{"id":7083,"date":"2011-10-16T23:15:48","date_gmt":"2011-10-16T22:15:48","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7083"},"modified":"2022-01-25T18:08:39","modified_gmt":"2022-01-25T18:08:39","slug":"das-raparigas-que-moram-em-vale-do-rei","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7083","title":{"rendered":"Das raparigas que moram em Vale do Rei"},"content":{"rendered":"<p><ul id='GTTabs_ul_7083' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7083' class='GTTabs_curr'><a  id=\"7083_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7083' ><a  id=\"7083_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_7083'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<ol>\n<li>Seja S o espa\u00e7o de resultados associado a uma experi\u00eancia aleat\u00f3ria.<br \/>\nSejam A e B dois acontecimentos ($A\\subset S$ e $B\\subset S$).<\/p>\n<p>Prove que: $P(\\overline{A}\\cap \\overline{B})=P(\\overline{A})-P(B)+P(A|B)\\times P(B)$.<\/p>\n<\/li>\n<li>Das raparigas que moram em Vale do Rei, sabe-se que:<br \/>\n&#8211; a quarta parte tem olhos verdes;<br \/>\n&#8211; a ter\u00e7a parte tem cabelo louro;<br \/>\n&#8211; das que t\u00eam cabelo louro, metade tem olhos verdes.<\/p>\n<p>Escolhendo aleatoriamente uma rapariga de Vale do Rei, qual \u00e9 a probabilidade de ela n\u00e3o ser loura nem ter os olhos verdes?<\/p>\n<p><strong>Sugest\u00e3o<\/strong>: se lhe for \u00fatil, pode utilizar a igualdade enunciada na al\u00ednea anterior para resolver o problema.<\/p>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7083' onClick='GTTabs_show(1,7083)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7083'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Aplicando propriedades das opera\u00e7\u00f5es entre conjuntos e das probabilidades, temos: \\[\\begin{array}{*{35}{l}}<br \/>\nP(\\overline{A}\\cap \\overline{B}) &amp; = &amp; P(\\overline{A\\cup B})\u00a0 \\\\<br \/>\n{} &amp; = &amp; 1-P(A\\cup B)\u00a0 \\\\<br \/>\n{} &amp; = &amp; 1-P(A)-P(B)+P(A\\cap B)\u00a0 \\\\<br \/>\n{} &amp; = &amp; P(\\overline{A})-P(B)+P(A|B)\\times P(B)\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/li>\n<li>Sendo\u00a0A: &#8220;A rapariga tem olhos verdes&#8221; e B: &#8220;A rapariga \u00e9 loura&#8221;, a probabilidade pedida \u00e9 $P(\\overline{A}\\cap \\overline{B})$.\n<p>Sabe-se que:<br \/>\n&#8211; a quarta parte tem olhos verdes; \u2192 $P(A)=\\frac{1}{4}$<br \/>\n&#8211; a ter\u00e7a parte tem cabelo louro; \u2192 $P(B)=\\frac{1}{3}$<br \/>\n&#8211; das que t\u00eam cabelo louro, metade tem olhos verdes. \u2192 $P(A|B)=\\frac{1}{2}$<\/p>\n<p>Utilizando a igualdade da al\u00ednea anterior, temos: \\[\\begin{array}{*{35}{l}}<br \/>\nP(\\overline{A}\\cap \\overline{B}) &amp; = &amp; (1-\\frac{1}{4})-\\frac{1}{3}+\\frac{1}{2}\\times \\frac{1}{3}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{3}{4}-\\frac{1}{3}+\\frac{1}{6}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{9-4+2}{12}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{7}{12}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Portanto, a probabilidade de, escolhendo aleatoriamente uma rapariga de Vale do Rei,\u00a0ela n\u00e3o ser loura nem ter os olhos verdes \u00e9 $\\frac{7}{12}$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7083' onClick='GTTabs_show(0,7083)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Seja S o espa\u00e7o de resultados associado a uma experi\u00eancia aleat\u00f3ria. Sejam A e B dois acontecimentos ($A\\subset S$ e $B\\subset S$). Prove que: $P(\\overline{A}\\cap \\overline{B})=P(\\overline{A})-P(B)+P(A|B)\\times P(B)$. Das raparigas que moram&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21004,"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-7083","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":2773,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/12-Das_raparigas_que_moram_em_Vale_do_Rei.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7083","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=7083"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7083\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21004"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7083"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7083"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7083"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7083"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}