{"id":33,"date":"2023-08-06T10:52:51","date_gmt":"2023-08-06T10:52:51","guid":{"rendered":"https:\/\/statorials.org\/pt\/operacoes-com-eventos\/"},"modified":"2023-08-06T10:52:51","modified_gmt":"2023-08-06T10:52:51","slug":"operacoes-com-eventos","status":"publish","type":"post","link":"https:\/\/statorials.org\/pt\/operacoes-com-eventos\/","title":{"rendered":"Opera\u00e7\u00f5es com eventos"},"content":{"rendered":"<p>Aqui explicamos quais opera\u00e7\u00f5es podem ser realizadas com eventos e como \u00e9 calculado cada tipo de opera\u00e7\u00e3o com eventos. Al\u00e9m disso, voc\u00ea pode praticar exerc\u00edcios passo a passo sobre opera\u00e7\u00f5es com eventos.<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"tipos-de-operaciones-con-sucesos\"><\/span> Tipos de opera\u00e7\u00f5es com eventos<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Na teoria das probabilidades, existem tr\u00eas tipos de opera\u00e7\u00f5es com eventos, que s\u00e3o:<\/p>\n<ul>\n<li> <strong><a href=\"https:\/\/statorials.org\/pt\/probabilidade-da-uniao-de-eventos\/\">Uni\u00e3o de eventos<\/a><\/strong> : \u00e9 a probabilidade de ocorrer um evento ou outro.<\/li>\n<li> <strong><a href=\"https:\/\/statorials.org\/pt\">Interse\u00e7\u00e3o de eventos<\/a><\/strong> : \u00e9 a probabilidade conjunta de dois ou mais eventos.<\/li>\n<li> <strong>Diferen\u00e7a de evento<\/strong> : Esta \u00e9 a probabilidade de um evento ocorrer, mas outro evento n\u00e3o ocorrer ao mesmo tempo.<\/li>\n<\/ul>\n<p> Simplesmente definindo cada tipo de opera\u00e7\u00e3o de evento, fica dif\u00edcil entender como cada tipo de opera\u00e7\u00e3o \u00e9 executado. Portanto, explicaremos as tr\u00eas opera\u00e7\u00f5es com mais detalhes a seguir.<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"union-de-sucesos\"><\/span> uni\u00e3o de eventos<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A <strong>uni\u00e3o de dois eventos<\/strong> A e B \u00e9 a probabilidade de que o evento A, o evento B ou ambos os eventos ocorram ao mesmo tempo.<\/p>\n<p> O s\u00edmbolo da uni\u00e3o de dois eventos diferentes \u00e9 um U, portanto a uni\u00e3o de dois eventos \u00e9 expressa por um U no meio das duas letras que representam os eventos.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-f50505c0ff1d2e55a78d3f7ff7f84346_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\\cup B\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> A <strong>probabilidade de uni\u00e3o de dois eventos<\/strong> \u00e9 igual \u00e0 soma da probabilidade de ocorr\u00eancia de cada evento menos a probabilidade de intersec\u00e7\u00e3o dos dois eventos.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-be7c6347c0f331eeb6540ac9e15081dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(A\\cup B)=P(A)+P(B)-P(A\\cap B)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"299\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Por exemplo, calcularemos a probabilidade dos eventos <em>\u201crolar um n\u00famero par\u201d<\/em> ou <em>\u201crolar um n\u00famero maior que 4\u201d<\/em> ao lan\u00e7ar um dado.<\/p>\n<p> Existem tr\u00eas possibilidades de obter um n\u00famero par ao lan\u00e7ar o dado (2, 4 e 6), ent\u00e3o a probabilidade do evento ocorrer \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-507520eabc66126944ccf6869984ce22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A=\\{2,4,6\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-8e7182c80426b8d6d491af6c679f0527_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(A)=\\cfrac{3}{6}=0,5\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"124\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Por outro lado, existem apenas dois n\u00fameros maiores que quatro (5 e 6), a sua probabilidade \u00e9 portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-141dd1e0fa8df6a6db4ccbb63cc89dd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B=\\{5,6\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-563300d3d19119cf838f3ce63d4bf789_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(B)=\\cfrac{2}{6}=0,33\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"135\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> E a intersec\u00e7\u00e3o dos dois eventos corresponde aos n\u00fameros que aparecem em ambos os eventos, ent\u00e3o:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-f0ba40be6117cc3e22d6d347389dea9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\\cap B=\\{6\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-65de1160fc1ca0bf26e9acf7f5fd8c9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(A\\cap B)=\\cfrac{1}{6}=0,167\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"177\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Resumindo, juntando