{"id":7053,"date":"2011-10-15T14:32:04","date_gmt":"2011-10-15T13:32:04","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7053"},"modified":"2022-01-25T17:17:11","modified_gmt":"2022-01-25T17:17:11","slug":"os-habitantes-de-vila-triste","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7053","title":{"rendered":"Os habitantes de Vila Triste"},"content":{"rendered":"<p><ul id='GTTabs_ul_7053' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7053' class='GTTabs_curr'><a  id=\"7053_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7053' ><a  id=\"7053_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_7053'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/village.gif\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7054\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7054\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/village.gif\" data-orig-size=\"482,360\" 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=\"Vila Triste\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/village.gif\" class=\"alignright size-full wp-image-7054\" title=\"Vila Triste\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/village.gif\" alt=\"\" width=\"231\" height=\"173\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/village.gif 482w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/village-300x224.gif 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/village-150x112.gif 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/village-400x298.gif 400w\" sizes=\"auto, (max-width: 231px) 100vw, 231px\" \/><\/a>Sobre os habitantes de Vila Triste com mais de 40 anos, sabe-se que:<\/p>\n<ul>\n<li>40% s\u00e3o homens;<\/li>\n<li>30% s\u00e3o analfabetos;<\/li>\n<li>um quinto dos homens s\u00e3o analfabetos.<\/li>\n<\/ul>\n<ol>\n<li>Mostre que a probabilidade do acontecimento &#8220;ser homem e n\u00e3o ser analfabeto&#8221; \u00e9 32%.<\/li>\n<li>Escolhendo ao acaso um habitante de Vila Triste, qual a probabilidade de:\n<p>a) ser homem ou analfabeto?<\/p>\n<p>b) ser analfabeto dado que \u00e9 mulher? (apresente o resultado em percentagem, arredondado \u00e0s unidades.)<\/p>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7053' onClick='GTTabs_show(1,7053)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7053'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Sejam H: &#8220;ser homem&#8221; e A: &#8220;ser analfabeto&#8221;.<\/p>\n<p>Sabe-se que:<\/p>\n<ul>\n<li>40% s\u00e3o homens; \u2192 $P(H)=0,4$<\/li>\n<li>30% s\u00e3o analfabetos; \u2192 $P(A)=0,3$<\/li>\n<li>um quinto dos homens s\u00e3o analfabetos. \u2192 $P(A|H)=\\frac{1}{5}=0,2$<\/li>\n<\/ul>\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;\">A<\/td>\n<td style=\"text-align: center;\">$\\overline{A}$<\/td>\n<td style=\"text-align: center;\"><strong>Total<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">H<\/td>\n<td style=\"text-align: center;\"><span style=\"color: #0000ff;\">0,08<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #0000ff;\">0,32<\/span><\/td>\n<td style=\"text-align: center;\">0,4<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$\\overline{H}$<\/td>\n<td style=\"text-align: center;\"><\/td>\n<td style=\"text-align: center;\"><\/td>\n<td style=\"text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>Total<\/strong><\/td>\n<td style=\"text-align: center;\">0,3<\/td>\n<td style=\"text-align: center;\"><\/td>\n<td style=\"text-align: center;\"><strong>1<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u00ad<\/p>\n<ol>\n<li>Mostre que a probabilidade do acontecimento &#8220;ser homem e n\u00e3o ser analfabeto&#8221; \u00e9 32%.\n<p>Ora, $P(\\overline{A}|H)=1-P(A|H)=1-0,2=0,8$.<\/p>\n<p>Logo, $P(H\\cap \\overline{A})=P(H)\\times P(\\overline{A}|H)=0,4\\times 0,8=0,32$.<\/p>\n<p>Portanto,\u00a0\u00a0<span style=\"color: #0000ff;\">$P(H\\cap \\overline{A})=32$%<\/span>.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Escolhendo ao acaso um habitante de Vila Triste, qual a probabilidade de:\n<p>a) ser homem ou analfabeto?<\/p>\n<p>Comecemos por determinar <span style=\"color: #0000ff;\">$P(H\\cap A)=P(H)-P(H\\cap \\overline{A})=0,4-0,32=0,08$<\/span>.<\/p>\n<p>Logo, a probabilidade pedida \u00e9 $P(H\\cup A)=P(H)+P(A)-P(H\\cap A)=0,4+0,3-0,08=0,62$.<\/p>\n<p>b) ser analfabeto dado que \u00e9 mulher? (apresente o resultado em percentagem, arredondado \u00e0s unidades.)<\/p>\n<p>Ora, \\[\\begin{array}{*{35}{l}}<br \/>\nP(A|\\overline{H}) &amp; = &amp; \\frac{P(A\\cap \\overline{H})}{P(\\overline{H})}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{P(A)-P(A\\cap H)}{1-P(H)}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{0,3-0,08}{1-0,4}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{11}{30}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Logo, $P(A|\\overline{H})\\approx 37$%.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7053' onClick='GTTabs_show(0,7053)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Sobre os habitantes de Vila Triste com mais de 40 anos, sabe-se que: 40% s\u00e3o homens; 30% s\u00e3o analfabetos; um quinto dos homens s\u00e3o analfabetos. Mostre que a probabilidade do acontecimento&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20992,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,217,215,235],"series":[],"class_list":["post-7053","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-probabilidade","tag-probabilidade-condicionada"],"views":1609,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/12V1Pag170-27_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7053","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=7053"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7053\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20992"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7053"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7053"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7053"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7053"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}