{"id":5586,"date":"2026-01-16T13:35:33","date_gmt":"2026-01-16T16:35:33","guid":{"rendered":"https:\/\/colegadeclasse.com.br\/blog\/?p=5586"},"modified":"2026-01-16T13:35:36","modified_gmt":"2026-01-16T16:35:36","slug":"probabilidade-conceitos-fundamentais-para-concursos-publicos","status":"publish","type":"post","link":"https:\/\/colegadeclasse.com.br\/blog\/2026\/01\/16\/probabilidade-conceitos-fundamentais-para-concursos-publicos\/","title":{"rendered":"Probabilidade: Conceitos Fundamentais para Concursos P\u00fablicos"},"content":{"rendered":"<div style=\"display:flex; gap:10px;justify-content:flex-end\" class=\"wps-pgfw-pdf-generate-icon__wrapper-frontend\">\n\t\t<a  href=\"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/posts\/5586?action=genpdf&amp;id=5586\" class=\"pgfw-single-pdf-download-button\" ><img src=\"https:\/\/colegadeclasse.com.br\/blog\/wp-content\/plugins\/pdf-generator-for-wp\/admin\/src\/images\/PDF_Tray.svg\" title=\"Gerar PDF  \" style=\"width:auto; height:45px;\"><\/a>\n\t\t<\/div>\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"\">A probabilidade \u00e9 o ramo da Matem\u00e1tica que estuda a chance de ocorr\u00eancia de determinados eventos. Em termos pr\u00e1ticos, \u00e9 a medida num\u00e9rica da possibilidade de algo acontecer, variando sempre entre 0 (evento imposs\u00edvel) e 1 (evento certo), podendo ser expressa tamb\u00e9m em forma percentual (0% a 100%).<\/p>\n\n\n\n<p class=\"\">A teoria das probabilidades surgiu no s\u00e9culo XVII com os estudos de Blaise Pascal e Pierre de Fermat sobre jogos de azar, mas hoje suas aplica\u00e7\u00f5es v\u00e3o muito al\u00e9m, permeando \u00e1reas como estat\u00edstica, f\u00edsica, economia, medicina e, evidentemente, quest\u00f5es de concursos p\u00fablicos.<\/p>\n\n\n\n<p class=\"ptt-red\">Em concursos, probabilidade \u00e9 frequentemente cobrada em provas de Racioc\u00ednio L\u00f3gico, Matem\u00e1tica e Estat\u00edstica. Dominar este conte\u00fado pode garantir pontos preciosos, pois as quest\u00f5es seguem padr\u00f5es recorrentes.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">Conceitos Fundamentais<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Experimento Aleat\u00f3rio<\/h3>\n\n\n\n<p class=\"\">\u00c9 qualquer processo cujo resultado n\u00e3o pode ser previsto com certeza antes de sua realiza\u00e7\u00e3o. Exemplos cl\u00e1ssicos:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">Lan\u00e7amento de um dado<\/li>\n\n\n\n<li class=\"\">Lan\u00e7amento de uma moeda<\/li>\n\n\n\n<li class=\"\">Retirada de uma carta de um baralho<\/li>\n\n\n\n<li class=\"\">Sorteio de n\u00fameros em uma loteria<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Espa\u00e7o Amostral (\u03a9)<\/h3>\n\n\n\n<p class=\"\">\u00c9 o conjunto de todos os resultados poss\u00edveis de um experimento aleat\u00f3rio.<\/p>\n\n\n\n<p class=\"\"><strong>Exemplos:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">Lan\u00e7amento de uma moeda: \u03a9 = {cara, coroa}<\/li>\n\n\n\n<li class=\"\">Lan\u00e7amento de um dado: \u03a9 = {1, 2, 3, 4, 5, 6}<\/li>\n\n\n\n<li class=\"\">Nascimento de um beb\u00ea (quanto ao sexo): \u03a9 = {masculino, feminino}<\/li>\n<\/ul>\n\n\n\n<p class=\"ptt-white\"><strong>\u26a0\ufe0f OBSERVA\u00c7\u00c3O <\/strong>probabilidade. Este \u00e9 o erro mais comum em provas!<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Evento<\/h3>\n\n\n\n<p class=\"\">\u00c9 qualquer subconjunto do espa\u00e7o amostral. Representa um resultado espec\u00edfico ou conjunto de resultados que nos interessa.<\/p>\n\n\n\n<p class=\"\"><strong>Exemplos:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">Evento A: &#8220;obter n\u00famero par no lan\u00e7amento de um dado&#8221; = {2, 4, 6}<\/li>\n\n\n\n<li class=\"\">Evento B: &#8220;obter n\u00famero maior que 4&#8221; = {5, 6}<\/li>\n\n\n\n<li class=\"\">Evento C: &#8220;obter cara no lan\u00e7amento de moeda&#8221; = {cara}<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">C\u00e1lculo da Probabilidade Cl\u00e1ssica<\/h2>\n\n\n\n<p class=\"\">A probabilidade de um evento A ocorrer \u00e9 dada pela f\u00f3rmula fundamental:<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$P(A) = \\frac{\\text{n\u00famero de casos favor\u00e1veis}}{\\text{n\u00famero de casos poss\u00edveis}} = \\frac{n(A)}{n(\\Omega)}$$<script src=\"https:\/\/colegadeclasse.com.br\/blog\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/colegadeclasse.com.br\/blog\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<p class=\"\">Onde:<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$P(A) = \\text{probabilidade do