{"id":333,"date":"2020-07-08T13:56:39","date_gmt":"2020-07-08T17:56:39","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&#038;p=333"},"modified":"2024-02-08T13:11:31","modified_gmt":"2024-02-08T18:11:31","slug":"3-1-basic-concepts-in-probability","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/3-1-basic-concepts-in-probability\/","title":{"raw":"3.1 Basic Concepts in Probability","rendered":"3.1 Basic Concepts in Probability"},"content":{"raw":"Let us first introduce some basic concepts in probability theory.\r\n<ul type=\"disc\">\r\n \t<li><strong>Chance experiment<\/strong> is a process producing outcomes that vary randomly when repeated.<\/li>\r\n \t<li><strong>Sample space, <\/strong>denoted as [latex] S[\/latex], is the collection of ALL possible outcomes of a chance experiment.<\/li>\r\n \t<li>Each possible outcome in the sample space is called a <strong>sample point<\/strong>.<\/li>\r\n \t<li><strong>An event<\/strong> is a combination of sample points; it is a subset of the sample space. We use capital letters A, B, C, \u2026, E, \u2026 to represent events.<\/li>\r\n<\/ul>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example: Basic Concepts<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div>\r\n<p style=\"text-align: center;\"><strong>Table 3.1<\/strong>: Examples of Sample Space and Events<\/p>\r\n\r\n<table class=\"aligncenter\" style=\"height: 128px; width: 100%;\">\r\n<thead>\r\n<tr class=\"shaded\" style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 27.5641%; vertical-align: top;\"><strong>Chance Experiment<\/strong><\/td>\r\n<td style=\"height: 15px; width: 27.6709%; vertical-align: top;\"><strong>Sample Space <em>S<\/em><\/strong><\/td>\r\n<td style=\"height: 15px; width: 44.6581%; vertical-align: top;\"><strong>Events<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 10px;\">\r\n<td style=\"height: 22px; width: 27.5641%; vertical-align: top;\">Flip a balanced coin<\/td>\r\n<td style=\"height: 22px; width: 27.6709%; vertical-align: top;\">{H, T} where H: head, T: tail<\/td>\r\n<td style=\"height: 22px; width: 44.6581%; vertical-align: top;\">E = observe a head = {H}<\/td>\r\n<\/tr>\r\n<tr style=\"height: 40px;\">\r\n<td style=\"height: 54px; width: 27.5641%; vertical-align: top;\">Roll a fair die<\/td>\r\n<td style=\"height: 54px; width: 27.6709%; vertical-align: top;\">{1, 2, 3, 4, 5, 6}<\/td>\r\n<td style=\"height: 54px; width: 44.6581%; vertical-align: top;\">E = observe a six = {6}\r\nA = observe even numbers = {2, 4, 6}\r\nB=outcome is less than 3 = {1, 2}<\/td>\r\n<\/tr>\r\n<tr style=\"height: 58px;\">\r\n<td style=\"height: 37px; width: 27.5641%; vertical-align: top;\">Flip a balanced coin twice<\/td>\r\n<td style=\"height: 37px; width: 27.6709%; vertical-align: top;\">{HH, HT, TT, TH}<\/td>\r\n<td style=\"height: 37px; width: 44.6581%; vertical-align: top;\">E = observe the same outcome = {HH, TT}\r\nA = observe at least one head = {HH, HT, TH}<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nFor example, consider rolling a fair die. The sample space [latex] S = \\{1, 2, 3, 4, 5, 6\\}[\/latex]\u00a0consists of six sample points, while the event observing even numbers [latex] A = \\{2, 4, 6 \\}[\/latex]<strong>\u00a0<\/strong>contains three sample points, which are part of the sample space.\r\n<div style=\"height: 55px; margin-top: 2.1428571429em;\">\r\n\r\n<img class=\"size-full wp-image-99 alignleft\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/06\/activity.png\" alt=\"\" width=\"250\" height=\"50\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise: Basic Concepts<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nConsider the chance experiment of rolling a fair die twice.\r\n<ol>\r\n \t<li>Identify the sample space <em>S<\/em>.<\/li>\r\n \t<li>List all possible outcomes of the event that the two rolls give the same result.<\/li>\r\n \t<li>List all possible outcomes of the event that at least one six is observed.<\/li>\r\n<\/ol>\r\n<details><summary>Show\/Hide Answer<\/summary>\r\n<ol>\r\n \t<li>The sample space [latex] S[\/latex] contains [latex] 6 \\times 6[\/latex] pairs in the form of (1, 1), (1, 2), \u2026, (1, 6), (2, 1), (2, 2), \u2026, (2, 6), \u2026, (6,1), (6, 2), \u2026, (6, 6).<\/li>\r\n \t<li>E = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}<\/li>\r\n \t<li>E = {(1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5)}<\/li>\r\n<\/ol>\r\n<\/details><\/div>\r\n<\/div>","rendered":"<p>Let us first introduce some basic concepts in probability theory.<\/p>\n<ul type=\"disc\">\n<li><strong>Chance experiment<\/strong> is a process producing outcomes that vary randomly when repeated.<\/li>\n<li><strong>Sample space, <\/strong>denoted as [latex]S[\/latex], is the collection of ALL possible outcomes of a chance experiment.