{"id":7244,"date":"2011-12-03T23:39:58","date_gmt":"2011-12-03T23:39:58","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7244"},"modified":"2022-01-25T23:56:21","modified_gmt":"2022-01-25T23:56:21","slug":"atividade-terapeutica-de-um-medicamento","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7244","title":{"rendered":"Atividade terap\u00eautica de um medicamento"},"content":{"rendered":"<p><ul id='GTTabs_ul_7244' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7244' class='GTTabs_curr'><a  id=\"7244_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7244' ><a  id=\"7244_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_7244'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/pills.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7245\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7245\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/pills.jpg\" data-orig-size=\"490,245\" 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=\"Medicamento\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/pills.jpg\" class=\"alignright size-full wp-image-7245\" title=\"Medicamento\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/pills.jpg\" alt=\"\" width=\"176\" height=\"88\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/pills.jpg 490w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/pills-300x150.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/pills-150x75.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/pills-400x200.jpg 400w\" sizes=\"auto, (max-width: 176px) 100vw, 176px\" \/><\/a>A tabela seguinte refere-se aos dados obtidos nos estudos cl\u00ednicos realizados para avaliar a atividade terap\u00eautica de um medicamento.<\/p>\n<table class=\" aligncenter\" style=\"width: 70%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"background-color: #ffefd5; border: #b22222 1px solid;\"><strong>Fases da experi\u00eancia<\/strong><\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\"><strong>1.\u00aa<\/strong><\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\"><strong>2.\u00aa<\/strong><\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\"><strong>3.\u00aa<\/strong><\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\"><strong>4.\u00aa<\/strong><\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\"><strong>5.\u00aa<\/strong><\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\"><strong>6.\u00aa<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"background-color: #ffefd5; border: #b22222 1px solid;\">N.\u00ba de doentes medicados<\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\">120<\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\">235<\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\">528<\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\">822<\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\">1099<\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\">2244<\/td>\n<\/tr>\n<tr>\n<td style=\"background-color: #ffefd5; border: #b22222 1px solid;\">N.\u00ba de doentes que melhoraram<\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\">52<\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\">126<\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\">310<\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\">490<\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\">659<\/td>\n<td style=\"text-align: center; background-color: #ffefd5; border: #b22222 1px solid;\">1346<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>Com base nos resultados obtidos, os investigadores conclu\u00edram que a probabilidade de obter \u00eaxito com o referido medicamento \u00e9 de 60%.\n<p>a) Comente a conclus\u00e3o a que chegaram os investigadores, referindo a lei em que se basearam;<\/p>\n<p>b) Indique a probabilidade de que o medicamento n\u00e3o tenha \u00eaxito e demonstre a propriedade que justifica a sua resposta;<\/p>\n<p>c) Calcule a probabilidade do medicamento ter \u00eaxito em pelo menos 8 de 10 doentes tratados, escolhidos ao acaso.<\/p>\n<\/li>\n<li>Os medicamentos em ensaio neste laborat\u00f3rio s\u00e3o identificados por c\u00f3digos que obedecem \u00e0s regras seguintes:<br \/>\n&#8211; t\u00eam cinco letras seguidas de dois algarismos;<br \/>\n&#8211; come\u00e7am por vogal;<br \/>\n&#8211; n\u00e3o podem ter duas vogais nem duas consoantes seguidas;<br \/>\n&#8211; o \u00faltimo algarismo \u00e9 0 ou 1.<\/p>\n<p>a) Calcule o n\u00famero m\u00e1ximo de c\u00f3digos diferentes que obedecem a estas regras (considere 23 letras e 10 algarismos).<\/p>\n<p>b) Escolhendo um c\u00f3digo ao acaso, calcule a probabilidade dele n\u00e3o ter letras nem algarismos repetidos.<\/p>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7244' onClick='GTTabs_show(1,7244)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7244'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>\n<table class=\" aligncenter\" style=\"width: 80%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td><strong>Fases da experi\u00eancia<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>1.\u00aa<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>2.\u00aa<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>3.\u00aa<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>4.\u00aa<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>5.\u00aa<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>6.