evento A}$$<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$n(A) = \\text{n\u00famero de elementos do evento A}$$<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$n(\\Omega) = \\text{n\u00famero de elementos do espa\u00e7o amostral}$$<\/div>\n\n\n\n<p class=\"\"><strong>Propriedades Fundamentais:<\/strong><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$0 \\leq P(A) \\leq 1 \\text{ (a probabilidade est\u00e1 sempre entre 0 e 1)}$$<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$P(\\Omega) = 1\\text{ (a probabilidade do evento certo \u00e9 1)}$$<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$P(\\emptyset) = 0$ \\text{(a probabilidade do evento imposs\u00edvel \u00e9 0)}$$<\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Exemplo Pr\u00e1tico 1<\/h3>\n\n\n\n<p class=\"\"><strong>Quest\u00e3o tipo concurso:<\/strong> Qual a probabilidade de, ao lan\u00e7ar um dado comum, obter um n\u00famero primo?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<ul class=\"wp-block-list\">\n<li class=\"\">Espa\u00e7o amostral: \u03a9 = {1, 2, 3, 4, 5, 6} \u2192 n(\u03a9) = 6<\/li>\n\n\n\n<li class=\"\">N\u00fameros primos no dado: A = {2, 3, 5} \u2192 n(A) = 3<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<div class=\"wp-block-mathml-mathmlblock\">$$P(A) = \\frac{3}{6} = \\frac{1}{2} = 0,5 = 50%$$<\/div>\n<\/div>\n<\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Exemplo Pr\u00e1tico 2<\/h3>\n\n\n\n<p class=\"\"><strong>Quest\u00e3o tipo concurso:<\/strong> Em uma urna h\u00e1 5 bolas vermelhas, 3 azuis e 2 brancas. Qual a probabilidade de retirar uma bola azul?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<ul class=\"wp-block-list\">\n<li class=\"\">Total de bolas: 5 + 3 + 2 = 10 \u2192 n(\u03a9) = 10<\/li>\n\n\n\n<li class=\"\">Bolas azuis: n(A) = 3<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<div class=\"wp-block-mathml-mathmlblock\">$$P(A) = \\frac{3}{10} = 0,3 = 30%$$<\/div>\n<\/div>\n<\/div>\n\n\n\n<p class=\"\"><strong>\u26a0\ufe0f PONTO DE ATEN\u00c7\u00c3O:<\/strong> Sempre identifique claramente o que a quest\u00e3o est\u00e1 pedindo. Leia com aten\u00e7\u00e3o se \u00e9 &#8220;pelo menos&#8221;, &#8220;exatamente&#8221;, &#8220;no m\u00e1ximo&#8221;, pois cada express\u00e3o altera o c\u00e1lculo.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">Probabilidade do Evento Complementar<\/h2>\n\n\n\n<p class=\"\"><strong>Propriedade fundamental:<\/strong><\/p>\n\n\n\n<p class=\"\">$$P(\\overline{A}) = 1 &#8211; P(A)$$<\/p>\n\n\n\n<p class=\"\"><strong>Exemplo:<\/strong> Se a probabilidade de chover amanh\u00e3 \u00e9 30%, a probabilidade de N\u00c3O chover \u00e9: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$P(\\overline{A}) = 1 &#8211; 0,30 = 0,70 = 70%$$<\/div>\n\n\n\n<p class=\"\"><strong>\ud83d\udca1 DICA ESTRAT\u00c9GICA:<\/strong> Em muitas quest\u00f5es de concurso, \u00e9 mais f\u00e1cil calcular a probabilidade do complementar e depois subtrair de 1. Especialmente em problemas com &#8220;pelo menos um&#8221;.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">Probabilidade da Uni\u00e3o de Eventos<\/h2>\n\n\n\n<p class=\"\">Para dois eventos A e B, a probabilidade de ocorrer A <strong>OU<\/strong> B \u00e9:<\/p>\n\n\n\n<p class=\"\"><span class=\"MathJax_Preview\" style=\"color: inherit;\"><\/span><span class=\"mjx-chtml MJXc-display\" style=\"text-align: center;\"><span id=\"MathJax-Element-7-Frame\" class=\"mjx-chtml MathJax_CHTML\" tabindex=\"0\" data-mathml=\"<math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;<mi&gt;P<\/mi&gt;<mo stretchy=&quot;false&quot;&gt;(<\/mo&gt;<mi&gt;A<\/mi&gt;<mo&gt;&#x222A;<\/mo&gt;<mi&gt;B<\/mi&gt;<mo stretchy=&quot;false&quot;&gt;)<\/mo&gt;<mo&gt;=<\/mo&gt;<mi&gt;P<\/mi&gt;<mo stretchy=&quot;false&quot;&gt;(<\/mo&gt;<mi&gt;A<\/mi&gt;<mo stretchy=&quot;false&quot;&gt;)<\/mo&gt;<mo&gt;+<\/mo&gt;<mi&gt;P<\/mi&gt;<mo stretchy=&quot;false&quot;&gt;(<\/mo&gt;<mi&gt;B<\/mi&gt;<mo stretchy=&quot;false&quot;&gt;)<\/mo&gt;<mo&gt;&#x2212;<\/mo&gt;<mi&gt;P<\/mi&gt;<mo stretchy=&quot;false&quot;&gt;(<\/mo&gt;<mi&gt;A<\/mi&gt;<mo&gt;&#x2229;<\/mo&gt;<mi&gt;B<\/mi&gt;<mo stretchy=&quot;false&quot;&gt;)<\/mo&gt;<\/math&gt;\" role=\"presentation\" style=\"font-size: 113%; text-align: center; position: relative;\"><span id=\"MJXc-Node-101\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-102\" class=\"mjx-mrow\"><span id=\"MJXc-Node-103\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.445em; padding-bottom: 0.297em; padding-right: 0.109em;\">P<\/span><\/span><span id=\"MJXc-Node-104\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">(<\/span><\/span><span