>\n<li>Construa um diagrama de <em>Venn<\/em> para ilustrar a situa\u00e7\u00e3o.<\/li>\n<li>Considere os acontecimentos F: &#8220;Matricular-se em F\u00edsica&#8221; e Q: &#8220;Matricular-se em Qu\u00edmica&#8221;.<br \/>\nDetermine a probabilidade de um desses alunos submetido ao estudo:<\/p>\n<p>a) matricular-se em F\u00edsica ou em Qu\u00edmica;<\/p>\n<p>b) matricular-se em F\u00edsica, dado que se matriculou em Qu\u00edmica;<\/p>\n<p>c) n\u00e3o se matricular em F\u00edsica, dado que se matriculou em Qu\u00edmica;<\/p>\n<p>d) matricular-se em Qu\u00edmica, dado que n\u00e3o se matriculou em F\u00edsica.<\/p>\n<\/li>\n<li>\u00c9 mais prov\u00e1vel um aluno matricular-se em F\u00edsica se se matriculou em Qu\u00edmica ou se n\u00e3o se matriculou em Qu\u00edmica?<\/li>\n<li>Justifique que os acontecimentos F e Q n\u00e3o s\u00e3o independentes.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7057' onClick='GTTabs_show(1,7057)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7057'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/diagrama-FQ.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7058\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7058\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/diagrama-FQ.jpg\" data-orig-size=\"294,163\" 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=\"Diagrama\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/diagrama-FQ.jpg\" class=\"alignright wp-image-7058 size-full\" title=\"Diagrama\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/diagrama-FQ.jpg\" alt=\"\" width=\"294\" height=\"163\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/diagrama-FQ.jpg 294w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/diagrama-FQ-150x83.jpg 150w\" sizes=\"auto, (max-width: 294px) 100vw, 294px\" \/><\/a>Apresenta-se ao lado um diagrama de <em>Venn<\/em> para ilustrar a situa\u00e7\u00e3o.<br \/>\n\u00ad<\/li>\n<li>Sejam F: &#8220;Matricular-se em F\u00edsica&#8221; e Q: &#8220;Matricular-se em Qu\u00edmica&#8221;.\n<p>a) matricular-se em F\u00edsica ou em Qu\u00edmica;<\/p>\n<p>Ora,\u00a0$P(F\\cup Q)=P(F)+P(Q)-P(F\\cap Q)=0,5+0,7-0,3=0,9$.<br \/>\nA probabilidade pedida \u00e9 90%.<\/p>\n<p>b) matricular-se em F\u00edsica, dado que se matriculou em Qu\u00edmica;<\/p>\n<p>A probabilidade pedida \u00e9: \\[P(F|Q)=\\frac{P(F\\cap Q)}{P(Q)}=\\frac{0,3}{0,7}=\\frac{3}{7}\\]<\/p>\n<p>c) n\u00e3o se matricular em F\u00edsica, dado que se matriculou em Qu\u00edmica;<\/p>\n<p>A probabilidade pedida \u00e9: \\[P(\\overline{F}|Q)=\\frac{P(\\overline{F}\\cap Q)}{P(Q)}=\\frac{0,4}{0,7}=\\frac{4}{7}\\]<\/p>\n<p>d) matricular-se em Qu\u00edmica, dado que n\u00e3o se matriculou em F\u00edsica.<\/p>\n<p>A probabilidade pedida \u00e9: \\[P(Q|\\overline{F})=\\frac{P(\\overline{F}\\cap Q)}{P(\\overline{F})}=\\frac{0,4}{1-0,5}=\\frac{4}{5}\\]<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\u00c9 mais prov\u00e1vel um aluno matricular-se em F\u00edsica se se matriculou em Qu\u00edmica ou se n\u00e3o se matriculou em Qu\u00edmica?\n<p>J\u00e1 vimos que $P(F|Q)=\\frac{3}{7}$.<br \/>\nPor outro lado: \\[P(F|\\overline{Q})=\\frac{P(F\\cap \\overline{Q})}{P(\\overline{Q})}=\\frac{0,2}{1-0,7}=\\frac{2}{3}\\]<br \/>\nLogo, \u00e9 mais prov\u00e1vel um aluno matricular-se em F\u00edsica se se n\u00e3o matriculou em Qu\u00edmica, pois $P(F|\\overline{Q})&gt;P(F|Q)$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Justifique que os acontecimentos F e Q n\u00e3o s\u00e3o independentes.\n<p>Como $P(F\\cap Q)=0,3$ e $P(F)\\times P(Q)=0,5\\times 0,7=0,35$, ent\u00e3o $P(F\\cap Q)\\ne P(F)\\times P(Q)$ e, por isso, os acontecimentos F e Q n\u00e3o s\u00e3o independentes.<\/p>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7057' onClick='GTTabs_show(0,7057)'>&lt;&lt; Enunciado<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7057' onClick='GTTabs_show(2,7057)'>ES2,3 da S\u00e9 &#8211; 3D &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_2_7057'>\n<span class='GTTabs_titles'><b>ES2,3 da S\u00e9 &#8211; 3D<\/b><\/span><\/p>\n<p><style>.embed-container { position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden; max-width: 100%; } .embed-container iframe, .embed-container object, .embed-container embed { position: absolute; top: 0; left: 0; width: 100%; height: 100%; }<\/style>\n<\/p>\n<div class=\"embed-container\"><iframe src=\"https:\/\/www.youtube-nocookie.com\/embed\/fCGAiU4Ndc4\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><br \/>\n\u00ad<\/div>\n<p>\u00ad<br \/>\nTrabalho realizado por Jo\u00e3o Melo, que frequenta(ou) a Escola Secund\u00e1ria\/2,3 da S\u00e9, em Lamego.<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7057' onClick='GTTabs_show(1,7057)'>&lt;&lt; Resolu\u00e7\u00e3o<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Numa Escola Secund\u00e1ria fez-se um estudo sobe o n\u00famero de alunos do 12.\u00ba ano que se matricularam nas disciplinas de F\u00edsica e de Qu\u00edmica, tendo-se conclu\u00eddo que: 30% dos alunos matricularam-se&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20994,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,217,236,215,235],"series":[],"class_list":["post-7057","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-acontecimento","tag-acontecimentos-independentes","tag-probabilidade","tag-probabilidade-condicionada"],"views":2733,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/12V1Pag171-29_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7057","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=7057"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7057\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20994"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7057"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7057"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7057"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7057"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}