{"id":7021,"date":"2011-10-12T17:08:40","date_gmt":"2011-10-12T16:08:40","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7021"},"modified":"2022-01-25T15:14:26","modified_gmt":"2022-01-25T15:14:26","slug":"uma-urna-contem-seis-bolas-numeradas","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7021","title":{"rendered":"Uma urna cont\u00e9m seis bolas numeradas"},"content":{"rendered":"<p><ul id='GTTabs_ul_7021' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7021' class='GTTabs_curr'><a  id=\"7021_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7021' ><a  id=\"7021_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_7021'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/Bolas-3v3p.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7022\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7022\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/Bolas-3v3p.jpg\" data-orig-size=\"235,156\" 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\/10\/Bolas-3v3p.jpg\" class=\"alignright size-full wp-image-7022\" title=\"Bolas\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/Bolas-3v3p.jpg\" alt=\"\" width=\"235\" height=\"156\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/Bolas-3v3p.jpg 235w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/Bolas-3v3p-150x99.jpg 150w\" sizes=\"auto, (max-width: 235px) 100vw, 235px\" \/><\/a>Uma urna cont\u00e9m seis bolas numeradas: tr\u00eas vermelhas, respetivamente, com os n\u00fameros 1, 3 e 5 e tr\u00eas pretas com os n\u00fameros 2, 4 e 6, respetivamente.<\/p>\n<p>Tiram-se, ao acaso, sucessivamente e sem reposi\u00e7\u00e3o, duas bolas da urna para formar um n\u00famero: a primeira bola extra\u00edda indica o algarismo das unidades e a segunda o algarismos das dezenas.<\/p>\n<ol>\n<li>Efetuando todas as extra\u00e7\u00f5es poss\u00edveis, quantos n\u00fameros diferentes podemos escrever? &nbsp;<\/li>\n<li>Qual a probabilidade do n\u00famero ser formado por bolas de cores diferentes? &nbsp;<\/li>\n<li>Qual a a probabilidade do n\u00famero ser divis\u00edvel por 3?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7021' onClick='GTTabs_show(1,7021)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7021'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/Bolas-3v3p.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignright\" title=\"Bolas\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/Bolas-3v3p.jpg\" alt=\"\" width=\"235\" height=\"156\"><\/a>A primeira bola pode ser extra\u00edda de 6 maneiras distintas e a segunda bola pode ser extra\u00edda de 5 maneiras distintas.<br>Logo, existem $6\\times 5=30$ maneiras de extrair sucessivamente duas bolas da urna, sem reposi\u00e7\u00e3o.<br>Como a cada bola corresponde um algarismo (significativo) diferente, podemos escrever 30 n\u00fameros diferentes, efetuadas todas as extra\u00e7\u00f5es poss\u00edveis.<br>\u00ad<\/li>\n<li>Podemos obter 2 bolas de cores diferentes em duas situa\u00e7\u00f5es distintas: VP e PV.<br>Cada uma dessas situa\u00e7\u00f5es pode ser obtida de $3\\times 3=9$ maneiras distintas (H\u00e1 3 possibilidades de tirar a 1.\u00aa bola de uma determinada cor, bem como de tirar a 2.\u00aa bola da outra cor).<br>Logo, existem 18 casos favor\u00e1veis a que o n\u00famero seja formado por bolas de cores diferentes.<br>Logo, a probabilidade pedida \u00e9 $p=\\frac{18}{30}=\\frac{3}{5}$.<br>\u00ad<\/li>\n<li>Ora, um n\u00famero \u00e9 divis\u00edvel por 3 quando a soma dos seus algarismos \u00e9 um m\u00faltiplo de tr\u00eas.<br>Os casos favor\u00e1veis s\u00e3o 10: $\\text{(1}\\text{,2)}\\text{, (1}\\text{,5)}\\text{, (2}\\text{,1)}\\text{, (2}\\text{,4)}\\text{, (3}\\text{,6)}\\text{, (4}\\text{,2)}\\text{, (4}\\text{,5)}\\text{, (5}\\text{,1)}\\text{, (5}\\text{,4) e (6}\\text{,3)}$.<br>Nota: (3,3) n\u00e3o \u00e9 resultado poss\u00edvel. (Porqu\u00ea?)<br>Logo, a probabilidade pedida \u00e9 $p=\\frac{10}{30}=\\frac{1}{3}$.<\/li>\n<\/ol>\n\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7021' onClick='GTTabs_show(0,7021)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Uma urna cont\u00e9m seis bolas numeradas: tr\u00eas vermelhas, respetivamente, com os n\u00fameros 1, 3 e 5 e tr\u00eas pretas com os n\u00fameros 2, 4 e 6, respetivamente. Tiram-se, ao acaso, sucessivamente&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20973,"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-7021","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":3741,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/12V1Pag167-14_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7021","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=7021"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7021\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20973"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7021"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7021"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7021"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7021"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}