id=\"MJXc-Node-105\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.494em; padding-bottom: 0.297em;\">A<\/span><\/span><span id=\"MJXc-Node-106\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.297em; padding-bottom: 0.347em;\">\u222a<\/span><\/span><span id=\"MJXc-Node-107\" class=\"mjx-mi MJXc-space2\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.445em; padding-bottom: 0.297em;\">B<\/span><\/span><span id=\"MJXc-Node-108\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">)<\/span><\/span><span id=\"MJXc-Node-109\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-110\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.445em; padding-bottom: 0.297em; padding-right: 0.109em;\">P<\/span><\/span><span id=\"MJXc-Node-111\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">(<\/span><\/span><span id=\"MJXc-Node-112\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.494em; padding-bottom: 0.297em;\">A<\/span><\/span><span id=\"MJXc-Node-113\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">)<\/span><\/span><span id=\"MJXc-Node-114\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.297em; padding-bottom: 0.445em;\">+<\/span><\/span><span id=\"MJXc-Node-115\" class=\"mjx-mi MJXc-space2\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.445em; padding-bottom: 0.297em; padding-right: 0.109em;\">P<\/span><\/span><span id=\"MJXc-Node-116\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">(<\/span><\/span><span id=\"MJXc-Node-117\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.445em; padding-bottom: 0.297em;\">B<\/span><\/span><span id=\"MJXc-Node-118\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">)<\/span><\/span><span id=\"MJXc-Node-119\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.297em; padding-bottom: 0.445em;\">\u2212<\/span><\/span><span id=\"MJXc-Node-120\" class=\"mjx-mi MJXc-space2\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.445em; padding-bottom: 0.297em; padding-right: 0.109em;\">P<\/span><\/span><span id=\"MJXc-Node-121\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">(<\/span><\/span><span id=\"MJXc-Node-122\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.494em; padding-bottom: 0.297em;\">A<\/span><\/span><span id=\"MJXc-Node-123\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.297em; padding-bottom: 0.347em;\">\u2229<\/span><\/span><span id=\"MJXc-Node-124\" class=\"mjx-mi MJXc-space2\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.445em; padding-bottom: 0.297em;\">B<\/span><\/span><span id=\"MJXc-Node-125\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">)<\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML MJX_Assistive_MathML_Block\" role=\"presentation\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mo>\u222a<\/mo><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mo>\u2229<\/mo><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/span><\/span><\/span><script>P(A \\cup B) = P(A) + P(B) - P(A \\cap B)<\/script><\/p>\n\n\n\n<p class=\"\"><strong>Caso especial &#8211; Eventos Mutuamente Exclusivos:<\/strong> Se A e B n\u00e3o podem ocorrer simultaneamente, ent\u00e3o A U B = 0, e a f\u00f3rmula simplifica para:<\/p>\n\n\n\n<p class=\"\">$$P(A \\cup B) = P(A) + P(B)$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Exemplo Pr\u00e1tico<\/h3>\n\n\n\n<p class=\"\">No lan\u00e7amento de um dado, qual a probabilidade de sair um n\u00famero par <strong>OU<\/strong> um n\u00famero maior que 4?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<ul class=\"wp-block-list\">\n<li class=\"\">A = {2, 4, 6} \u2192 P(A) = 3\/6<\/li>\n\n\n\n<li class=\"\">B = {5, 6} \u2192 P(B) = 2\/6<\/li>\n\n\n\n<li class=\"\">A \u2229 B = {6} \u2192 P(A \u2229 B) = 1\/6<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<div class=\"wp-block-mathml-mathmlblock\">$$P(A \\cup B) = \\frac{3}{6} + \\frac{2}{6} &#8211; \\frac{1}{6} = \\frac{4}{6} = \\frac{2}{3}$$<\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">Probabilidade Condicional<\/h2>\n\n\n\n<p class=\"\">A probabilidade condicional mede a chance de um evento A ocorrer <strong>dado que<\/strong> outro evento B j\u00e1 ocorreu. Nota\u00e7\u00e3o: $P(A|B)$ (l\u00ea-se: &#8220;probabilidade de A dado B&#8221;).