{"id":25394,"date":"2021-03-11T00:00:17","date_gmt":"2021-03-11T00:00:17","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25394"},"modified":"2024-06-05T04:16:12","modified_gmt":"2024-06-05T04:16:12","slug":"discover-the-sample-space-of-probability-theory","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/discover-the-sample-space-of-probability-theory\/","title":{"rendered":"Discover the Sample Space of Probability Theory"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Understanding the Sample Space in Probability Theory<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>Summary<\/strong><br \/><em>In this lesson, we explore the concept of Probability Space, a mathematical structure consisting of a Sample Space, Sigma-Algebra, and Probability Measure. We examine the Sample Space in detail, understood as the collection of all possible states of a random process. Through practical examples, we illustrate the construction of discrete and continuous sample spaces, and explain how measurable events are constructed from them and how probability measures are calculated. This lesson is fundamental to understanding the foundations of Probability Theory and lays the groundwork for its application in various fields.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>LEARNING OBJECTIVES:<\/strong><br \/>\nUpon completing this lesson, students will be able to:\n<\/p>\n<ol>\n<li><strong>Understand<\/strong> the concept of Probability Space.<\/li>\n<li><strong>Identify<\/strong> the elements that make up the Probability Space.<\/li>\n<li><strong>Differentiate<\/strong> between discrete and continuous sample spaces.<\/li>\n<li><strong>Construct<\/strong> discrete and continuous sample spaces.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>TABLE OF CONTENTS<\/strong><br \/>\n<a href=\"#1\">THE PROBABILITY SPACE<\/a><br \/>\n<a href=\"#2\">EXAMPLES OF SAMPLE SPACES<\/a><br \/>\n<a href=\"#3\">DISCRETE AND CONTINUOUS SAMPLE SPACES<\/a><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/NgYvWIWC_c4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p><a name=\"1\"><\/a><\/p>\n<h2>The Probability Space<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=NgYvWIWC_c4&amp;t=89s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">Probability Theory is founded on an object called Probability Space.<\/span><\/strong><\/a> This is a mathematical structure composed of: (i) a Sample Space <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span>, (ii) a Sigma-Algebra <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma<\/span>, and (iii) a Probability Measure <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span>. To construct the probability space, we first review the concept of <strong>sample space.<\/strong><\/p>\n<p style=\"text-align: justify; color: #000000;\">The union of all <strong>possible states<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\omega<\/span> of a <strong><a href=\"https:\/\/toposuranos.com\/probabilidades-deterministas-indeterministas-azar\/\" rel=\"noopener\" target=\"_blank\">random process<\/a><\/strong> forms a non-empty set <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span> which we call <strong>Sample Space.<\/strong><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Examples of Sample Spaces<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; background-color: #dddddd;\"><a href=\"https:\/\/www.youtube.com\/watch?v=NgYvWIWC_c4&amp;t=136s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">EXAMPLE 1<\/span><\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify; background-color: #ffffff; color: #000000;\">If we toss a coin, then we have two possible outcomes: Heads (H) and Tails (T). Therefore, the sample space will be<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1m}=\\{H,T\\}<\/span>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; background-color: #dddddd; color: #008000;\"><strong>EXAMPLE 2<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify; background-color: #ffffff; color: #000000;\">If the previous experiment is repeated, but now with two tosses, then we will have:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2m}=\\{(H,H);(H,T);(T,H);(T,T)\\}<\/span>\n<p>That is, all possible ways of arranging heads and tails in groups of two.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; background-color: #dddddd; color: #008000;\"><strong>EXAMPLE 3<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify; background-color: #ffffff; color: #000000;\">Rolling a six-faced die has the following sample space:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1d6}=\\{1,2,3,4,5,6\\}<\/span>\n<p>That is, the number indicated on each of its faces.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; background-color: #dddddd; color: #008000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=NgYvWIWC_c4&amp;t=213s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">EXAMPLE 4<\/span><\/strong><\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify; background-color: #ffffff; color: #000000;\">The lifespan of an electrical appliance (measured in hours) has a sample space of the form<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{ae}=\\{t\\in \\mathbb{R} \\;|\\; t\\geq 0\\}<\/span>\n<p>That is, the lifespan of the appliance is a number <span class=\"katex-eq\" data-katex-display=\"false\">t<\/span> contained in the interval <span class=\"katex-eq\" data-katex-display=\"false\"> [0,+\\infty[ <\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Discrete and Continuous Sample Spaces<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=NgYvWIWC_c4&amp;t=285s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">From these examples, we can distinguish between two types of sample spaces,<\/span><\/strong><\/a> these are discrete and continuous. Discrete sample spaces, as in the first three examples, consist of finite sets, although they can also be infinite and countable (like any subset of <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{N}<\/span>). On the other hand, continuous sample spaces are infinite and uncountable sets; they are generally represented by sub-intervals of <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span>.<\/p>\n<p style=\"text-align: justify; color: #000000;\">From the elements of the sample space (the possible states), <strong>measurable events<\/strong> (objects of the sigma-algebra) of the probability space are constructed, and it is on these objects that <strong>probability measures<\/strong> are calculated.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Understanding the Sample Space in Probability Theory SummaryIn this lesson, we explore the concept of Probability Space, a mathematical structure consisting of a Sample Space, Sigma-Algebra, and Probability Measure. We examine the Sample Space in detail, understood as the collection of all possible states of a random process. Through practical examples, we illustrate the construction [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":25391,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":5,"footnotes":""},"categories":[567,670],"tags":[],"class_list":["post-25394","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","category-probabilities-and-statistics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Discover the Sample Space of Probability Theory - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Find out all the details about the Sample Space that you might not have known and get ready to advance in probability theory.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/en\/discover-the-sample-space-of-probability-theory\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Discover the Sample Space of Probability Theory\" \/>\n<meta property=\"og:description\" content=\"Find out all the details about the Sample Space that you might not have known and get ready to advance in probability theory.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/discover-the-sample-space-of-probability-theory\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-03-11T00:00:17+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-06-05T04:16:12+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/espaciomuestral-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Discover the Sample Space of Probability Theory\" \/>\n<meta name=\"twitter:description\" content=\"Find out all the details about the Sample Space that you might not have known and get ready to advance in probability theory.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/espaciomuestral.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/discover-the-sample-space-of-probability-theory\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/discover-the-sample-space-of-probability-theory\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Discover the Sample Space of Probability Theory\",\"datePublished\":\"2021-03-11T00:00:17+00:00\",\"dateModified\":\"2024-06-05T04:16:12+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/discover-the-sample-space-of-probability-theory\\\/\"},\"wordCount\":491,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/discover-the-sample-space-of-probability-theory\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/12\\\/espaciomuestral.jpg\",\"articleSection\":[\"Mathematics\",\"Probabilities and Statistics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/discover-the-sample-space-of-probability-theory\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/discover-the-sample-space-of-probability-theory\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/discover-the-sample-space-of-probability-theory\\\/\",\"name\":\"Discover the Sample Space of Probability Theory - 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