os eventos A e B, a probabilidade de ocorr\u00eancia ser\u00e1: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-09c6f02f4584314058aaadd171152410_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}P(A\\cup B)&amp; =P(A)+P(B)-P(A\\cap B)\\\\[2ex] &amp; =0,5+0,33-0,167\\\\[2ex] &amp;=0,67\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"102\" width=\"299\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"interseccion-de-sucesos\"><\/span> interse\u00e7\u00e3o de eventos<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A <strong>interse\u00e7\u00e3o de dois eventos<\/strong> A e B \u00e9 a probabilidade de que ambos os eventos A e B ocorram ao mesmo tempo.<\/p>\n<p> O s\u00edmbolo da intersec\u00e7\u00e3o de dois eventos \u00e9 representado por um U invertido.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-d142a3ddc9de576e468ef331890da9e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\\cap B\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> A <strong>probabilidade de intersec\u00e7\u00e3o de dois eventos<\/strong> \u00e9 igual ao produto das probabilidades de cada evento separadamente.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-593a1a131cab23d805d8324e21e6b4ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(A\\cap B)=P(A)\\cdot P(B)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"193\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Obviamente, para calcular a probabilidade de intersec\u00e7\u00e3o de dois eventos, estes dois eventos devem ser compat\u00edveis. <\/p>\n<div style=\"background-color:#FFFDE7; padding-top: 10px; padding-bottom: 10px; padding-right: 20px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\"> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <a href=\"https:\/\/statorials.org\/pt\/eventos-compativeis\/\">Quais eventos s\u00e3o compat\u00edveis?<\/a><\/div>\n<p> Como exemplo, encontraremos a probabilidade de os eventos <em>\u201cobter um n\u00famero par\u201d<\/em> e <em>\u201cobter um n\u00famero maior que 4\u201d<\/em> se cruzarem durante o lan\u00e7amento de um dado.<\/p>\n<p> Como calculamos acima, a probabilidade de cada evento ocorrer separadamente \u00e9: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-3 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-507520eabc66126944ccf6869984ce22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A=\\{2,4,6\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-8e7182c80426b8d6d491af6c679f0527_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(A)=\\cfrac{3}{6}=0,5\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"124\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-141dd1e0fa8df6a6db4ccbb63cc89dd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B=\\{5,6\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-563300d3d19119cf838f3ce63d4bf789_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(B)=\\cfrac{2}{6}=0,33\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"135\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> Assim, a probabilidade de intersec\u00e7\u00e3o dos dois eventos ser\u00e1 a multiplica\u00e7\u00e3o das probabilidades de cada evento: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-467241b200c59905ecdcb42d834bc7ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}P(A\\cap B)&amp; =P(A)\\cdot P(B)\\\\[2ex] &amp; =0,5\\cdot 0,33\\\\[2ex] &amp;=0,167\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"102\" width=\"193\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"diferencia-de-sucesos\"><\/span> diferen\u00e7a de eventos<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A <strong>diferen\u00e7a de dois eventos<\/strong> A menos B corresponde a todos os eventos elementares de A que n\u00e3o est\u00e3o em B. Em outras palavras, na diferen\u00e7a de dois eventos A menos B, o evento A \u00e9 satisfeito, mas o evento B n\u00e3o pode ser satisfeito simultaneamente.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-52a50ded14e604d3d89d5df4d87449e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A-B\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> A <strong>probabilidade de diferen\u00e7a entre dois eventos<\/strong> A e B \u00e9 igual \u00e0 probabilidade de ocorr\u00eancia do evento A menos a probabilidade de ocorr\u00eancia de eventos elementares compartilhada por A e B.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-6629f4be6b709ade16ee97ae8a42d8f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(A-B)=P(A)-P(A\\cap B)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"237\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Seguindo o mesmo exemplo dos dois tipos de opera\u00e7\u00f5es anteriores, determinaremos a probabilidade de isso acontecer a partir da diferen\u00e7a do evento <em>\u201cobter um n\u00famero par\u201d<\/em> menos <em>\u201cobter um n\u00famero maior que 4\u201d<\/em> no lan\u00e7amento dos dados.