<\/p>\n\n\n\n<p class=\"\"><strong>F\u00f3rmula:<\/strong><\/p>\n\n\n\n<p class=\"\"><span class=\"MathJax_Preview\" style=\"color: inherit;\"><\/span><span class=\"mjx-chtml MJXc-display\" style=\"text-align: center;\"><span id=\"MathJax-Element-9-Frame\" class=\"mjx-chtml MathJax_CHTML\" tabindex=\"0\" data-mathml=\"<math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;<mi&gt;P<\/mi&gt;<mo stretchy=&quot;false&quot;&gt;(<\/mo&gt;<mi&gt;A<\/mi&gt;<mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;<mo stretchy=&quot;false&quot;&gt;|<\/mo&gt;<\/mrow&gt;<mi&gt;B<\/mi&gt;<mo stretchy=&quot;false&quot;&gt;)<\/mo&gt;<mo&gt;=<\/mo&gt;<mfrac&gt;<mrow&gt;<mi&gt;P<\/mi&gt;<mo stretchy=&quot;false&quot;&gt;(<\/mo&gt;<mi&gt;A<\/mi&gt;<mo&gt;&#x2229;<\/mo&gt;<mi&gt;B<\/mi&gt;<mo stretchy=&quot;false&quot;&gt;)<\/mo&gt;<\/mrow&gt;<mrow&gt;<mi&gt;P<\/mi&gt;<mo stretchy=&quot;false&quot;&gt;(<\/mo&gt;<mi&gt;B<\/mi&gt;<mo stretchy=&quot;false&quot;&gt;)<\/mo&gt;<\/mrow&gt;<\/mfrac&gt;<\/math&gt;\" role=\"presentation\" style=\"font-size: 113%; text-align: center; position: relative;\"><span id=\"MJXc-Node-144\" class=\"mjx-math\" aria-hidden=\"true\"><span id=\"MJXc-Node-145\" class=\"mjx-mrow\"><span id=\"MJXc-Node-146\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.445em; padding-bottom: 0.297em; padding-right: 0.109em;\">P<\/span><\/span><span id=\"MJXc-Node-147\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">(<\/span><\/span><span id=\"MJXc-Node-148\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.494em; padding-bottom: 0.297em;\">A<\/span><\/span><span id=\"MJXc-Node-149\" class=\"mjx-texatom\"><span id=\"MJXc-Node-150\" class=\"mjx-mrow\"><span id=\"MJXc-Node-151\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">|<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-152\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.445em; padding-bottom: 0.297em;\">B<\/span><\/span><span id=\"MJXc-Node-153\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">)<\/span><\/span><span id=\"MJXc-Node-154\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.101em; padding-bottom: 0.297em;\">=<\/span><\/span><span id=\"MJXc-Node-155\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-stacked\" style=\"width: 4.349em; padding: 0px 0.12em;\"><span class=\"mjx-numerator\" style=\"width: 4.349em; top: -1.514em;\"><span id=\"MJXc-Node-156\" class=\"mjx-mrow\"><span id=\"MJXc-Node-157\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.445em; padding-bottom: 0.297em; padding-right: 0.109em;\">P<\/span><\/span><span id=\"MJXc-Node-158\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">(<\/span><\/span><span id=\"MJXc-Node-159\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.494em; padding-bottom: 0.297em;\">A<\/span><\/span><span id=\"MJXc-Node-160\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.297em; padding-bottom: 0.347em;\">\u2229<\/span><\/span><span id=\"MJXc-Node-161\" class=\"mjx-mi MJXc-space2\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.445em; padding-bottom: 0.297em;\">B<\/span><\/span><span id=\"MJXc-Node-162\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">)<\/span><\/span><\/span><\/span><span class=\"mjx-denominator\" style=\"width: 4.349em; bottom: -1.014em;\"><span id=\"MJXc-Node-163\" class=\"mjx-mrow\"><span id=\"MJXc-Node-164\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.445em; padding-bottom: 0.297em; padding-right: 0.109em;\">P<\/span><\/span><span id=\"MJXc-Node-165\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">(<\/span><\/span><span id=\"MJXc-Node-166\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\" style=\"padding-top: 0.445em; padding-bottom: 0.297em;\">B<\/span><\/span><span id=\"MJXc-Node-167\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\" style=\"padding-top: 0.445em; padding-bottom: 0.592em;\">)<\/span><\/span><\/span><\/span><span class=\"mjx-line\" style=\"border-bottom: 1.3px solid; top: -0.281em; width: 4.349em;\"><\/span><\/span><span class=\"mjx-vsize\" style=\"height: 2.527em; vertical-align: -1.014em;\"><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML MJX_Assistive_MathML_Block\" role=\"presentation\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mrow class=\"MJX-TeXAtom-ORD\"><mo stretchy=\"false\">|<\/mo><\/mrow><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mo>\u2229<\/mo><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>P<\/mi><mo stretchy=\"false\">(<\/mo><mi>B<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/math><\/span><\/span><\/span><script>P(A|B) = \\frac{P(A \\cap B)}{P(B)}<\/script><\/p>\n\n\n\n<p class=\"\">desde que P(B) > 0.<\/p>\n\n\n\n<p class=\"ptt-white\"><strong>\u26a0\ufe0f OBSERVA\u00c7\u00c3O CR\u00cdTICA:<\/strong> A probabilidade condicional \u00e9 um dos t\u00f3picos mais cobrados em concursos de n\u00edvel superior. O espa\u00e7o amostral &#8220;se reduz&#8221; ao evento condicionante.