{"id":7060,"date":"2011-10-15T22:56:29","date_gmt":"2011-10-15T21:56:29","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7060"},"modified":"2022-01-25T17:37:55","modified_gmt":"2022-01-25T17:37:55","slug":"infetado-por-um-virus","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7060","title":{"rendered":"Infetado por um v\u00edrus"},"content":{"rendered":"<p><ul id='GTTabs_ul_7060' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7060' class='GTTabs_curr'><a  id=\"7060_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7060' ><a  id=\"7060_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_7060'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/virus.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7061\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7061\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/virus.jpg\" data-orig-size=\"164,170\" 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=\"V\u00edrus\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/virus.jpg\" class=\"alignright wp-image-7061 size-full\" title=\"V\u00edrus\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/virus.jpg\" alt=\"\" width=\"164\" height=\"170\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/virus.jpg 164w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/virus-144x150.jpg 144w\" sizes=\"auto, (max-width: 164px) 100vw, 164px\" \/><\/a>Sabe-se, por inqu\u00e9ritos m\u00e9dicos, que um indiv\u00edduo pertencente a uma determinada popula\u00e7\u00e3o tem probabilidade 0,01 de ser infetado por um v\u00edrus R e probabilidade 0,05 de ser contaminado pelo v\u00edrus S.<\/p>\n<p>Sabendo que estas contamina\u00e7\u00f5es s\u00e3o independentes, quantos indiv\u00edduos contaminados encontramos (aproximadamente) por um, pelo menos, destes v\u00edrus, se escolhermos ao acaso uma amostra de 10.000 indiv\u00edduos desta popula\u00e7\u00e3o?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7060' onClick='GTTabs_show(1,7060)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7060'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Sejam R: &#8220;Ser infetado pelo v\u00edrus R&#8221; e S: &#8220;Ser infetado pelo v\u00edrus S&#8221;.<\/p>\n<p>Sabe-se:<\/p>\n<ul>\n<li>$P(R)=0,01$<\/li>\n<li>$P(S)=0,05$<\/li>\n<li>R e S s\u00e3o independentes<\/li>\n<\/ul>\n<p>Como os acontecimentos R e S s\u00e3o independentes, ent\u00e3o $P(R\\cap S)=P(R)\\times P(S)=0,01\\times 0,05=0,0005$.<\/p>\n<p>A probabilidade de um desses indiv\u00edduos, escolhido ao acaso, se encontrar contaminado pelo menos por um desses v\u00edrus \u00e9 dada por:<\/p>\n<p style=\"text-align: center;\">$p=P((R\\cap \\overline{S})\\cup (\\overline{R}\\cap S)\\cup (R\\cap S))=P(R\\cup S)=1-P(\\overline{R\\cup S})=1-P(\\overline{R}\\cap \\overline{S})$.<\/p>\n<p>Ent\u00e3o, essa probabilidade \u00e9 $p=P(R\\cup S)=P(R)+P(R)-P(R\\cap S)=0,01+0,05-0,0005=0,0595$.<\/p>\n<p>Como $10000\\times p=10000\\times 0,0595=595$, \u00e9 de esperar encontrar contaminados 595 desses indiv\u00edduos.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7060' onClick='GTTabs_show(0,7060)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Sabe-se, por inqu\u00e9ritos m\u00e9dicos, que um indiv\u00edduo pertencente a uma determinada popula\u00e7\u00e3o tem probabilidade 0,01 de ser infetado por um v\u00edrus R e probabilidade 0,05 de ser contaminado pelo v\u00edrus S.&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20995,"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-7060","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":2222,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/12V1Pag171-30_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7060","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=7060"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7060\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20995"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7060"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7060"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7060"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7060"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}