\u00aa<\/strong><\/td>\n<\/tr>\n<tr>\n<td>N.\u00ba de doentes medicados<\/td>\n<td style=\"text-align: center;\">120<\/td>\n<td style=\"text-align: center;\">235<\/td>\n<td style=\"text-align: center;\">528<\/td>\n<td style=\"text-align: center;\">822<\/td>\n<td style=\"text-align: center;\">1099<\/td>\n<td style=\"text-align: center;\">2244<\/td>\n<\/tr>\n<tr>\n<td>N.\u00ba de doentes que melhoraram<\/td>\n<td style=\"text-align: center;\">52<\/td>\n<td style=\"text-align: center;\">126<\/td>\n<td style=\"text-align: center;\">310<\/td>\n<td style=\"text-align: center;\">490<\/td>\n<td style=\"text-align: center;\">659<\/td>\n<td style=\"text-align: center;\">1346<\/td>\n<\/tr>\n<tr>\n<td>Percentagem dos\u00a0doentes que melhoraram<\/td>\n<td style=\"text-align: center;\">43,33%<\/td>\n<td style=\"text-align: center;\">53,62%<\/td>\n<td style=\"text-align: center;\">58,71%<\/td>\n<td style=\"text-align: center;\">59,61%<\/td>\n<td style=\"text-align: center;\">59,96%<\/td>\n<td style=\"text-align: center;\">59,98%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>a)<br \/>\nO conceito frequencista de probabilidade permitiu os investigadores conclur\u00edrem que \u00e9 de 60% a probabilidade de obter \u00eaxito com o referido medicamento.<\/p>\n<p>b)<br \/>\nA probabilidade de que o medicamento n\u00e3o tenha \u00eaxito \u00e9 $P(\\overline{E})=1-P(E)=1-0,6=0,4$.<br \/>\nOs acontecimentos $E$ e $\\overline{E}$ s\u00e3o contr\u00e1rios, ou seja, $E$ e $\\overline{E}$ s\u00e3o disjuntos e $E\\cup \\overline{E}=S$. Logo, $$\\begin{array}{*{35}{l}}<br \/>\nP(E\\cup \\overline{E})=P(S) &amp; \\Leftrightarrow\u00a0 &amp; P(E\\cup \\overline{E})=1\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; P(E)+P(\\overline{E})=1\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; P(\\overline{E})=1-P(E)\u00a0 \\\\<br \/>\n\\end{array}$$<br \/>\nc)<br \/>\nA vari\u00e1vel aleat\u00f3ria $X$: &#8220;<em>N\u00famero de doentes tratados com \u00eaxito, num grupo de 10 doentes<\/em>&#8221; tem distribui\u00e7\u00e3o binomial de par\u00e2metros $n=10$ e $p=\\frac{3}{5}=0,6$.<br \/>\nAssim, a probabilidade pedida \u00e9 $$\\begin{array}{*{35}{l}}<br \/>\nP(8\\le X\\le 10) &amp; = &amp; {}^{10}{{C}_{8}}\\times {{\\left( \\frac{3}{5} \\right)}^{8}}\\times {{\\left( \\frac{2}{5} \\right)}^{2}}+{}^{10}{{C}_{9}}\\times {{\\left( \\frac{3}{5} \\right)}^{9}}\\times {{\\left( \\frac{2}{5} \\right)}^{1}}+{}^{10}{{C}_{10}}\\times {{\\left( \\frac{3}{5} \\right)}^{10}}\\times {{\\left( \\frac{2}{5} \\right)}^{0}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 45\\times {{\\left( \\frac{3}{5} \\right)}^{8}}\\times {{\\left( \\frac{2}{5} \\right)}^{2}}+10\\times {{\\left( \\frac{3}{5} \\right)}^{9}}\\times {{\\left( \\frac{2}{5} \\right)}^{1}}+1\\times {{\\left( \\frac{3}{5} \\right)}^{10}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{1633689}{9765625}\u00a0 \\\\<br \/>\n{} &amp; \\approx\u00a0 &amp; 0,167\u00a0 \\\\<br \/>\n\\end{array}$$<\/p>\n<\/li>\n<li>a)<br \/>\nConjugando as resti\u00e7\u00f5es, obtemos:<\/p>\n<table class=\" aligncenter\" style=\"width: 80%;\" border=\"0\" align=\"center\">\n<caption>C\u00f3digo<\/caption>\n<tbody>\n<tr>\n<td style=\"border: #191970 1px solid;\">vogal<\/td>\n<td style=\"border: #191970 1px solid;\">consoante<\/td>\n<td style=\"border: #191970 1px solid;\">vogal<\/td>\n<td style=\"border: #191970 1px solid;\">consoante<\/td>\n<td style=\"border: #191970 1px solid;\">vogal<\/td>\n<td style=\"border: #191970 1px solid;\">algarismo<\/td>\n<td style=\"border: #191970 1px solid;\">0 ou 1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Portanto, o n\u00famero m\u00e1ximo de c\u00f3digos diferentes que obedecem a essas regras \u00e9 $N=5\\times 18\\times 5\\times 18\\times 5\\times 10\\times 2=810000$.<\/p>\n<p>b)<br \/>\nO n\u00famero de casos favor\u00e1veis \u00e9 $NCF=5\\times 18\\times 4\\times 17\\times 3\\times 9\\times 2=330480$.<br \/>\nLogo, a probabilidade pedida \u00e9 $$p=\\frac{330480}{810000}=\\frac{51}{125}=0,408$$<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7244' onClick='GTTabs_show(0,7244)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A tabela seguinte refere-se aos dados obtidos nos estudos cl\u00ednicos realizados para avaliar a atividade terap\u00eautica de um medicamento. Fases da experi\u00eancia 1.\u00aa 2.\u00aa 3.\u00aa 4.\u00aa 5.\u00aa 6.\u00aa N.\u00ba de doentes&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21059,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[255,251,215],"series":[],"class_list":["post-7244","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-probabilidades-e-combinatoria","tag-analise-combinatoria","tag-distribuicao-binomial","tag-probabilidade"],"views":2111,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12V1Pag179-67_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7244","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=7244"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7244\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21059"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7244"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7244"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7244"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7244"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}