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Exemplo Pr\u00e1tico<\/h3>\n\n\n\n<p class=\"\">Em uma empresa, 60% dos funcion\u00e1rios s\u00e3o homens e 40% s\u00e3o mulheres. Entre os homens, 30% t\u00eam curso superior; entre as mulheres, 50% t\u00eam curso superior. Se escolhermos aleatoriamente um funcion\u00e1rio com curso superior, qual a probabilidade de ser mulher?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<ul class=\"wp-block-list\">\n<li class=\"\">P(Homem) = 0,6 e P(Superior|Homem) = 0,3<\/li>\n\n\n\n<li class=\"\">P(Mulher) = 0,4 e P(Superior|Mulher) = 0,5<\/li>\n\n\n\n<li class=\"\">P(Homem e Superior) = 0,6 \u00d7 0,3 = 0,18<\/li>\n\n\n\n<li class=\"\">P(Mulher e Superior) = 0,4 \u00d7 0,5 = 0,20<\/li>\n\n\n\n<li class=\"\">P(Superior) = 0,18 + 0,20 = 0,38<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<div class=\"wp-block-mathml-mathmlblock\">$$P(\\text{Mulher}|\\text{Superior}) = \\frac{0,20}{0,38} = \\frac{10}{19} \\approx 52,63%$$<\/div>\n<\/div>\n<\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">Eventos Independentes<\/h2>\n\n\n\n<p class=\"\">Dois eventos A e B s\u00e3o independentes quando a ocorr\u00eancia de um n\u00e3o afeta a probabilidade do outro.<\/p>\n\n\n\n<p class=\"\"><strong>Condi\u00e7\u00e3o matem\u00e1tica:<\/strong><\/p>\n\n\n\n<p class=\"\">$$P(A \\cap B) = P(A) \\times P(B)$$<\/p>\n\n\n\n<p class=\"\">Ou equivalentemente: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$P(A|B) = P(A) \\text{ e } P(B|A) = P(B)$$<\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Exemplo Pr\u00e1tico<\/h3>\n\n\n\n<p class=\"\">No lan\u00e7amento de dois dados, os resultados s\u00e3o independentes. A probabilidade de obter 6 no primeiro <strong>E<\/strong> 6 no segundo \u00e9:<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$P(6 \\text{ e } 6) = \\frac{1}{6} \\times \\frac{1}{6} = \\frac{1}{36}$$<\/div>\n\n\n\n<p class=\"\"><strong>\ud83d\udca1 REGRA DE OURO:<\/strong> Para eventos sucessivos &#8220;E&#8221; que s\u00e3o independentes, <strong>MULTIPLIQUE<\/strong> as probabilidades.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">T\u00e9cnicas de Contagem Aplicadas \u00e0 Probabilidade<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Princ\u00edpio Fundamental da Contagem (PFC)<\/h3>\n\n\n\n<p class=\"\">Se uma decis\u00e3o pode ser tomada de m maneiras e outra de n maneiras, o total de formas de tomar ambas as decis\u00f5es \u00e9 :<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$m \\times n$$<\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Permuta\u00e7\u00e3o Simples<\/h3>\n\n\n\n<p class=\"\">N\u00famero de maneiras de ordenar n elementos distintos:<\/p>\n\n\n\n<p class=\"\">$$P_n = n!$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Arranjo<\/h3>\n\n\n\n<p class=\"\">N\u00famero de maneiras de escolher e ordenar k elementos de um conjunto de n elementos:<\/p>\n\n\n\n<p class=\"\">$$A_{n,k} = \\frac{n!}{(n-k)!}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Combina\u00e7\u00e3o<\/h3>\n\n\n\n<p class=\"\">N\u00famero de maneiras de escolher k elementos de um conjunto de n elementos (sem importar a ordem):<\/p>\n\n\n\n<p class=\"\">$$C_{n,k} = \\binom{n}{k} = \\frac{n!}{k!(n-k)!}$$<\/p>\n\n\n\n<p class=\"ptt-yellow\"><strong>\u26a0\ufe0f FUNDAMENTAL:<\/strong> Use COMBINA\u00c7\u00c3O quando a ordem n\u00e3o importa; use ARRANJO quando a ordem importa.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Exemplo Pr\u00e1tico Completo<\/h3>\n\n\n\n<p class=\"\">Uma comiss\u00e3o de 3 pessoas ser\u00e1 formada entre 10 candidatos, sendo 6 homens e 4 mulheres. Qual a probabilidade de a comiss\u00e3o ter exatamente 2 homens?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<p class=\"\">Total de comiss\u00f5es poss\u00edveis: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$C_{10,3} = \\frac{10!}{3!7!} = \\frac{10 \\times 9 \\times 8}{3 \\times 2 \\times 1} = 120$$<\/div>\n\n\n\n<p class=\"\">Comiss\u00f5es com 2 homens e 1 mulher: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$C_{6,2} \\times C_{4,1} = 15 \\times 4 = 60$$<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$P = \\frac{60}{120} = \\frac{1}{2} = 50%$$<\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">Problemas Cl\u00e1ssicos de Concursos<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Problema 1: &#8220;Pelo Menos Um&#8221;<\/h3>\n\n\n\n<p class=\"\"><strong>Estrat\u00e9gia:<\/strong> Use o complementar!<\/p>\n\n\n\n<p class=\"\">Para &#8220;pelo menos um&#8221;, calcule a probabilidade de &#8220;nenhum&#8221; e subtraia de 1.<\/p>\n\n\n\n<p class=\"\"><strong>Exemplo:<\/strong> Lan\u00e7am-se 3 moedas. Qual a probabilidade de sair pelo menos uma cara?