{"id":7219,"date":"2011-11-28T16:34:11","date_gmt":"2011-11-28T16:34:11","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7219"},"modified":"2022-01-25T23:03:59","modified_gmt":"2022-01-25T23:03:59","slug":"num-saco-existem-doze-cartoes","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7219","title":{"rendered":"Num saco existem doze cart\u00f5es"},"content":{"rendered":"<p><ul id='GTTabs_ul_7219' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7219' class='GTTabs_curr'><a  id=\"7219_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7219' ><a  id=\"7219_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_7219'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Num saco existem doze cart\u00f5es de igual forma e material, dos quais quatro s\u00e3o verdes, quatro s\u00e3o azuis e quatro s\u00e3o pretos.<\/p>\n<p>Para cada uma das cores, os cart\u00f5es est\u00e3o numerados de 1 a 4.<\/p>\n<ol>\n<li>Retirando do saco, um a um, todos os cart\u00f5es e dispondo-os em fila, qual \u00e9 a probabilidade dos cart\u00f5es com os mesmos algarismos ficarem todos juntos?<br \/>\nApresente o resultado na forma de fra\u00e7\u00e3o irredut\u00edvel.<\/li>\n<li>Suponha que se retiraram do saco alguns cart\u00f5es.<br \/>\nSabe-se que se extrairmos, ao acaso, um cart\u00e3o do saco:<br \/>\n&#8211; a probabilidade desse cart\u00e3o ser verde \u00e9 60%;<br \/>\n&#8211; a probabilidade desse cart\u00e3o ter o n\u00famero 4 \u00e9 30%;<br \/>\n&#8211; a probabilidade desse cart\u00e3o ser verde ou ter o n\u00famero 4 \u00e9 75%.<\/p>\n<p>Numa pequena composi\u00e7\u00e3o comente a afirma\u00e7\u00e3o:<br \/>\n&#8220;Pode afirmar-se, perante as condi\u00e7\u00f5es enunciadas, que o cart\u00e3o verde n\u00famero 4 est\u00e1 no saco.&#8221;<\/p>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7219' onClick='GTTabs_show(1,7219)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7219'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<table class=\" aligncenter\" style=\"border: 0px solid currentColor; width: 80%;\" border=\"0\" cellspacing=\"2\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"background-color: #006400; border: #ffffff 1px solid;\"><span style=\"color: #ffffff;\">1<\/span><\/td>\n<td style=\"background-color: #006400; border: #ffffff 1px solid;\"><span style=\"color: #ffffff;\">2<\/span><\/td>\n<td style=\"background-color: #006400; border: #ffffff 1px solid;\"><span style=\"color: #ffffff;\">3<\/span><\/td>\n<td style=\"background-color: #006400; border: #ffffff 1px solid;\"><span style=\"color: #ffffff;\">4<\/span><\/td>\n<td style=\"background-color: #0000ff; border: #ffffff 1px solid;\"><span style=\"color: #ffffff;\">1<\/span><\/td>\n<td style=\"background-color: #0000ff; border: #ffffff 1px solid;\"><span style=\"color: #ffffff;\">2<\/span><\/td>\n<td style=\"background-color: #0000ff; border: #ffffff 1px solid;\"><span style=\"color: #ffffff;\">3<\/span><\/td>\n<td style=\"background-color: #0000ff; border: #ffffff 1px solid;\"><span style=\"color: #ffffff;\">4<\/span><\/td>\n<td style=\"background-color: #000000; border: #ffffff 1px solid;\"><span style=\"color: #ffffff;\">1<\/span><\/td>\n<td style=\"background-color: #000000; border: #ffffff 1px solid;\"><span style=\"color: #ffffff;\">2<\/span><\/td>\n<td style=\"background-color: #000000; border: #ffffff 1px solid;\"><span style=\"color: #ffffff;\">3<\/span><\/td>\n<td style=\"background-color: #000000; border: #ffffff 1px solid;\"><span style=\"color: #ffffff;\">4<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>Os doze cart\u00f5es podem ser dispostos em fila de $NCP={{P}_{12}}$ modos diferentes.\n<p>Para que os cart\u00f5es com o mesmo algarismo fiquem juntos, estes t\u00eam de ficar agrupados em 4 blocos de 3 cart\u00f5es com o mesmo algarismo. Estes 4 blocos podem ser dispostos de ${{P}_{4}}$ modos diferentes.<\/p>\n<p>Para cada uma dessas configura\u00e7\u00f5es, os tr\u00eas cart\u00f5es com o mesmo algarismo, em cada um dos 4 blocos, podem ser dispostos de ${{P}_{3}}$ modos diferentes.<\/p>\n<p>Logo, o n\u00famero de casos favor\u00e1veis \u00e9 $NCF=4!\\times 3!\\times 3!\\times 3!\\times 3!$.<br \/>\nPortanto, a probabilidade pedida \u00e9 $$p=\\frac{4!\\times 3!\\times 3!\\times 3!\\times 3!}{12!}=\\frac{1}{15400}$$<\/p>\n<\/li>\n<li>Consideremos os seguintes acontecimentos:\n<p>A: &#8220;O cart\u00e3o extra\u00eddo \u00e9 verde&#8221;;<\/p>\n<p>B: &#8220;O cart\u00e3o extra\u00eddo tem o n\u00famero 4&#8221;.<\/p>\n<p>Como sabemos, $P(A\\cup B)=P(A)+P(B)-P(A\\cap B)$. Substituindo os valores conhecidos, vem: $$\\begin{matrix}<br \/>\n0,75=0,6+0,3-P(A\\cap B) &amp; \\Leftrightarrow\u00a0 &amp; P(A\\cap B)=0,15\u00a0 \\\\<br \/>\n\\end{matrix}$$<br \/>\nOu seja, a probabilidade de o cart\u00e3o extra\u00eddo ser verde com o n\u00famero 4 \u00e9 15%.<\/p>\n<p>Logo, podemos afirmar, perante as condi\u00e7\u00f5es enunciadas, que o cart\u00e3o verde n\u00famero 4 est\u00e1 no saco.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7219' onClick='GTTabs_show(0,7219)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Num saco existem doze cart\u00f5es de igual forma e material, dos quais quatro s\u00e3o verdes, quatro s\u00e3o azuis e quatro s\u00e3o pretos. Para cada uma das cores, os cart\u00f5es est\u00e3o numerados&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21043,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,255,215],"series":[],"class_list":["post-7219","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-analise-combinatoria","tag-probabilidade"],"views":2245,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/12V1Pag176-56_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7219","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=7219"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7219\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21043"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7219"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7219"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7219"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7219"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}