{"id":7027,"date":"2011-10-12T20:10:00","date_gmt":"2011-10-12T19:10:00","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7027"},"modified":"2022-01-25T15:43:57","modified_gmt":"2022-01-25T15:43:57","slug":"lancaram-se-simultaneamente-tres-dados","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7027","title":{"rendered":"Lan\u00e7aram-se simultaneamente tr\u00eas dados"},"content":{"rendered":"<p><ul id='GTTabs_ul_7027' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7027' class='GTTabs_curr'><a  id=\"7027_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7027' ><a  id=\"7027_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_7027'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/3dados.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7029\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7029\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/3dados.jpg\" data-orig-size=\"237,213\" 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=\"Dados\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/3dados.jpg\" class=\"alignright size-full wp-image-7029\" title=\"Dados\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/3dados.jpg\" alt=\"\" width=\"237\" height=\"213\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/3dados.jpg 237w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/3dados-150x134.jpg 150w\" sizes=\"auto, (max-width: 237px) 100vw, 237px\" \/><\/a>Lan\u00e7aram-se simultaneamente tr\u00eas dados: um vermelho, um verde e um azul.<\/p>\n<p>Representa-se cada lan\u00e7amento pelo terno (a,b,c) em que <strong>a<\/strong> designa a pontua\u00e7\u00e3o do dado vermelho, <strong>b<\/strong> a do dado verde e <strong>c<\/strong> a do dado azul.<\/p>\n<p>Determine:<\/p>\n<ol>\n<li>o n\u00famero de ternos diferentes que se pode obter;<\/li>\n<li>a probabilidade de $a+b+c$ ser igual a 9.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7027' onClick='GTTabs_show(1,7027)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7027'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>A pontua\u00e7\u00e3o obtida no dado vermelho pode ocorrer de 6 maneiras diferentes, o mesmo acontecendo nos dois restantes dados.<br \/>\nAssim, a, b e c assumem os valores inteiros compreendidos entre 1 e 6 inclusive.<br \/>\nLogo, podemos obter $6\\times 6\\times 6=216$ ternos ordenados.<br \/>\n\u00ad<\/li>\n<li>Comecemos por identificar os conjuntos de 1, 2 ou 3 n\u00fameros inteiros compreendidos entre 1 e 6 inclusive, que, com os seus elementos, se possa obter soma 9 com 3 parcelas: \\[\\left\\{ \\text{3} \\right\\}\\text{, }\\left\\{ \\text{1}\\text{,4} \\right\\}\\text{, }\\left\\{ \\text{2}\\text{,5} \\right\\}\\text{, }\\left\\{ \\text{1}\\text{,2}\\text{,6} \\right\\}\\text{, }\\left\\{ \\text{1}\\text{,3}\\text{,5} \\right\\}\\text{ e }\\left\\{ \\text{2}\\text{,3}\\text{,4} \\right\\}\\]Passemos a identificar esses ternos ordenados:\\[\\begin{matrix}<br \/>\n\\text{Conjunto} &amp; {} &amp; \\text{Ternos} &amp; {} &amp; \\text{N}\\text{. }\\!\\!{}^\\text{o}\\!\\!\\text{\u00a0 de ternos}\u00a0 \\\\<br \/>\n\\left\\{ 3 \\right\\} &amp; \\Rightarrow\u00a0 &amp; (3,3,3) &amp; \\to\u00a0 &amp; 1\u00a0 \\\\<br \/>\n\\left\\{ 1,4 \\right\\} &amp; \\Rightarrow\u00a0 &amp; (1,4,4),(4,1,4),(4,4,1) &amp; \\to\u00a0 &amp; 3\u00a0 \\\\<br \/>\n\\left\\{ 2,5 \\right\\} &amp; \\Rightarrow\u00a0 &amp; (2,2,5),(2,5,2),(5,2,2) &amp; \\to\u00a0 &amp; 3\u00a0 \\\\<br \/>\n\\left\\{ 1,2,6 \\right\\} &amp; \\Rightarrow\u00a0 &amp; (1,2,6),(1,6,2),(2,1,6),(6,1,2),(2,6,1),(6,2,1) &amp; \\to\u00a0 &amp; 6\u00a0 \\\\<br \/>\n\\left\\{ 1,3,5 \\right\\} &amp; \\Rightarrow\u00a0 &amp; &#8230; &amp; \\to\u00a0 &amp; 6\u00a0 \\\\<br \/>\n\\left\\{ 2,3,4 \\right\\} &amp; \\Rightarrow\u00a0 &amp; &#8230; &amp; \\to\u00a0 &amp; 6\u00a0 \\\\<br \/>\n{} &amp; {} &amp; {} &amp; {} &amp; {}\u00a0 \\\\<br \/>\n{} &amp; {} &amp; {} &amp; Total &amp; 25\u00a0 \\\\<br \/>\n\\end{matrix}\\]<br \/>\nPortanto, existem 25 casos favor\u00e1veis ao acontecimento considerado.<br \/>\nLogo, a probabilidade pedida \u00e9 $p=\\frac{25}{216}$.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7027' onClick='GTTabs_show(0,7027)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Lan\u00e7aram-se simultaneamente tr\u00eas dados: um vermelho, um verde e um azul. Representa-se cada lan\u00e7amento pelo terno (a,b,c) em que a designa a pontua\u00e7\u00e3o do dado vermelho, b a do dado verde&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20978,"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-7027","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":2561,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/12V1Pag167-17_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7027","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=7027"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7027\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20978"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7027"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7027"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7027"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7027"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}