{"id":7045,"date":"2011-10-14T13:28:13","date_gmt":"2011-10-14T12:28:13","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7045"},"modified":"2022-01-25T16:42:29","modified_gmt":"2022-01-25T16:42:29","slug":"estudo-de-linguas-estrangeiras-2","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7045","title":{"rendered":"Estudo de l\u00ednguas estrangeiras (2)"},"content":{"rendered":"<p><ul id='GTTabs_ul_7045' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7045' class='GTTabs_curr'><a  id=\"7045_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7045' ><a  id=\"7045_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_7045'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/alunos.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7044\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7044\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/alunos.jpg\" data-orig-size=\"301,343\" 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=\"Alunos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/alunos.jpg\" class=\"alignright size-full wp-image-7044\" title=\"Alunos\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/alunos.jpg\" alt=\"\" width=\"181\" height=\"206\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/alunos.jpg 301w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/alunos-263x300.jpg 263w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/alunos-131x150.jpg 131w\" sizes=\"auto, (max-width: 181px) 100vw, 181px\" \/><\/a>Numa turma, todos os alunos estudam duas, e s\u00f3 duas, l\u00ednguas estrangeiras: ou Ingl\u00eas (I) e Alem\u00e3o (A) ou Ingl\u00eas (I) e Espanhol (E).<\/p>\n<ul>\n<li>60% dos alunos da turma estudam Alem\u00e3o;<\/li>\n<li>75% dos alunos da turma s\u00e3o raparigas;<\/li>\n<li>metade dos alunos da turma de Espanhol s\u00e3o rapazes.<\/li>\n<\/ul>\n<ol>\n<li>Calcule a probabilidade dos acontecimentos:\n<p>a) M: &#8220;o aluno \u00e9 um rapaz&#8221;;<\/p>\n<p>b) E: &#8220;o aluno estuda Espanhol&#8221;;<\/p>\n<p>c) $F\\cap A$: &#8220;o aluno \u00e9 uma rapariga que estuda alem\u00e3o&#8221;.<\/p>\n<\/li>\n<li>Determine a probabilidade de interrogar:\n<p>a) um estudante de Alem\u00e3o, sabendo que \u00e9 rapariga;<\/p>\n<p>b) uma rapariga, sabendo que estuda alem\u00e3o.<\/p>\n<\/li>\n<li>No corredor passamos por um rapaz da classe.<br \/>\nQual \u00e9 a probabilidade de ele estudar Espanhol?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7045' onClick='GTTabs_show(1,7045)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7045'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ul>\n<li>60% dos alunos da turma estudam Alem\u00e3o; \u2192 $P(A)=0,6$<\/li>\n<li>75% dos alunos da turma s\u00e3o raparigas; \u2192 $P(F)=0,75$<\/li>\n<li>metade dos alunos da turma de Espanhol s\u00e3o rapazes. \u2192 $P(M|E)=0,5$<\/li>\n<\/ul>\n<p>No sentido de facilitar a resolu\u00e7\u00e3o do problema, vamos usar a tabela seguinte, onde se anotaram os dados, alguns dos valores calculados\u00a0e foi feita a indica\u00e7\u00e3o da ordem de alguns c\u00e1lculos no sentido da determina\u00e7\u00e3o de $P(F\\cap A)$:<\/p>\n<table class=\"aligncenter\" style=\"width: 30%;\" border=\"1\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><\/td>\n<td style=\"text-align: center;\"><strong>M<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>F<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>Total<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>A<\/strong><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #000000;\">(2)<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #000000;\">(?)