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{"id":7003,"date":"2011-10-11T02:31:03","date_gmt":"2011-10-11T01:31:03","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7003"},"modified":"2022-01-25T13:56:41","modified_gmt":"2022-01-25T13:56:41","slug":"um-problema-que-se-deve-a-moivre","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7003","title":{"rendered":"Um problema que se deve a Moivre"},"content":{"rendered":"<p><ul id='GTTabs_ul_7003' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7003' class='GTTabs_curr'><a  id=\"7003_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7003' ><a  id=\"7003_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_7003'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Paulo lan\u00e7a duas moedas e Renato lan\u00e7a tr\u00eas. Cada jogada \u00e9 ganha por aquele que obtiver o maior n\u00famero de faces &#8220;euro&#8221;. No caso de obterem o mesmo n\u00famero, a jogada \u00e9 nula.<\/p>\n<p>Qual \u00e9 a probabilidade de Renato ganhar?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7003' onClick='GTTabs_show(1,7003)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7003'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>As experi\u00eancias individuais dos dois jogadores possuem os seguintes espa\u00e7os de resultados:<\/p>\n<ul>\n<li>${{S}_{P}}=\\left\\{ \\text{EE}\\text{, EN}\\text{, NE}\\text{, NN} \\right\\}$<\/li>\n<li>${{S}_{R}}=\\left\\{ \\text{EEE}\\text{, EEN}\\text{, ENE}\\text{, NEE}\\text{, ENN}\\text{, NEN}\\text{, NNE}\\text{, NNN} \\right\\}$<\/li>\n<\/ul>\n<p>De seguida apresenta-se uma tabela de dupla entrada, que reflete os resultados poss\u00edveis do jogo, em termos de vit\u00f3ria de cada um dos advers\u00e1rios:<\/p>\n<table class=\"aligncenter\" style=\"width: 70%;\" border=\"1\" align=\"center\">\n<tbody>\n<tr>\n<td colspan=\"2\" rowspan=\"2\"><\/td>\n<td style=\"text-align: center;\" colspan=\"8\"><span style=\"color: #ff0000;\">Renato<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">EEE<\/td>\n<td style=\"text-align: center;\">EEN<\/td>\n<td style=\"text-align: center;\">ENE<\/td>\n<td style=\"text-align: center;\">NEE<\/td>\n<td style=\"text-align: center;\">ENN<\/td>\n<td style=\"text-align: center;\">NEN<\/td>\n<td style=\"text-align: center;\">NNE<\/td>\n<td style=\"text-align: center;\">NNN<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" rowspan=\"4\"><span style=\"color: #0000ff;\">Paulo<\/span><\/td>\n<td style=\"text-align: center;\">EE<\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\"><span style=\"color: #0000ff;\">P<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #0000ff;\">P<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #0000ff;\">P<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #0000ff;\">P<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">EN<\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\"><span style=\"color: #0000ff;\">P<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">NE<\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<td style=\"text-align: center;\"><span style=\"color: #0000ff;\">P<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">NN<\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #ff0000;\">R<\/span><\/td>\n<td style=\"text-align: center;\">&#8211;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u00a0O jogo \u00e9 favor\u00e1vel ao Renato.<\/p>\n<p>A probabilidade de Renato ganhar \u00e9 $P=\\frac{16}{22}=\\frac{8}{11}$.<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7003' onClick='GTTabs_show(0,7003)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Paulo lan\u00e7a duas moedas e Renato lan\u00e7a tr\u00eas. Cada jogada \u00e9 ganha por aquele que obtiver o maior n\u00famero de faces &#8220;euro&#8221;. No caso de obterem o mesmo n\u00famero, a jogada&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20967,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,217,216,215],"series":[],"class_list":["post-7003","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-espaco-de-resultados","tag-probabilidade"],"views":1944,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/12V1Pag166-10_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7003","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=7003"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7003\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20967"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7003"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7003"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7003"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7003"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}