{"id":6939,"date":"2011-09-28T00:36:28","date_gmt":"2011-09-27T23:36:28","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6939"},"modified":"2022-01-16T02:35:53","modified_gmt":"2022-01-16T02:35:53","slug":"tres-bolas-pretas-e-duas-bolas-brancas","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6939","title":{"rendered":"Tr\u00eas bolas pretas e duas bolas brancas"},"content":{"rendered":"<p><ul id='GTTabs_ul_6939' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6939' class='GTTabs_curr'><a  id=\"6939_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6939' ><a  id=\"6939_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<li id='GTTabs_li_2_6939' ><a  id=\"6939_2\" onMouseOver=\"GTTabsShowLinks('Diagrama de \u00e1rvore'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Diagrama de \u00e1rvore<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_6939'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6940\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6940\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas.jpg\" data-orig-size=\"457,84\" 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=\"Bolas\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas.jpg\" class=\"alignright size-medium wp-image-6940\" title=\"Bolas\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas-300x55.jpg\" alt=\"\" width=\"300\" height=\"55\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas-300x55.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas-150x27.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas-400x73.jpg 400w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas.jpg 457w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Um saco cont\u00e9m tr\u00eas bolas pretas e duas bolas brancas.<\/p>\n<p>Calcula a probabilidade de tirar (sem reposi\u00e7\u00e3o):<\/p>\n<ol>\n<li>uma bola branca;<\/li>\n<li>tr\u00eas bolas brancas (em 3 extra\u00e7\u00f5es consecutivas);<\/li>\n<li>tr\u00eas bolas pretas (em 3 extra\u00e7\u00f5es consecutivas);<\/li>\n<li>uma bola azul;<\/li>\n<li>uma bola branca ou preta (numa s\u00f3 extra\u00e7\u00e3o).<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6939' onClick='GTTabs_show(1,6939)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6939'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6940\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6940\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas.jpg\" data-orig-size=\"457,84\" 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=\"Bolas\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas.jpg\" class=\"aligncenter size-medium wp-image-6940\" title=\"Bolas\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas-300x55.jpg\" alt=\"\" width=\"300\" height=\"55\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas-300x55.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas-150x27.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas-400x73.jpg 400w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/5bolas.jpg 457w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<ol>\n<li>Nesta experi\u00eancia, o espa\u00e7o de resultados \u00e9 $S=\\left\\{ {{P}_{1}},{{P}_{2}},{{P}_{3}},{{B}_{1}},{{B}_{2}} \\right\\}$. Logo, $NCP=5$.<br \/>\nComo h\u00e1 duas maneiras distintas de retirar uma bola branca, ent\u00e3o $NCF=2$.<br \/>\nLogo, a probabilidade de tirar uma bola branca \u00e9 $P=\\frac{2}{5}$.<br \/>\n\u00ad<\/li>\n<li>O acontecimento considerado \u00e9 imposs\u00edvel, visto haver apenas duas bolas brancas e n\u00e3o se efetuar a reposi\u00e7\u00e3o da bola extra\u00edda. Assim, a probabilidade pedida \u00e9 nula.<br \/>\n\u00ad<\/li>\n<li>Consideremos o acontecimento X: &#8220;tirar tr\u00eas bolas pretas&#8221; (em 3 extra\u00e7\u00f5es consecutivas, com reposi\u00e7\u00e3o).<br \/>\nComecemos por identificar o conjunto dos casos favor\u00e1veis: \\[X=\\left\\{ {{P}_{1}}{{P}_{2}}{{P}_{3}},{{P}_{1}}{{P}_{3}}{{P}_{2}},{{P}_{2}}{{P}_{1}}{{P}_{3}},{{P}_{3}}{{P}_{1}}{{P}_{2}},{{P}_{2}}{{P}_{3}}{{P}_{1}},{{P}_{3}}{{P}_{2}}{{P}_{1}} \\right\\}\\]<br \/>\nLogo, $NCF=6$.<\/p>\n<p>Nesta experi\u00eancia aleat\u00f3ria, o espa\u00e7o de resultados \u00e9 bastante numeroso. Ent\u00e3o, como contar o n\u00famero de casos poss\u00edveis?<br \/>\nUma possibilidade passar\u00e1 por recorrer a um diagrama de \u00e1rvore, contudo \u00e9 uma tarefa algo demorada e, por outro lado, esse diagrama ocupar\u00e1 um espa\u00e7o significativo na folha de papel. (Experimenta!)<\/p>\n<p><strong>Vamos imaginar apenas como ser\u00e1 o aspeto desse diagrama de \u00e1rvore<\/strong>:<\/p>\n<p>Relativamente \u00e0 primeira extra\u00e7\u00e3o de uma bola do saco, surgem 5 ramos, correspondentes aos cinco resultados poss\u00edveis de obter nessa extra\u00e7\u00e3o (${{P}_{1}},{{P}_{2}},{{P}_{3}},{{B}_{1}},{{B}_{2}}$).