<\/li>\n<li>Each possible outcome in the sample space is called a <strong>sample point<\/strong>.<\/li>\n<li><strong>An event<\/strong> is a combination of sample points; it is a subset of the sample space. We use capital letters A, B, C, \u2026, E, \u2026 to represent events.<\/li>\n<\/ul>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example: Basic Concepts<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div>\n<p style=\"text-align: center;\"><strong>Table 3.1<\/strong>: Examples of Sample Space and Events<\/p>\n<table class=\"aligncenter\" style=\"height: 128px; width: 100%;\">\n<thead>\n<tr class=\"shaded\" style=\"height: 15px;\">\n<td style=\"height: 15px; width: 27.5641%; vertical-align: top;\"><strong>Chance Experiment<\/strong><\/td>\n<td style=\"height: 15px; width: 27.6709%; vertical-align: top;\"><strong>Sample Space <em>S<\/em><\/strong><\/td>\n<td style=\"height: 15px; width: 44.6581%; vertical-align: top;\"><strong>Events<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 10px;\">\n<td style=\"height: 22px; width: 27.5641%; vertical-align: top;\">Flip a balanced coin<\/td>\n<td style=\"height: 22px; width: 27.6709%; vertical-align: top;\">{H, T} where H: head, T: tail<\/td>\n<td style=\"height: 22px; width: 44.6581%; vertical-align: top;\">E = observe a head = {H}<\/td>\n<\/tr>\n<tr style=\"height: 40px;\">\n<td style=\"height: 54px; width: 27.5641%; vertical-align: top;\">Roll a fair die<\/td>\n<td style=\"height: 54px; width: 27.6709%; vertical-align: top;\">{1, 2, 3, 4, 5, 6}<\/td>\n<td style=\"height: 54px; width: 44.6581%; vertical-align: top;\">E = observe a six = {6}<br \/>\nA = observe even numbers = {2, 4, 6}<br \/>\nB=outcome is less than 3 = {1, 2}<\/td>\n<\/tr>\n<tr style=\"height: 58px;\">\n<td style=\"height: 37px; width: 27.5641%; vertical-align: top;\">Flip a balanced coin twice<\/td>\n<td style=\"height: 37px; width: 27.6709%; vertical-align: top;\">{HH, HT, TT, TH}<\/td>\n<td style=\"height: 37px; width: 44.6581%; vertical-align: top;\">E = observe the same outcome = {HH, TT}<br \/>\nA = observe at least one head = {HH, HT, TH}<\/td>\n<\/tr>\n<\/thead>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>For example, consider rolling a fair die. The sample space [latex]S = \\{1, 2, 3, 4, 5, 6\\}[\/latex]\u00a0consists of six sample points, while the event observing even numbers [latex]A = \\{2, 4, 6 \\}[\/latex]<strong>\u00a0<\/strong>contains three sample points, which are part of the sample space.<\/p>\n<div style=\"height: 55px; margin-top: 2.1428571429em;\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-99 alignleft\" src=\"https:\/\/openbooks.macewan.ca\/rcommander\/wp-content\/uploads\/sites\/8\/2020\/06\/activity.png\" alt=\"\" width=\"250\" height=\"50\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/06\/activity.png 250w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/06\/activity-65x13.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/06\/activity-225x45.png 225w\" sizes=\"auto, (max-width: 250px) 100vw, 250px\" \/><\/p>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise: Basic Concepts<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Consider the chance experiment of rolling a fair die twice.<\/p>\n<ol>\n<li>Identify the sample space <em>S<\/em>.<\/li>\n<li>List all possible outcomes of the event that the two rolls give the same result.<\/li>\n<li>List all possible outcomes of the event that at least one six is observed.<\/li>\n<\/ol>\n<details>\n<summary>Show\/Hide Answer<\/summary>\n<ol>\n<li>The sample space [latex]S[\/latex] contains [latex]6 \\times 6[\/latex] pairs in the form of (1, 1), (1, 2), \u2026, (1, 6), (2, 1), (2, 2), \u2026, (2, 6), \u2026, (6,1), (6, 2), \u2026, (6, 6).<\/li>\n<li>E = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}<\/li>\n<li>E = {(1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5)}<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n","protected":false},"author":19,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-333","chapter","type-chapter","status-publish","hentry"],"part":327,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/333","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/users\/19"}],"version-history":[{"count":17,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/333\/revisions"}],"predecessor-version":[{"id":5276,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/333\/revisions\/5276"}],"part":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/parts\/327"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/333\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=333"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=333"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=333"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=333"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}