{"id":6999,"date":"2011-10-10T01:58:41","date_gmt":"2011-10-10T00:58:41","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6999"},"modified":"2022-01-25T13:46:44","modified_gmt":"2022-01-25T13:46:44","slug":"um-portugues-um-frances-um-ingles-e-um-belga","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6999","title":{"rendered":"Um portugu\u00eas, um franc\u00eas, um ingl\u00eas e um belga"},"content":{"rendered":"<p><ul id='GTTabs_ul_6999' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6999' class='GTTabs_curr'><a  id=\"6999_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6999' ><a  id=\"6999_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<li id='GTTabs_li_2_6999' ><a  id=\"6999_2\" onMouseOver=\"GTTabsShowLinks('Diagrama'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Diagrama<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_6999'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/4cadeiras2b.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7000\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7000\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/4cadeiras2b.jpg\" data-orig-size=\"250,120\" 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=\"4 cadeiras\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/4cadeiras2b.jpg\" class=\"alignright wp-image-7000 size-full\" title=\"4 cadeiras\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/4cadeiras2b.jpg\" alt=\"\" width=\"250\" height=\"120\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/4cadeiras2b.jpg 250w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/4cadeiras2b-150x72.jpg 150w\" sizes=\"auto, (max-width: 250px) 100vw, 250px\" \/><\/a>Considere a experi\u00eancia aleat\u00f3ria que consiste em sentar, ao acaso, um portugu\u00eas, um franc\u00eas, um ingl\u00eas e um belga em quatro cadeiras dispostas em fila e registar o modo como se dispuseram.<\/p>\n<ol>\n<li>Qual \u00e9 o espa\u00e7o de resultados desta experi\u00eancia aleat\u00f3ria?<\/li>\n<li>Supondo que os acontecimentos elementares s\u00e3o equiprov\u00e1veis, calcule a probabilidade de:\n<p>a) A: &#8220;o portugu\u00eas ficar numa das extremidades&#8221;;<\/p>\n<p>b) B: &#8220;o portugu\u00eas ficar ao lado do franc\u00eas&#8221;;<\/p>\n<p>c) C: &#8220;o portugu\u00eas ficar \u00e0 esquerda do franc\u00eas&#8221;.<\/p>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6999' onClick='GTTabs_show(1,6999)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6999'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>O espa\u00e7o de resultados \u00e9 constitu\u00eddo por $4\\times 3\\times 2\\times 1=24$ elementos, visto existirem 4 possibilidades para sentar uma pessoa na cadeira da esquerda, 3 possibilidades para sentar outra pessoa na cadeira imediatamente \u00e0 direita, 2 possibilidades para sentar outra pessoa na segunda cadeira \u00e0 direita e, finalmente, 1 possibilidade para sentar outra pessoa na cadeira da extremidade direita. O espa\u00e7o de resultados \u00e9 o seguinte conjunto:\\[\\begin{matrix}<br \/>\nS &amp; = &amp; \\{ &amp; \\text{(P}\\text{,F}\\text{,I}\\text{,B)}\\text{, (P}\\text{,F}\\text{,B}\\text{,I)}\\text{, (P}\\text{,I}\\text{,F}\\text{,B)}\\text{, (P}\\text{,I}\\text{,B}\\text{,F)}\\text{, (P}\\text{,B}\\text{,F}\\text{,I)}\\text{, (P}\\text{,B}\\text{,I}\\text{,F)}\\text{,} &amp; {} \\\\<br \/>\n{} &amp; {} &amp; {} &amp; \\text{(F}\\text{,P}\\text{,I}\\text{,B)}\\text{, (F}\\text{,P}\\text{,B}\\text{,I)}\\text{, (F}\\text{,I}\\text{,P}\\text{,B)}\\text{, (F}\\text{,I}\\text{,B}\\text{,P)}\\text{, (F}\\text{,B}\\text{,P}\\text{,I)}\\text{, (F}\\text{,B}\\text{,I}\\text{,P)}\\text{,} &amp; {} \\\\<br \/>\n{} &amp; {} &amp; {} &amp; \\text{(I}\\text{,P}\\text{,F}\\text{,B)}\\text{, (I}\\text{,P}\\text{,B}\\text{,F)}\\text{, (I}\\text{,F}\\text{,P}\\text{,B)}\\text{, (I}\\text{,F}\\text{,B}\\text{,P)}\\text{, (I}\\text{,B}\\text{,P}\\text{,F)}\\text{, (I}\\text{,B}\\text{,F}\\text{,P)}\\text{,} &amp; {} \\\\<br \/>\n{} &amp; {} &amp; {} &amp; \\text{(B}\\text{,P}\\text{,F}\\text{,I)}\\text{, (B}\\text{,P}\\text{,I}\\text{,F)}\\text{, (B}\\text{,F}\\text{,P}\\text{,I)}\\text{, (B}\\text{,F}\\text{,I}\\text{,P)}\\text{, (B}\\text{,I}\\text{,P}\\text{,F)}\\text{, (B}\\text{,I}\\text{,F}\\text{,P)} &amp; \\} \\\\<br \/>\n\\end{matrix}\\]<br \/>\nO diagrama de \u00e1rvore pode ser consultado na sec\u00e7\u00e3o seguinte.