{"id":7088,"date":"2011-10-17T01:19:15","date_gmt":"2011-10-17T00:19:15","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7088"},"modified":"2022-01-25T18:17:06","modified_gmt":"2022-01-25T18:17:06","slug":"lanca-se-um-dado-equilibrado","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7088","title":{"rendered":"Lan\u00e7a-se um dado equilibrado"},"content":{"rendered":"<p><ul id='GTTabs_ul_7088' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7088' class='GTTabs_curr'><a  id=\"7088_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7088' ><a  id=\"7088_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_7088'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/dado.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7039\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7039\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/dado.jpg\" data-orig-size=\"193,204\" 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=\"Dado\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/dado.jpg\" class=\"alignright size-full wp-image-7039\" title=\"Dado\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/dado.jpg\" alt=\"\" width=\"116\" height=\"122\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/dado.jpg 193w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/dado-141x150.jpg 141w\" sizes=\"auto, (max-width: 116px) 100vw, 116px\" \/><\/a>Lan\u00e7a-se um dado equilibrado, com as faces numeradas de 1 a 6.<\/p>\n<ol>\n<li>Considere os acontecimentos A e B.<br \/>\nA: \u00absair face par\u00bb<br \/>\nB: \u00absair um n\u00famero menor do que 4\u00bb<\/p>\n<p>Indique o valor da probabilidade condicionada $P(B|A)$.<br \/>\nJustifique a sua resposta.<\/p>\n<\/li>\n<li>Considere agora que o dado \u00e9 lan\u00e7ado tr\u00eas vezes.\n<p>Qual \u00e9 a probabilidade de a face 6 sair, pela primeira vez, precisamente no terceiro lan\u00e7amento?<br \/>\nApresente o resultado sob a forma de percentagem, arredondada \u00e0s d\u00e9cimas.<\/p>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7088' onClick='GTTabs_show(1,7088)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7088'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>$P(B|A)$ \u00e9 a probabilidade de \u00absair um n\u00famero menor do que 4\u00bb, sabendo que \u00absaiu face par\u00bb.\n<p>Ora, se \u00absaiu face par\u00bb, o conjunto dos resultados poss\u00edveis \u00e9 $\\left\\{ 2,4,6 \\right\\}$. Logo, h\u00e1 apenas um resultado favor\u00e1vel ao acontecimento \u00absair um n\u00famero menor do que 4\u00bb.<\/p>\n<p>Assim, $P(B|A)=\\frac{1}{3}$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Seja C: &#8220;sair face 6&#8221;.\n<p>A probabilidade pedida \u00e9 $P(\\overline{{{C}_{1}}}\\cap \\overline{{{C}_{2}}}\\cap {{C}_{3}})$, onde os \u00edndices se referem \u00e0 ordem do lan\u00e7amento do dado.<\/p>\n<p>Nessa experi\u00eancia existem $6\\times 6\\times 6={{6}^{3}}$ resultados poss\u00edveis e apenas $5\\times 5\\times 1={{5}^{2}}$ favor\u00e1veis.<\/p>\n<p>Logo, pela Lei de Laplace, \u00e9 $P(\\overline{{{C}_{1}}}\\cap \\overline{{{C}_{2}}}\\cap {{C}_{3}})=\\frac{25}{216}$, isto \u00e9, aproximadamente, 11,6%.<\/p>\n<p>Considerando que os acontecimentos ${{C}_{1}}$, ${{C}_{2}}$ e ${{C}_{3}}$ s\u00e3o independentes, podemos ainda calcular essa probabiladade da seguinte forma: \\[P(\\overline{{{C}_{1}}}\\cap \\overline{{{C}_{2}}}\\cap {{C}_{3}})=P(\\overline{{{C}_{1}}})\\times P(\\overline{{{C}_{2}}})\\times P({{C}_{3}})=\\frac{5}{6}\\times \\frac{5}{6}\\times \\frac{1}{6}=\\frac{25}{216}\\]<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7088' onClick='GTTabs_show(0,7088)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Lan\u00e7a-se um dado equilibrado, com as faces numeradas de 1 a 6. Considere os acontecimentos A e B. A: \u00absair face par\u00bb B: \u00absair um n\u00famero menor do que 4\u00bb Indique&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21008,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,236,215,235],"series":[],"class_list":["post-7088","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-acontecimentos-independentes","tag-probabilidade","tag-probabilidade-condicionada"],"views":7182,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/12V1Pag168-20_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\/7088","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=7088"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7088\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21008"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7088"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7088"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7088"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7088"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}