{"id":7237,"date":"2011-12-03T19:26:51","date_gmt":"2011-12-03T19:26:51","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7237"},"modified":"2022-01-25T23:41:47","modified_gmt":"2022-01-25T23:41:47","slug":"um-teste-de-escolha-multipla","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7237","title":{"rendered":"Um teste de escolha m\u00faltipla"},"content":{"rendered":"<p><ul id='GTTabs_ul_7237' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7237' class='GTTabs_curr'><a  id=\"7237_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7237' ><a  id=\"7237_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_7237'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/MultipleChoiceTest.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7238\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7238\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/MultipleChoiceTest.jpg\" data-orig-size=\"320,249\" 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=\"Escolha m\u00faltipla\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/MultipleChoiceTest.jpg\" class=\"alignright size-full wp-image-7238\" title=\"Escolha m\u00faltipla\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/MultipleChoiceTest.jpg\" alt=\"\" width=\"192\" height=\"149\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/MultipleChoiceTest.jpg 320w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/MultipleChoiceTest-300x233.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/MultipleChoiceTest-150x116.jpg 150w\" sizes=\"auto, (max-width: 192px) 100vw, 192px\" \/><\/a>Num teste de escolha m\u00faltipla com cinco quest\u00f5es em que, para cada quest\u00e3o, existem tr\u00eas respostas poss\u00edveis, s\u00f3 uma sendo correta, um aluno, que n\u00e3o tinha estudado, decide responder ao acaso.<\/p>\n<p>Qual a probabilidade de:<\/p>\n<ol>\n<li>n\u00e3o acertar nenhuma?<\/li>\n<li>acertar em pelo menos uma?<\/li>\n<li>acertar em todas?<\/li>\n<li>acertar em 3, no m\u00e1ximo?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7237' onClick='GTTabs_show(1,7237)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7237'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>\u00a0Cada quest\u00e3o permite tr\u00eas respostas, logo o n\u00famero de casos poss\u00edveis \u00e9 $NCP=3\\times 3\\times 3\\times 3\\times 3={}^{3}A{{&#8216;}_{5}}={{3}^{5}}=243$.<\/p>\n<ol>\n<li>O n\u00famero de casos favor\u00e1veis \u00e9 $NCF=2\\times 2\\times 2\\times 2\\times 2={}^{2}A{{&#8216;}_{5}}={{2}^{5}}=32$.<br \/>\nLogo, a probabilidade pedida \u00e9 $p=\\frac{32}{243}$.<\/li>\n<li>O n\u00famero de casos\u00a0favor\u00e1veis \u00e9 $NCF={}^{3}A{{&#8216;}_{5}}-{}^{2}A{{&#8216;}_{5}}=243-32=211$.<br \/>\n(O acontecimento \u00e9 contr\u00e1rio do da al\u00ednea anterior).<br \/>\nLogo, a probabilidade pedida \u00e9 $p=\\frac{211}{243}$.<\/li>\n<li>O n\u00famero de casos\u00a0favor\u00e1veis \u00e9 $NCF=1\\times 1\\times 1\\times 1\\times 1={}^{1}A{{&#8216;}_{5}}={{1}^{5}}=1$.<br \/>\nLogo, a probabilidade pedida \u00e9 $p=\\frac{1}{243}$.<\/li>\n<li>O n\u00famero de casos favor\u00e1veis \u00e9 $NCF=243-{}^{5}{{C}_{4}}\\times 2\\times 1\\times 1\\times 1\\times 1-{}^{5}{{C}_{5}}\\times 1\\times 1\\times 1\\times 1\\times 1-=243-10-1=232$.<br \/>\nLogo, a probabilidade pedida \u00e9 $p=\\frac{232}{243}$.<\/li>\n<\/ol>\n<p><strong>Alternativa<\/strong>:<\/p>\n<p>Consideremos a vari\u00e1vel aleat\u00f3ria $X$: &#8220;<em>n\u00famero de respostas certas \u00e0s cinco quest\u00f5es<\/em>&#8220;.<\/p>\n<p>A vari\u00e1vel aleat\u00f3ria tem distribui\u00e7\u00e3o binomial de par\u00e2metros $n=5$ e $p=\\frac{1}{3}$.<\/p>\n<p>Assim, temos:<\/p>\n<ol>\n<li>$$P(X=0)={}^{5}{{C}_{0}}\\times {{\\left( \\frac{1}{3} \\right)}^{0}}\\times {{\\left( \\frac{2}{3} \\right)}^{5}}={{\\left( \\frac{2}{3} \\right)}^{5}}=\\frac{32}{243}$$<\/li>\n<li>$$p=1-P(X=0)=1-{}^{5}{{C}_{0}}\\times {{\\left( \\frac{1}{3} \\right)}^{0}}\\times {{\\left( \\frac{2}{3} \\right)}^{5}}=\\frac{211}{243}$$<\/li>\n<li>$$P(X=5)={}^{5}{{C}_{5}}\\times {{\\left( \\frac{1}{3} \\right)}^{5}}\\times {{\\left( \\frac{2}{3} \\right)}^{0}}={{\\left( \\frac{1}{3} \\right)}^{5}}=\\frac{1}{243}$$<\/li>\n<li>$$P(X\\le 3)=1-P(4\\le X\\le 5)=1-{}^{5}{{C}_{4}}\\times {{\\left( \\frac{1}{3} \\right)}^{4}}\\times {{\\left( \\frac{2}{3} \\right)}^{1}}-{}^{5}{{C}_{5}}\\times {{\\left( \\frac{1}{3} \\right)}^{5}}\\times {{\\left( \\frac{2}{3} \\right)}^{0}}=1-\\frac{10}{243}-\\frac{1}{243}=\\frac{232}{243}$$<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7237' onClick='GTTabs_show(0,7237)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Num teste de escolha m\u00faltipla com cinco quest\u00f5es em que, para cada quest\u00e3o, existem tr\u00eas respostas poss\u00edveis, s\u00f3 uma sendo correta, um aluno, que n\u00e3o tinha estudado, decide responder ao acaso.&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21055,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,255,251,215],"series":[],"class_list":["post-7237","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-analise-combinatoria","tag-distribuicao-binomial","tag-probabilidade"],"views":6839,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12V1Pag178-64_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7237","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=7237"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7237\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21055"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7237"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7237"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7237"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7237"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}