<br \/>\n\u00ad<\/li>\n<li>a)<br \/>\n&#8220;O portugu\u00eas ficar numa das extremidades&#8221; pode ocorrer em duas situa\u00e7\u00f5es distintas: &#8220;ficar na extremidade esquerda&#8221; (Situa\u00e7\u00e3o 1) ou &#8220;ficar na extremidade direita&#8221; (Situa\u00e7\u00e3o 2).<\/p>\n<p>Situa\u00e7\u00e3o 1: P __ __ __. Existem ${{N}_{1}}=1\\times 3\\times 2\\times 1=6$ resultados favor\u00e1veis a esta situa\u00e7\u00e3o.<\/p>\n<p>Situa\u00e7\u00e3o 2: __ __ __ P. Existem ${{N}_{2}}=3\\times 2\\times 1\\times 1=6$ resultados favor\u00e1veis a esta situa\u00e7\u00e3o.<\/p>\n<p>Portanto, o n\u00famero de resultados favor\u00e1veis ao acontecimento A: &#8220;o portugu\u00eas ficar numa das extremidades&#8221; \u00e9 $NCF={{N}_{1}}+{{N}_{2}}=6+6=12$.<\/p>\n<p>Logo, $P(A)=\\frac{12}{24}=\\frac{1}{2}$.<\/p>\n<p>b)<br \/>\n&#8220;O portugu\u00eas ficar ao lado do franc\u00eas&#8221; pode ocorrer em 6 situa\u00e7\u00f5es distintas:<\/p>\n<p>PF __ __, FP __ __, __ PF __ __, __ FP __, __ __ PF ou __ __ FP.<\/p>\n<p>Existem 2 resultados favor\u00e1veis para cada uma dessas situa\u00e7\u00f5es (por exemplo, para 1.\u00aa situa\u00e7\u00e3o: $N=1\\times 1\\times 2\\times 1=2$).<\/p>\n<p>Portanto, o n\u00famero de resultados favor\u00e1veis ao acontecimento B: &#8220;o portugu\u00eas ficar ao lado do franc\u00eas&#8221; \u00e9 $NCF=6\\times N=6\\times 2=12$.<\/p>\n<p>Logo, $P(B)=\\frac{12}{24}=\\frac{1}{2}$.<\/p>\n<p>c)<br \/>\n&#8220;O portugu\u00eas ficar \u00e0 esquerda do franc\u00eas&#8221; pode ocorrer em 3 situa\u00e7\u00f5es distintas: PF __ __, __ PF __ ou __ __ PF.<\/p>\n<p>Existem 2 resultados favor\u00e1veis para cada uma dessas situa\u00e7\u00f5es (por exemplo, para a 1.\u00aa situa\u00e7\u00e3o: $N=1\\times 1\\times 2\\times 1=2$).<\/p>\n<p>Portanto, o n\u00famero de resultados favor\u00e1veis ao acontecimento C: &#8220;o portugu\u00eas ficar \u00e0 esquerda do franc\u00eas&#8221; \u00e9 $NCF=3\\times N=3\\times 2=6$.<\/p>\n<p>Logo, $P(C)=\\frac{6}{24}=\\frac{1}{4}$.<\/p>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6999' onClick='GTTabs_show(0,6999)'>&lt;&lt; Enunciado<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6999' onClick='GTTabs_show(2,6999)'>Diagrama &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_2_6999'>\n<span class='GTTabs_titles'><b>Diagrama<\/b><\/span><\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/4cadeirasPFIB.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7001\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7001\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/4cadeirasPFIB.jpg\" data-orig-size=\"624,601\" 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=\"Diagrama\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/4cadeirasPFIB.jpg\" class=\"aligncenter wp-image-7001 size-full\" title=\"Diagrama\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/4cadeirasPFIB.jpg\" alt=\"\" width=\"624\" height=\"601\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/4cadeirasPFIB.jpg 624w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/4cadeirasPFIB-300x288.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/4cadeirasPFIB-150x144.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/4cadeirasPFIB-400x385.jpg 400w\" sizes=\"auto, (max-width: 624px) 100vw, 624px\" \/><\/a><\/p>\n<p style=\"text-align: center;\">\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6999' onClick='GTTabs_show(1,6999)'>&lt;&lt; Resolu\u00e7\u00e3o<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere a experi\u00eancia aleat\u00f3ria que consiste em sentar, ao acaso, um portugu\u00eas, um franc\u00eas, um ingl\u00eas e um belga em quatro cadeiras dispostas em fila e registar o modo como se&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20964,"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-6999","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":1552,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/12V1Pag166-9_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6999","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=6999"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6999\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20964"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6999"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6999"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6999"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6999"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}