<\/span><\/td>\n<td style=\"text-align: center;\">60%<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>E<\/strong><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #000000;\">(1)<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #000000;\">\u00a0<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #0000ff;\">40%<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>Total<\/strong><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #0000ff;\">25%<\/span><\/td>\n<td style=\"text-align: center;\">75%<\/td>\n<td style=\"text-align: center;\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u00ad<\/p>\n<ol>\n<li>a)<br \/>\nM: &#8220;o aluno \u00e9 um rapaz&#8221;.<\/p>\n<p>A probabilidade pedida \u00e9 <span style=\"color: #0000ff;\">$P(M)=1-P(F)=1-0,75=0,25$<\/span>.<\/p>\n<p>b)<br \/>\nE: &#8220;o aluno estuda Espanhol&#8221;.<\/p>\n<p>A probabilidade pedida \u00e9 <span style=\"color: #0000ff;\">$P(E)=1-P(A)=1-0,6=0,4$<\/span>.<\/p>\n<p>c)<br \/>\n$F\\cap A$: &#8220;o aluno \u00e9 uma rapariga que estuda alem\u00e3o&#8221;.<\/p>\n<p>J\u00e1 que sabemos $P(M|E)=0,5$, podemos seguir a sequ\u00eancia (1), (2) e (?).<br \/>\nOra, \\[\\begin{array}{*{35}{l}}<br \/>\nP(M\\cap E) &amp; = &amp; P(E)\\times P(M|E) &amp; {} &amp; (1)\u00a0 \\\\<br \/>\n{} &amp; = &amp; 0,4\\times 0,5 &amp; {} &amp; {}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 0,2 &amp; {} &amp; {}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\ne, como $P(M)=P((A\\cap M)\\cup (E\\cap M))=P(A\\cap M)+P(E\\cap M)$, pois $(A\\cap M)$ e $(E\\cap M)$ s\u00e3o acontecimentos disjuntos, vem: \\[\\begin{array}{*{35}{l}}<br \/>\nP(A\\cap M) &amp; = &amp; P(M)-P(E\\cap M) &amp; {} &amp; (2)\u00a0 \\\\<br \/>\n{} &amp; = &amp; 0,25-0,2 &amp; {} &amp; {}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 0,05 &amp; {} &amp; {}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nAssim, considerando tamb\u00e9m que os acontecimentos $(F\\cap A)$ e $(M\\cap A)$ s\u00e3o disjuntos, temos: \\[\\begin{array}{*{35}{l}}<br \/>\nP(F\\cap A) &amp; = &amp; P(A)-P(M\\cap A) &amp; {} &amp; (?)\u00a0 \\\\<br \/>\n{} &amp; = &amp; 0,6-0,05 &amp; {} &amp; {}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 0,55 &amp; {} &amp; {}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nPortanto a probabilidade pedida \u00e9\u00a0$P(F\\cap A)=0,55$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>a)<br \/>\n&#8220;um estudante de Alem\u00e3o, sabendo que \u00e9 rapariga&#8221;.<\/p>\n<p>A probabilidade pedida \u00e9: \\[P(A|F)=\\frac{P(A\\cap F)}{P(F)}=\\frac{0,55}{0,75}=\\frac{11}{15}\\]<\/p>\n<p>b)<br \/>\n&#8220;uma rapariga, sabendo que estuda alem\u00e3o&#8221;.<\/p>\n<p>A probabilidade pedida \u00e9: \\[P(F|A)=\\frac{P(A\\cap F)}{P(A)}=\\frac{0,55}{0,6}=\\frac{11}{12}\\]<br \/>\n\u00ad<\/p>\n<\/li>\n<li>No corredor passamos por um rapaz da classe.<br \/>\nQual \u00e9 a probabilidade de ele estudar Espanhol?<\/p>\n<p>A probabilidade pedida \u00e9: \\[P(E|M)=\\frac{P(E\\cap M)}{P(M)}=\\frac{0,2}{0,25}=\\frac{4}{5}\\]<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7045' onClick='GTTabs_show(0,7045)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Numa turma, todos os alunos estudam duas, e s\u00f3 duas, l\u00ednguas estrangeiras: ou Ingl\u00eas (I) e Alem\u00e3o (A) ou Ingl\u00eas (I) e Espanhol (E). 60% dos alunos da turma estudam Alem\u00e3o;&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20988,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,217,216,215,235],"series":[],"class_list":["post-7045","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-espaco-de-resultados","tag-probabilidade","tag-probabilidade-condicionada"],"views":2353,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/12V1Pag168-22_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\/7045","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=7045"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7045\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20988"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7045"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7045"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7045"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7045"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}