<\/p>\n\n\n\n<p class=\"\">P(nenhuma cara) = P(3 coroas) = <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$(1\/2)^3 = 1\/8$$<\/div>\n\n\n\n<p class=\"\">P(pelo menos 1 cara) = <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$1 &#8211; 1\/8 = 7\/8 = 87,5%$$<\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Problema 2: Extra\u00e7\u00e3o com e sem Reposi\u00e7\u00e3o<\/h3>\n\n\n\n<p class=\"\"><strong>Com reposi\u00e7\u00e3o:<\/strong> O elemento retirado retorna ao conjunto (eventos independentes)<\/p>\n\n\n\n<p class=\"\"><strong>Sem reposi\u00e7\u00e3o:<\/strong> O elemento n\u00e3o retorna (eventos dependentes, use probabilidade condicional)<\/p>\n\n\n\n<p class=\"\"><strong>Exemplo:<\/strong> Dois cart\u00f5es s\u00e3o retirados de 10 cart\u00f5es numerados de 1 a 10.<\/p>\n\n\n\n<p class=\"\"><strong>SEM reposi\u00e7\u00e3o:<\/strong> P(ambos pares) = <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$\\frac{5}{10} \\times \\frac{4}{9} = \\frac{20}{90} = \\frac{2}{9}$$<\/div>\n\n\n\n<p class=\"\"><strong>COM reposi\u00e7\u00e3o:<\/strong> P(ambos pares) = <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$\\frac{5}{10} \\times \\frac{5}{10} = \\frac{1}{4}$$<\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Problema 3: Baralho (muito frequente!)<\/h3>\n\n\n\n<p class=\"\">Baralho padr\u00e3o: 52 cartas<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">4 naipes: \u2660 (espadas), \u2665 (copas), \u2666 (ouros), \u2663 (paus)<\/li>\n\n\n\n<li class=\"\">13 cartas por naipe: A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K<\/li>\n\n\n\n<li class=\"\">26 cartas vermelhas (copas e ouros)<\/li>\n\n\n\n<li class=\"\">26 cartas pretas (espadas e paus)<\/li>\n<\/ul>\n\n\n\n<p class=\"\"><strong>Exemplo:<\/strong> Probabilidade de tirar um \u00c1s: <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$P = \\frac{4}{52} = \\frac{1}{13}$$<\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">Teorema de Bayes (N\u00edvel Avan\u00e7ado)<\/h2>\n\n\n\n<p class=\"\">O Teorema de Bayes permite calcular probabilidades &#8220;inversas&#8221;, fundamentado na probabilidade condicional:<\/p>\n\n\n\n<p class=\"\">$$P(A|B) = \\frac{P(B|A) \\times P(A)}{P(B)}$$<\/p>\n\n\n\n<p class=\"\">Ou na forma expandida:<\/p>\n\n\n\n<p class=\"\">$$P(A_i|B) = \\frac{P(B|A_i) \\times P(A_i)}{\\sum_{j=1}^{n} P(B|A_j) \\times P(A_j)}$$<\/p>\n\n\n\n<p class=\"\"><strong>\u26a0\ufe0f ATEN\u00c7\u00c3O:<\/strong> Este teorema aparece em concursos de n\u00edvel superior, especialmente para cargos de Analista, Auditor e \u00e1reas de Estat\u00edstica.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Exemplo Pr\u00e1tico (Tipo ESAF\/CESPE\/FCC)<\/h3>\n\n\n\n<p class=\"\">Uma doen\u00e7a rara afeta 0,1% da popula\u00e7\u00e3o. Um teste detecta a doen\u00e7a em 99% dos casos positivos, mas d\u00e1 falso positivo em 2% dos casos negativos. Se uma pessoa testou positivo, qual a probabilidade de realmente ter a doen\u00e7a?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">P(Doen\u00e7a) = 0,001<\/li>\n\n\n\n<li class=\"\">P(Positivo|Doen\u00e7a) = 0,99<\/li>\n\n\n\n<li class=\"\">P(Positivo|Sem doen\u00e7a) = 0,02<\/li>\n\n\n\n<li class=\"\">P(Sem doen\u00e7a) = 0,999<\/li>\n<\/ul>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$P(\\text{Doen\u00e7a}|\\text{Positivo}) = \\frac{0,99 \\times 0,001}{0,99 \\times 0,001 + 0,02 \\times 0,999}$$<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$= \\frac{0,00099}{0,00099 + 0,01998} = \\frac{0,00099}{0,02097} \\approx 0,0472 = 4,72%$$<\/div>\n\n\n\n<p class=\"\"><strong>Resultado surpreendente:<\/strong> Mesmo testando positivo, a chance de ter a doen\u00e7a \u00e9 apenas 4,72%! Isso ocorre porque a doen\u00e7a \u00e9 muito rara.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">Distribui\u00e7\u00f5es de Probabilidade Elementares<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Distribui\u00e7\u00e3o Uniforme Discreta<\/h3>\n\n\n\n<p class=\"\">Todos os resultados t\u00eam a mesma probabilidade. Exemplo: dado honesto.<\/p>\n\n\n\n<p class=\"\">$$P(X = x_i) = \\frac{1}{n}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Distribui\u00e7\u00e3o Binomial (Bernoulli Repetido)<\/h3>\n\n\n\n<p class=\"\">Usada quando temos n repeti\u00e7\u00f5es independentes de um experimento com apenas dois resultados (sucesso\/fracasso).