{"id":7046,"date":"2011-10-14T16:37:52","date_gmt":"2011-10-14T15:37:52","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7046"},"modified":"2022-01-25T16:48:30","modified_gmt":"2022-01-25T16:48:30","slug":"jovens-frequentadores-de-um-ginasio","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7046","title":{"rendered":"Jovens frequentadores de um gin\u00e1sio"},"content":{"rendered":"<p><ul id='GTTabs_ul_7046' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7046' class='GTTabs_curr'><a  id=\"7046_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7046' ><a  id=\"7046_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_7046'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/surf.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7047\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7047\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/surf.png\" data-orig-size=\"250,131\" 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=\"Surf\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/surf.png\" class=\"alignright size-full wp-image-7047\" title=\"Surf\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/surf.png\" alt=\"\" width=\"250\" height=\"131\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/surf.png 250w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/surf-150x78.png 150w\" sizes=\"auto, (max-width: 250px) 100vw, 250px\" \/><\/a>Entre 80 jovens frequentadores de um gin\u00e1sio, fez-se um inqu\u00e9rito sobre a pr\u00e1tica de surf e registaram-se os resultados: vinte das trinta raparigas disseram que praticam surf e 30 rapazes disseram n\u00e3o ser praticantes de surf.<\/p>\n<ol>\n<li>Escolhendo um destes jovens ao acaso, qual \u00e9 a probabilidade de:\n<p>a) praticar surf e ser rapaz?<\/p>\n<p>b) ser rapariga, sabendo-se que faz surf?<\/p>\n<\/li>\n<li>Mostre que, neste grupo do gin\u00e1sio, praticar surf n\u00e3o \u00e9 independente dos acontecimentos ser rapaz ou ser rapariga.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7046' onClick='GTTabs_show(1,7046)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7046'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Sejam:<\/p>\n<ul>\n<li>M: &#8220;ser rapaz&#8221;;<\/li>\n<li>F: &#8220;ser rapariga&#8221;;<\/li>\n<li>S: &#8220;praticar surf&#8221;.<\/li>\n<\/ul>\n<p>Sabe-se:<\/p>\n<ul>\n<li>Entre 80 jovens&#8230;;<\/li>\n<li>vinte das trinta raparigas disseram que praticam surf ;<\/li>\n<li>30 rapazes disseram n\u00e3o ser praticantes de surf.<\/li>\n<\/ul>\n<table class=\"aligncenter\" style=\"width: 30%;\" border=\"1\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center;\"><\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">M<\/td>\n<td style=\"text-align: center;\"><strong>Total<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">S<\/td>\n<td style=\"text-align: center;\">\u00a020<\/td>\n<td style=\"text-align: center;\"><span style=\"color: #008000;\">20<\/span><\/td>\n<td style=\"text-align: center;\"><span style=\"color: #800080;\">40<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">$\\overline{S}$<\/td>\n<td style=\"text-align: center;\"><\/td>\n<td style=\"text-align: center;\">\u00a030<\/td>\n<td style=\"text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\"><strong>Total<\/strong><\/td>\n<td style=\"text-align: center;\">\u00a030<\/td>\n<td style=\"text-align: center;\"><span style=\"color: #0000ff;\">50<\/span><\/td>\n<td style=\"text-align: center;\"><strong>\u00a080<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u00ad<\/p>\n<ol>\n<li>a)<br \/>\n&#8220;Praticar surf e ser rapaz&#8221;<\/p>\n<p>Comecemos por determinar <span style=\"color: #0000ff;\">$P(M)=1-P(F)=1-\\frac{30}{80}=\\frac{5}{8}$<\/span>.<br \/>\nAssim, <span style=\"color: #0000ff;\">$\\#M=50$<\/span> e, consequentemente:<br \/>\n<span style=\"color: #008000;\">$\\#(M\\cap S)=\\#M-\\#(M\\cap \\overline{S})=50-30=20$<\/span> e<br \/>\n<span style=\"color: #800080;\">$\\#(S)=\\#(S\\cap F)+\\#(S\\cap M)=20+20=40$<\/span>.<\/p>\n<p>Logo, \\[\\begin{array}{*{35}{l}}<br \/>\nP(M\\cap S) &amp; = &amp; P(M)-P(M\\cap \\overline{S})\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{50}{80}-\\frac{30}{80}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{1}{4}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\npois os acontecimentos $(M\\cap S)$ e $(M\\cap \\overline{S})$ s\u00e3o disjuntos.<\/p>\n<p>b)<br \/>\n&#8220;Ser rapariga, sabendo-se que faz surf&#8221;<\/p>\n<p>A probabilidade pedida \u00e9: \\[\\begin{array}{*{35}{l}}<br \/>\nP(F|S) &amp; = &amp; \\frac{P(F\\cap S)}{S}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{\\frac{20}{80}}{\\frac{40}{80}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{1}{2}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n\u00ad<\/p>\n<\/li>\n<li>J\u00e1 vimos, em 1.a), que $P(M\\cap S)=\\frac{1}{4}$.<br \/>\nPor outro lado, $P(M)\\times P(S)=\\frac{50}{80}\\times \\frac{40}{80}=\\frac{5}{16}$.<br \/>\nLogo, os acontecimentos M e S n\u00e3o s\u00e3o independentes, pois $P(M\\cap S)\\ne P(M)\\times P(S)$.<\/p>\n<p>Ora, $P(F\\cap S)=\\frac{20}{80}=\\frac{1}{4}$.<br \/>\nPor outro lado, $P(F)\\times P(S)=\\frac{30}{80}\\times \\frac{40}{80}=\\frac{3}{16}$.<br \/>\nLogo, os acontecimentos F e S n\u00e3o s\u00e3o independentes, pois $P(F\\cap S)\\ne P(M)\\times P(S)$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7046' onClick='GTTabs_show(0,7046)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Entre 80 jovens frequentadores de um gin\u00e1sio, fez-se um inqu\u00e9rito sobre a pr\u00e1tica de surf e registaram-se os resultados: vinte das trinta raparigas disseram que praticam surf e 30 rapazes disseram&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20990,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,217,236,215,235],"series":[],"class_list":["post-7046","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-acontecimentos-independentes","tag-probabilidade","tag-probabilidade-condicionada"],"views":2693,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/10\/12V1Pag169-24_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7046","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=7046"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7046\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20990"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7046"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7046"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7046"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7046"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}