<\/p>\n<p> As probabilidades de ocorr\u00eancia dos eventos A, B e sua interse\u00e7\u00e3o s\u00e3o as seguintes (voc\u00ea pode ver o c\u00e1lculo detalhado acima): <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-507520eabc66126944ccf6869984ce22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A=\\{2,4,6\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-8e7182c80426b8d6d491af6c679f0527_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(A)=\\cfrac{3}{6}=0,5\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"124\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-141dd1e0fa8df6a6db4ccbb63cc89dd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B=\\{5,6\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-563300d3d19119cf838f3ce63d4bf789_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(B)=\\cfrac{2}{6}=0,33\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"135\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-632af7dc98e8dfe1d290dad299b7ed8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\\cap B= \\{6\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-5534b4bfc1a446e367ee89c8b92a210e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(A\\cap B)=\\cfrac{1}{6}= 0,167\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"177\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> A probabilidade de a diferen\u00e7a entre os dois eventos aparecer \u00e9, portanto:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-e7aed40cdbbe256ec9b19c670f1d7607_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}P(A-B)&amp;=P(A)-P(A\\cap B)\\\\[2ex] &amp; =0,5-0,167\\\\[2ex] &amp; =0,33\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"102\" width=\"237\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> A t\u00edtulo de curiosidade, a diferen\u00e7a dos acontecimentos AB tem a propriedade de tamb\u00e9m ser equivalente \u00e0 intersec\u00e7\u00e3o entre o acontecimento A e o acontecimento complementar (ou oposto) de B. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-ef7d6b43d04514a954789f1ca5ef085d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A-B=A\\cap\\overline{B}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"121\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div style=\"background-color:#FFFDE7; padding-top: 10px; padding-bottom: 10px; padding-right: 20px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\"> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>Veja:<\/strong> <a href=\"https:\/\/statorials.org\/pt\/evento-oposto-evento-complementar\/\">O que \u00e9 evento complementar?<\/a> <\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-operaciones-con-sucesos\"><\/span> Exerc\u00edcios resolvidos sobre opera\u00e7\u00f5es com eventos<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Se voc\u00ea lan\u00e7ar um dado de seis faces, qual \u00e9 a probabilidade de obter um n\u00famero \u00edmpar ou menor que 3? <\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__FFF8E1\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#FFF8E1\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Neste exerc\u00edcio devemos calcular a probabilidade de ocorrer um evento ou outro, portanto devemos encontrar a probabilidade de uni\u00e3o dos dois eventos.<\/p>\n<p class=\"has-text-align-left\"> Portanto, primeiro calculamos a probabilidade de obter um n\u00famero \u00edmpar aplicando a lei de Laplace:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-979ad7431176d3876eefe36603c62fb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\" P(\\text{n\\'umero impar})=\\cfrac{3}{6}=0,5\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"220\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Em segundo lugar, determinamos a probabilidade de obter um n\u00famero menor que 3:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-f5bbf47b98e14d8d013618a4febadd1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\" P(\\text{n\\'umero menor que 3})=\\cfrac{2}{6}=0,33\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"281\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos calcular a probabilidade de eventos elementares que se repetem em eventos, que \u00e9 apenas o n\u00famero 1 (apenas \u00edmpar menor que 3):<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-def5fea23c1336469a00fb1d7f8a1c9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\" P(\\text{n\\'umero impar y menor que 3})=\\cfrac{1}{6}=0,167\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"356\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E