<\/p>\n\n\n\n<p class=\"\">$$P(X = k) = \\binom{n}{k} p^k (1-p)^{n-k}$$<\/p>\n\n\n\n<p class=\"\">Onde:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">n = n\u00famero de tentativas<\/li>\n\n\n\n<li class=\"\">k = n\u00famero de sucessos<\/li>\n\n\n\n<li class=\"\">p = probabilidade de sucesso em cada tentativa<\/li>\n<\/ul>\n\n\n\n<p class=\"\"><strong>Exemplo:<\/strong> Lan\u00e7ar uma moeda 5 vezes e calcular a probabilidade de dar cara exatamente 3 vezes:<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$P(X = 3) = \\binom{5}{3} (0,5)^3 (0,5)^2 = 10 \\times 0,125 \\times 0,25 = 0,3125 = 31,25%$$<\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">Principais Erros em Provas de Concurso<\/h2>\n\n\n\n<p class=\"\"><strong>\u274c ERRO 1:<\/strong> Confundir eventos independentes com mutuamente exclusivos<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li class=\"\">Independentes: A ocorr\u00eancia de um n\u00e3o afeta o outro (multiplica probabilidades no &#8220;E&#8221;)<\/li>\n\n\n\n<li class=\"\">Mutuamente exclusivos: N\u00e3o podem ocorrer simultaneamente (soma probabilidades no &#8220;OU&#8221;)<\/li>\n<\/ul>\n\n\n\n<p class=\"\"><strong>\u274c ERRO 2:<\/strong> Esquecer de subtrair a interse\u00e7\u00e3o na uni\u00e3o de eventos<\/p>\n\n\n\n<p class=\"\"><strong>\u274c ERRO 3:<\/strong> Em problemas &#8220;sem reposi\u00e7\u00e3o&#8221;, calcular como se fosse &#8220;com reposi\u00e7\u00e3o&#8221;<\/p>\n\n\n\n<p class=\"\"><strong>\u274c ERRO 4:<\/strong> Usar permuta\u00e7\u00e3o quando deveria usar combina\u00e7\u00e3o (ou vice-versa)<\/p>\n\n\n\n<p class=\"\"><strong>\u274c ERRO 5:<\/strong> N\u00e3o identificar corretamente o espa\u00e7o amostral reduzido em probabilidade condicional<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">Estrat\u00e9gias para Resolu\u00e7\u00e3o em Concursos<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Passo 1: Leia com extrema aten\u00e7\u00e3o<\/h3>\n\n\n\n<p class=\"\">Identifique palavras-chave: &#8220;pelo menos&#8221;, &#8220;no m\u00e1ximo&#8221;, &#8220;exatamente&#8221;, &#8220;ou&#8221;, &#8220;e&#8221;<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Passo 2: Determine o espa\u00e7o amostral<\/h3>\n\n\n\n<p class=\"\">Conte todos os casos poss\u00edveis<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Passo 3: Conte os casos favor\u00e1veis<\/h3>\n\n\n\n<p class=\"\">Identifique quais resultados atendem ao evento desejado<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Passo 4: Aplique a f\u00f3rmula adequada<\/h3>\n\n\n\n<p class=\"\">Probabilidade b\u00e1sica, condicional, uni\u00e3o, ou use o complementar<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Passo 5: Simplifique a fra\u00e7\u00e3o<\/h3>\n\n\n\n<p class=\"\">Apresente o resultado na forma mais simples ou em percentual<\/p>\n\n\n\n<p class=\"\"><strong>\ud83d\udca1 DICA FINAL:<\/strong> Fa\u00e7a uma estimativa mental antes de calcular. Se a probabilidade calculada for maior que 1 ou menor que 0, h\u00e1 erro no racioc\u00ednio!<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">Exerc\u00edcios Resolvidos Estilo Concurso<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 1 (N\u00edvel B\u00e1sico &#8211; CESGRANRIO)<\/h3>\n\n\n\n<p class=\"\">Em um grupo de 100 pessoas, 60 s\u00e3o homens e 40 s\u00e3o mulheres. Escolhendo uma pessoa ao acaso, qual a probabilidade de ser mulher?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong> <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$P(\\text{Mulher}) = \\frac{40}{100} = 0,4 = 40%$$<\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 2 (N\u00edvel M\u00e9dio &#8211; FCC)<\/h3>\n\n\n\n<p class=\"\">Tr\u00eas candidatos A, B e C disputam uma vaga. As probabilidades de aprova\u00e7\u00e3o s\u00e3o: P(A) = 0,2; P(B) = 0,3; P(C) = 0,4. Qual a probabilidade de pelo menos um ser aprovado?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong> P(nenhum aprovado) = (1 &#8211; 0,2) \u00d7 (1 &#8211; 0,3) \u00d7 (1 &#8211; 0,4) = 0,8 \u00d7 0,7 \u00d7 0,6 = 0,336<\/p>\n\n\n\n<p class=\"\">P(pelo menos um) = 1 &#8211; 0,336 = 0,664 = 66,4%<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 3 (N\u00edvel Dif\u00edcil &#8211; CESPE\/CEBRASPE)<\/h3>\n\n\n\n<p class=\"\">Uma caixa cont\u00e9m 5 bolas brancas e 3 pretas. Retiram-se duas bolas sucessivamente, sem reposi\u00e7\u00e3o. Qual a probabilidade de ambas serem da mesma cor?