{"id":7258,"date":"2011-12-04T22:43:39","date_gmt":"2011-12-04T22:43:39","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7258"},"modified":"2022-01-26T01:23:36","modified_gmt":"2022-01-26T01:23:36","slug":"um-fabricante-de-bolas-de-tenis","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7258","title":{"rendered":"Um fabricante de bolas de t\u00e9nis"},"content":{"rendered":"<p><ul id='GTTabs_ul_7258' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7258' class='GTTabs_curr'><a  id=\"7258_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7258' ><a  id=\"7258_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_7258'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/tennisBalls.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7259\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7259\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/tennisBalls.jpg\" data-orig-size=\"600,400\" 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 de t\u00e9nis\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/tennisBalls.jpg\" class=\"alignright size-full wp-image-7259\" title=\"Bolas de t\u00e9nis\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/tennisBalls.jpg\" alt=\"\" width=\"216\" height=\"144\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/tennisBalls.jpg 600w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/tennisBalls-300x200.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/tennisBalls-150x100.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/tennisBalls-400x266.jpg 400w\" sizes=\"auto, (max-width: 216px) 100vw, 216px\" \/><\/a>Um fabricante de bolas de t\u00e9nis possui tr\u00eas m\u00e1quinas A, B e C que fornecem respetivamente 10%, 40% e 50% da produ\u00e7\u00e3o total da sua f\u00e1brica.<\/p>\n<p>Um estudo mostrou que a percentagem de bolas defeituosas \u00e9 3,5% para a m\u00e1quina A, 1,5% para a m\u00e1quina B e 2,2% para a m\u00e1quina C.<\/p>\n<p>Retira-se, ao acaso, uma bola de uma embalagem.<\/p>\n<ol>\n<li>Mostre que a probabilidade de que esta bola provenha da m\u00e1quina C e seja defeituosa \u00e9 $0,011$.<\/li>\n<li>Calcule a probabilidade da bola escolhida ser defeituosa.<\/li>\n<li>Calcule a probabilidade de que esta bola provenha da m\u00e1quina C, sabendo que ela \u00e9 defeituosa.<\/li>\n<li>Tiram-se, sucessivamente, de um lote, 10 bolas de t\u00e9nis, repondo de cada vez a bola retirada.<br \/>\nCalcule a probabilidade de obter, pelo menos, uma bola defeituosa nas 10 tiragens.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7258' onClick='GTTabs_show(1,7258)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7258'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<table class=\" aligncenter\" style=\"width: 50%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"background-color: #fffacd; border: #ffd700 1px solid;\">M\u00e1quina<\/td>\n<td style=\"text-align: center; background-color: #fffacd; border: #ffd700 1px solid;\"><strong>A<\/strong><\/td>\n<td style=\"text-align: center; background-color: #fffacd; border: #ffd700 1px solid;\"><strong>B<\/strong><\/td>\n<td style=\"text-align: center; background-color: #fffacd; border: #ffd700 1px solid;\"><strong>C<\/strong><\/td>\n<td style=\"text-align: center; background-color: #fffacd; border: #ffd700 1px solid;\"><strong>Total<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"background-color: #fffacd; border: #ffd700 1px solid;\">Percentagem da produ\u00e7\u00e3o<\/td>\n<td style=\"text-align: center; background-color: #fffacd; border: #ffd700 1px solid;\">10%<\/td>\n<td style=\"text-align: center; background-color: #fffacd; border: #ffd700 1px solid;\">40%<\/td>\n<td style=\"text-align: center; background-color: #fffacd; border: #ffd700 1px solid;\">50%<\/td>\n<td style=\"text-align: center; background-color: #fffacd; border: #ffd700 1px solid;\">100%<\/td>\n<\/tr>\n<tr>\n<td style=\"background-color: #fffacd; border: #ffd700 1px solid;\">Percentagem de bolas defeituosas<\/td>\n<td style=\"text-align: center; background-color: #fffacd; border: #ffd700 1px solid;\">3,5%<\/td>\n<td style=\"text-align: center; background-color: #fffacd; border: #ffd700 1px solid;\">1,5%<\/td>\n<td style=\"text-align: center; background-color: #fffacd; border: #ffd700 1px solid;\">2,2%<\/td>\n<td style=\"text-align: center; background-color: #fffacd; border: #ffd700 1px solid;\">&#8212;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>Sejam os acontecimentos:<br \/>\nA: &#8220;a bola prov\u00e9m da m\u00e1quina A&#8221;<br \/>\nB: &#8220;a bola prov\u00e9m da m\u00e1quina B&#8221;<br \/>\nC: &#8220;a bola prov\u00e9m da m\u00e1quina C&#8221;<br \/>\nD: &#8220;a bola \u00e9 defeituosa&#8221;.<\/p>\n<p>A probabilidade pedida \u00e9 $P(C\\cap D)=P(C)\\times P(D|C)=0,5\\times 0,022=0,011$.<\/p>\n<\/li>\n<li>A probabilidade pedida \u00e9 $$\\begin{array}{*{35}{l}}<br \/>\nP(D) &amp; = &amp; P(A\\cap D)+P(B\\cap D)+P(C\\cap D)\u00a0 \\\\<br \/>\n{} &amp; = &amp; P(A)\\times P(D|A)+P(B)\\times P(D|B)+P(C)\\times P(D|C)\u00a0 \\\\<br \/>\n{} &amp; = &amp; 0,1\\times 0,035+0,4\\times 0,015+0,5\\times 0,022\u00a0 \\\\<br \/>\n{} &amp; = &amp; 0,0205\u00a0 \\\\<br \/>\n\\end{array}$$<\/li>\n<li>A probabilidade pedida \u00e9 $$P(C|D)=\\frac{P(C\\cap D)}{P(D)}=\\frac{0,011}{0,0205}=\\frac{110}{205}=\\frac{22}{41}$$<\/li>\n<li>A probabilidade pedida \u00e9 $p=1-P(X=0)=1-{}^{10}{{C}_{0}}\\times {{(0,0205)}^{0}}\\times {{(1-0,0205)}^{10}}\\approx 0,19$.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7258' onClick='GTTabs_show(0,7258)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Um fabricante de bolas de t\u00e9nis possui tr\u00eas m\u00e1quinas A, B e C que fornecem respetivamente 10%, 40% e 50% da produ\u00e7\u00e3o total da sua f\u00e1brica. Um estudo mostrou que a&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21068,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,234,251,215,235],"series":[],"class_list":["post-7258","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-axiomatica","tag-distribuicao-binomial","tag-probabilidade","tag-probabilidade-condicionada"],"views":2730,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/12\/12V1Pag181-73_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7258","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=7258"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7258\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21068"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7258"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7258"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7258"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7258"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}