{"id":7205,"date":"2011-11-26T22:56:06","date_gmt":"2011-11-26T22:56:06","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7205"},"modified":"2022-01-25T22:45:58","modified_gmt":"2022-01-25T22:45:58","slug":"uma-mao-de-5-cartas","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7205","title":{"rendered":"Uma m\u00e3o de 5 cartas"},"content":{"rendered":"<p><ul id='GTTabs_ul_7205' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7205' class='GTTabs_curr'><a  id=\"7205_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7205' ><a  id=\"7205_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_7205'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/easy-card-tricks-2.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"12283\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=12283\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/easy-card-tricks-2.jpg\" data-orig-size=\"434,356\" 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;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Cartas\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/easy-card-tricks-2.jpg\" class=\"alignright size-medium wp-image-12283\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/easy-card-tricks-2-300x246.jpg\" alt=\"Cartas\" width=\"300\" height=\"246\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/easy-card-tricks-2-300x246.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/easy-card-tricks-2.jpg 434w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Qual a probabilidade de um jogador, numa m\u00e3o de 5 cartas extra\u00eddas ao acaso de um baralho de 40 cartas, receber:<\/p>\n<ol>\n<li>o \u00e1s de copas;<\/li>\n<li>exatamente dois valetes;<\/li>\n<li>exatamente quatro ouros;<\/li>\n<li>pelo menos dois reis.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7205' onClick='GTTabs_show(1,7205)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7205'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Uma m\u00e3o de 5 cartas pode ser obtida, ao acaso,\u00a0de um baralho de 40 cartas de $$^{40}{{C}_{5}}=\\frac{40\\times 39\\times 38\\times 37\\times 36\\times 35!}{35!\\times 5!}=658008$$ maneiras diferentes.<\/p>\n<ol>\n<li>As m\u00e3os de cinco cartas que cont\u00eam o \u00e1s de copas s\u00e3o $${{N}_{1}}{{=}^{1}}{{C}_{1}}{{\\times }^{39}}{{C}_{4}}=1\\times \\frac{39\\times 38\\times 37\\times 36\\times 35!}{35!\\times 4!}=82251$$<br \/>\nLogo, a probabilidade pedida \u00e9 $$p=\\frac{^{1}{{C}_{1}}{{\\times }^{39}}{{C}_{4}}}{^{40}{{C}_{5}}}=\\frac{\\frac{39\\times 38\\times 37\\times 36\\times 35!}{35!\\times 4!}}{\\frac{40\\times 39\\times 38\\times 37\\times 36\\times 35!}{35!\\times 5!}}=\\frac{5}{40}=\\frac{1}{8}$$<\/li>\n<li>As m\u00e3os de cinco cartas que cont\u00eam dois valetes s\u00e3o $${{N}_{2}}{{=}^{4}}{{C}_{2}}{{\\times }^{36}}{{C}_{3}}=\\frac{4\\times 3\\times 2!}{2!\\times 2!}\\times \\frac{36\\times 35\\times 34\\times 33!}{33!\\times 3!}=42840$$<br \/>\nLogo, a probabilidade pedida \u00e9 $$p=\\frac{^{4}{{C}_{2}}{{\\times }^{36}}{{C}_{3}}}{^{40}{{C}_{5}}}=\\frac{\\frac{4\\times 3\\times 2!}{2!\\times 2!}\\times \\frac{36\\times 35\\times 34\\times 33!}{33!\\times 3!}}{\\frac{40\\times 39\\times 38\\times 37\\times 36\\times 35!}{35!\\times 5!}}=\\frac{36\\times 35\\times 34}{39\\times 38\\times 37\\times 12}=\\frac{595}{9139}$$<\/li>\n<li>As m\u00e3os de cinco cartas que cont\u00eam quatro ouros s\u00e3o $${{N}_{3}}{{=}^{10}}{{C}_{4}}{{\\times }^{30}}{{C}_{1}}=\\frac{10\\times 9\\times 8\\times 7\\times 6!}{6!\\times 4!}\\times 30=6300$$<br \/>\nLogo, a probabilidade pedida \u00e9 $$p=\\frac{^{10}{{C}_{4}}{{\\times }^{30}}{{C}_{1}}}{^{40}{{C}_{5}}}=\\frac{\\frac{10\\times 9\\times 8\\times 7\\times 6!}{6!\\times 4!}\\times 30}{\\frac{40\\times 39\\times 38\\times 37\\times 36\\times 35!}{35!\\times 5!}}=\\frac{10\\times 3\\times 7\\times 30}{39\\times 38\\times 37\\times 12}=\\frac{175}{18278}$$<\/li>\n<li>As m\u00e3os de cinco cartas que cont\u00eam pelo menos dois reis s\u00e3o $${{N}_{4}}{{=}^{4}}{{C}_{2}}{{\\times }^{36}}{{C}_{3}}{{+}^{4}}{{C}_{3}}{{\\times }^{36}}{{C}_{2}}{{+}^{4}}{{C}_{4}}{{\\times }^{36}}{{C}_{1}}=45396$$<br \/>\nLogo, a probabilidade pedida \u00e9 $$p=\\frac{^{10}{{C}_{4}}{{\\times }^{4}}{{C}_{2}}{{\\times }^{36}}{{C}_{3}}{{+}^{4}}{{C}_{3}}{{\\times }^{36}}{{C}_{2}}{{+}^{4}}{{C}_{4}}{{\\times }^{36}}{{C}_{1}}^{30}{{C}_{1}}}{^{40}{{C}_{5}}}=\\frac{45396}{658008}=\\frac{97}{1406}$$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7205' onClick='GTTabs_show(0,7205)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Qual a probabilidade de um jogador, numa m\u00e3o de 5 cartas extra\u00eddas ao acaso de um baralho de 40 cartas, receber: o \u00e1s de copas; exatamente dois valetes; exatamente quatro ouros;&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21039,"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],"series":[],"class_list":["post-7205","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"],"views":1892,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/12V1Pag174-47_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7205","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=7205"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7205\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21039"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7205"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7205"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7205"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7205"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}