<\/p>\n\n\n\n<p class=\"\"><strong>Resolu\u00e7\u00e3o:<\/strong><\/p>\n\n\n\n<p class=\"\">P(ambas brancas) = <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$\\frac{5}{8} \\times \\frac{4}{7} = \\frac{20}{56}$$<\/div>\n\n\n\n<p class=\"\">P(ambas pretas) = <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$\\frac{3}{8} \\times \\frac{2}{7} = \\frac{6}{56}$$<\/div>\n\n\n\n<p class=\"\">P(mesma cor) = <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">$$\\frac{20}{56} + \\frac{6}{56} = \\frac{26}{56} = \\frac{13}{28} \\approx 46,43%$$<\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h2 class=\"wp-block-heading\">Refer\u00eancias Bibliogr\u00e1ficas Confi\u00e1veis<\/h2>\n\n\n\n<p class=\"\">As informa\u00e7\u00f5es apresentadas neste material t\u00eam fundamento em obras consagradas da matem\u00e1tica e estat\u00edstica:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Obras Cl\u00e1ssicas<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li class=\"\"><strong>MORGADO, A. C.; CARVALHO, J. B. P.; CARVALHO, P. C. P.; FERNANDEZ, P.<\/strong> <em>An\u00e1lise Combinat\u00f3ria e Probabilidade<\/em>. Rio de Janeiro: SBM (Sociedade Brasileira de Matem\u00e1tica), 2006.<\/li>\n\n\n\n<li class=\"\"><strong>DANTE, L. R.<\/strong><em>Matem\u00e1tica: Contexto e Aplica\u00e7\u00f5es<\/em>. S\u00e3o Paulo: Editora \u00c1tica, 2018.\n<ul class=\"wp-block-list\">\n<li class=\"\">Cita\u00e7\u00e3o relevante: &#8220;A probabilidade de um evento \u00e9 um n\u00famero que expressa a chance de esse evento ocorrer, variando de 0 (imposs\u00edvel) a 1 (certo).&#8221;<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li class=\"\"><strong>ROSS, S.<\/strong> <em>Probabilidade: Um Curso Moderno com Aplica\u00e7\u00f5es<\/em>. Porto Alegre: Bookman, 2010.<\/li>\n\n\n\n<li class=\"\"><strong>MEYER, P. L.<\/strong> <em>Probabilidade: Aplica\u00e7\u00f5es \u00e0 Estat\u00edstica<\/em>. Rio de Janeiro: LTC, 2012.<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Fontes para Concursos P\u00fablicos<\/h3>\n\n\n\n<ol start=\"5\" class=\"wp-block-list\">\n<li class=\"\"><strong>WEBER, D.<\/strong> <em>Matem\u00e1tica para Concursos P\u00fablicos<\/em>. Rio de Janeiro: Elsevier, 2019.<\/li>\n\n\n\n<li class=\"\"><strong>Quest\u00f5es de bancas organizadoras:<\/strong> CESPE\/CEBRASPE, FCC, FGV, VUNESP, CESGRANRIO\n<ul class=\"wp-block-list\">\n<li class=\"\">Todas as bancas cobram probabilidade seguindo os fundamentos apresentados neste material<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"\">A Probabilidade \u00e9 um conte\u00fado fundamental e recorrente em concursos p\u00fablicos de todos os n\u00edveis. O dom\u00ednio dos conceitos apresentados, aliado \u00e0 pr\u00e1tica constante de exerc\u00edcios, garantir\u00e1 seu sucesso neste t\u00f3pico. Lembre-se:<\/p>\n\n\n\n<p class=\"\">\u2705 <strong>Domine as f\u00f3rmulas b\u00e1sicas<\/strong> \u2705 <strong>Identifique corretamente o tipo de problema<\/strong> \u2705 <strong>Pratique com quest\u00f5es de provas anteriores<\/strong> \u2705 <strong>Aten\u00e7\u00e3o redobrada com leitura do enunciado<\/strong> \u2705 <strong>Use estrat\u00e9gias inteligentes (complementar, contagem)<\/strong><\/p>\n\n\n\n<p class=\"\">O conhecimento s\u00f3lido em probabilidade n\u00e3o apenas garante pontos nas provas, mas desenvolve o racioc\u00ednio l\u00f3gico essencial para diversas outras disciplinas. Continue praticando e bons estudos!<\/p>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"<p>A probabilidade \u00e9 o ramo da Matem\u00e1tica que estuda a chance de ocorr\u00eancia de determinados eventos. Em termos pr\u00e1ticos, \u00e9 a medida num\u00e9rica da possibilidade de algo acontecer, variando sempre entre 0 (evento imposs\u00edvel) e 1 (evento certo), podendo ser expressa tamb\u00e9m em forma percentual (0% a 100%). A teoria das probabilidades surgiu no s\u00e9culo [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":688,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"nf_dc_page":"","footnotes":""},"categories":[107],"tags":[46,197,211,357],"class_list":["post-5586","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-matematica","tag-estudo-completo","tag-resumos_esquematizados","tag-questoes","tag-dicas"],"acf":[],"_links":{"self":[{"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/posts\/5586","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/comments?post=5586"}],"version-history":[{"count":1,"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/posts\/5586\/revisions"}],"predecessor-version":[{"id":5587,"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/posts\/5586\/revisions\/5587"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/media\/688"}],"wp:attachment":[{"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/media?parent=5586"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/categories?post=5586"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/colegadeclasse.com.br\/blog\/wp-json\/wp\/v2\/tags?post=5586"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}