<\/p>\n<p>Quando vamos extrair a segunda bola, h\u00e1 agora apenas quatro bolas dentro do saco, pois a primeira bola extra\u00edda n\u00e3o \u00e9 reposta. Por isso, feita a segunda extra\u00e7\u00e3o, crescem quatro ramos a partir de cada um dos cinco ramos correspondentes \u00e0 primeira extra\u00e7\u00e3o. Assim, ap\u00f3s a 2.\u00aa extra\u00e7\u00e3o a \u00e1rvore possui $5\\times 4=20$ ramos.<\/p>\n<p>Efetuada a 3.\u00aa extra\u00e7\u00e3o, crescem 3 ramos (porqu\u00ea?) a partir de cada um dos 20 ramos correspondentes \u00e0 segunda extra\u00e7\u00e3o. Logo, o diagrama em \u00e1rvore correspondente \u00e0 experi\u00eancia em causa possui $20\\times 3=60$ ramos.<\/p>\n<p>Consequentemente, o n\u00famero de casos poss\u00edveis \u00e9 $NCP=5\\times 4\\times 3=60$.<\/p>\n<p>Repara, ainda: \\[\\begin{matrix}<br \/>\nNCP &amp; = &amp; \\text{5} &amp; \\times\u00a0 &amp; 4 &amp; \\times\u00a0 &amp; 3 &amp; {} &amp; {}\u00a0 \\\\<br \/>\n{} &amp; {} &amp; \\Downarrow\u00a0 &amp; {} &amp; \\Downarrow\u00a0 &amp; {} &amp; \\Downarrow\u00a0 &amp; \\Rightarrow\u00a0 &amp; \\text{N}\\text{. }\\!\\!{}^\\text{o}\\!\\!\\text{\u00a0 de possibilidades de extrair a 3}\\text{. }\\!\\!{}^\\text{a}\\!\\!\\text{\u00a0 bola}\u00a0 \\\\<br \/>\n{} &amp; {} &amp; \\Downarrow\u00a0 &amp; {} &amp; \\Downarrow\u00a0 &amp; \\Rightarrow\u00a0 &amp; \\Rightarrow\u00a0 &amp; \\Rightarrow\u00a0 &amp; \\text{N}\\text{. }\\!\\!{}^\\text{o}\\!\\!\\text{\u00a0 de possibilidades de extrair a 2}\\text{. }\\!\\!{}^\\text{a}\\!\\!\\text{\u00a0 bola}\u00a0 \\\\<br \/>\n{} &amp; {} &amp; \\Downarrow\u00a0 &amp; \\Rightarrow\u00a0 &amp; \\Rightarrow\u00a0 &amp; \\Rightarrow\u00a0 &amp; \\Rightarrow\u00a0 &amp; \\Rightarrow\u00a0 &amp; \\text{N}\\text{. }\\!\\!{}^\\text{o}\\!\\!\\text{\u00a0 de possibilidades de extrair a 1}\\text{. }\\!\\!{}^\\text{a}\\!\\!\\text{\u00a0 bola}\u00a0 \\\\<br \/>\n\\end{matrix}\\]<\/p>\n<p>Logo, $P(X)=\\frac{6}{60}=\\frac{1}{10}$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>A probabilidade pedida \u00e9 nula, pois o acontecimento considerado \u00e9 imposs\u00edvel (o saco n\u00e3o cont\u00e9m qualquer bola azul).<br \/>\n\u00ad<\/li>\n<li>A probabilidade pedida \u00e9 1, pois o acontecimento considerado \u00e9 certo. (Porqu\u00ea?)<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6939' onClick='GTTabs_show(0,6939)'>&lt;&lt; Enunciado<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6939' onClick='GTTabs_show(2,6939)'>Diagrama de \u00e1rvore &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_2_6939'>\n<span class='GTTabs_titles'><b>Diagrama de \u00e1rvore<\/b><\/span><\/p>\n<p>Apresenta-se seguidamente o diagrama de \u00e1rvore considerado na quest\u00e3o 3:<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/3p2b.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6941\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6941\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/3p2b.jpg\" data-orig-size=\"606,872\" 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\/09\/3p2b.jpg\" class=\"aligncenter size-full wp-image-6941\" title=\"Diagrama\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/3p2b.jpg\" alt=\"\" width=\"606\" height=\"872\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/3p2b.jpg 606w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/3p2b-208x300.jpg 208w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/3p2b-104x150.jpg 104w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/3p2b-400x575.jpg 400w\" sizes=\"auto, (max-width: 606px) 100vw, 606px\" \/><\/a><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6939' onClick='GTTabs_show(1,6939)'>&lt;&lt; Resolu\u00e7\u00e3o<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Um saco cont\u00e9m tr\u00eas bolas pretas e duas bolas brancas. Calcula a probabilidade de tirar (sem reposi\u00e7\u00e3o): uma bola branca; tr\u00eas bolas brancas (em 3 extra\u00e7\u00f5es consecutivas); tr\u00eas bolas pretas (em&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20340,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[213,97,214],"tags":[426],"series":[],"class_list":["post-6939","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-9--ano","category-aplicando","category-estatistica-e-probabilidades","tag-9-o-ano"],"views":3243,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/09\/9V1Pag022-16_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6939","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=6939"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6939\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20340"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6939"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6939"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6939"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6939"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}