por fim, aplicamos a f\u00f3rmula da uni\u00e3o de dois eventos para descobrir sua probabilidade: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-09c6f02f4584314058aaadd171152410_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}P(A\\cup B)&amp; =P(A)+P(B)-P(A\\cap B)\\\\[2ex] &amp; =0,5+0,33-0,167\\\\[2ex] &amp;=0,67\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"102\" width=\"299\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 2<\/h3>\n<p> Numa caixa colocamos 3 bolas laranja, 2 bolas azuis e 5 bolas brancas. Fazemos a experi\u00eancia aleat\u00f3ria de pegar uma bola, coloc\u00e1-la de volta na caixa e depois remover outra bola. Qual \u00e9 a probabilidade de tirar uma bola azul na primeira e uma bola laranja na segunda? <\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__FFF8E1\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#FFF8E1\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para resolver este problema, devemos calcular a intersec\u00e7\u00e3o dos dois acontecimentos, porque queremos que ambos os acontecimentos elementares sejam verdadeiros.<\/p>\n<p class=\"has-text-align-left\"> Portanto, primeiro calculamos a probabilidade de pegar uma bola azul aplicando a regra de Laplace:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-2115f0ea9c6fe50a8dab8d498b4ef634_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(\\text{sacar bola azul})=\\cfrac{2}{3+2+5}=0,2\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"287\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Encontramos ent\u00e3o a probabilidade de obter uma bola laranja:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-edba4b58924cad3279e8aef1c629e272_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(\\text{sacar bola naranja})=\\cfrac{3}{3+2+5}=0,3\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"316\" style=\"vertical-align: -14px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E, por fim, calculamos a probabilidade de intersec\u00e7\u00e3o dos dois eventos multiplicando as duas probabilidades encontradas:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-cfcbe0cbee264116460aa623fe22b8f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}P(A\\cap B)&amp; =P(A)\\cdot P(B)\\\\[2ex] &amp; =0,2\\cdot 0,3\\\\[2ex] &amp;=0,06\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"102\" width=\"193\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Concluindo, h\u00e1 apenas 6% de chance de pegar uma bola azul na primeira tentativa e uma bola laranja na segunda tentativa.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 3<\/h3>\n<p> A probabilidade de Marta passar em um exame \u00e9 de 1\/3 e a probabilidade de Juan passar no mesmo exame \u00e9 de 2\/5. Qual \u00e9 a probabilidade de Marta ter sucesso e Juan falhar? <\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__FFF8E1\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#FFF8E1\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Neste exerc\u00edcio precisamos calcular a diferen\u00e7a entre os dois eventos, porque queremos que Marta aprove, mas n\u00e3o Juan. Para isso, basta utilizar a f\u00f3rmula deste tipo de opera\u00e7\u00e3o com eventos:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-d0e95b0b85a9436f42caf8eaa44f2a38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}\\displaystyle A-B =A\\cap\\overline{B}=\\\\[2ex]\\displaystyle =\\frac{1}{3}\\cdot \\left(1-\\frac{2}{5}\\right) = \\frac{1}{3} \\cdot \\frac{3}{5}=\\\\[3ex] =\\cfrac{3}{15} = 0,2\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"134\" width=\"194\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A probabilidade de Marta ter sucesso e Juan falhar ao mesmo tempo \u00e9, portanto, de 20%.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Aqui explicamos quais opera\u00e7\u00f5es podem ser realizadas com eventos e como \u00e9 calculado cada tipo de opera\u00e7\u00e3o com eventos. Al\u00e9m disso, voc\u00ea pode praticar exerc\u00edcios passo a passo sobre opera\u00e7\u00f5es com eventos. Tipos de opera\u00e7\u00f5es com eventos Na teoria das probabilidades, existem tr\u00eas tipos de opera\u00e7\u00f5es com eventos, que s\u00e3o: Uni\u00e3o de eventos : \u00e9 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[],"class_list":["post-33","post","type-post","status-publish","format-standard","hentry","category-probabilidade"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Opera\u00e7\u00f5es com eventos: uni\u00e3o, intersec\u00e7\u00e3o e diferen\u00e7a<\/title>\n<meta name=\"description\" content=\"Explicamos as opera\u00e7\u00f5es com eventos e como cada tipo de opera\u00e7\u00e3o \u00e9 calculado. 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