{"id":1747,"date":"2024-05-24T23:20:06","date_gmt":"2024-05-25T03:20:06","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&#038;p=1747"},"modified":"2024-06-12T14:49:03","modified_gmt":"2024-06-12T18:49:03","slug":"calculating-the-probability-above-or-below-values","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/calculating-the-probability-above-or-below-values\/","title":{"raw":"Calculating Probabilities Above\/Below Values","rendered":"Calculating Probabilities Above\/Below Values"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nCalculate the probability of [latex]x[\/latex] being at least, more than, at most or less than a value.\r\n\r\n<\/div>\r\n<\/div>\r\n<h2>Probability of Exactly One Value<\/h2>\r\nFor continuous distributions:\r\n<ul>\r\n \t<li>The probability of being exactly at one value is zero.<\/li>\r\n \t<li>Ie: [latex]P(x= X)=0[\/latex]<\/li>\r\n<\/ul>\r\nBecause of this:\r\n<ul>\r\n \t<li>[latex]P(x\\ge X)=P(x= X)+P(x\\gt X) = 0+P(x\\gt X) = P(x\\gt X)[\/latex]<\/li>\r\n \t<li>[latex]P(x\\le X)= P(x= X)+P(x\\lt X) = 0+P(x\\lt X) = P(x\\lt X) [\/latex]<\/li>\r\n<\/ul>\r\n<h2>At Least or More Than<\/h2>\r\nFor the probability of at least or more than [latex]X[\/latex], ie: [latex]P(x\\ge X)[\/latex] or [latex]P(x\\gt X)[\/latex]\r\n<ul>\r\n \t<li>[latex]x_1 = X [\/latex] (the lowest value in the [latex]x[\/latex]-range)<\/li>\r\n \t<li>[latex]x_2 = b [\/latex] (the highest possible [latex]x[\/latex]-value)<\/li>\r\n<\/ul>\r\nThis gives [latex]P(x_1 \\le x \\le x_2) =\u00a0 \\frac{x_2 - x_1}{b-a} = \\frac{b-X}{b-a}[\/latex].\r\n<h2>At Most or Less Than<\/h2>\r\nFor the probability of at most or less than [latex]X[\/latex], ie: [latex]P(x\\le X)[\/latex] or [latex]P(x\\lt X)[\/latex]:\r\n<ul>\r\n \t<li>[latex]x_1 = a [\/latex] (the lowest possible [latex]x[\/latex]-value)<\/li>\r\n \t<li>[latex]x_2 = X [\/latex] (the highest value in the [latex]x[\/latex]-range)<\/li>\r\n<\/ul>\r\nThis gives [latex]P(x_1 \\le x \\le x_2) =\u00a0 \\frac{x_2 - x_1}{b-a} = \\frac{X-a}{b-a}[\/latex]\r\n<h1>Travel Times (At Least Exercise)<\/h1>\r\nLet us look at calculating the probabilities of at least or more than a certain [latex]x[\/latex] value.\r\n<h2>Example 35.1.1<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: During the morning commute, the time it takes to drive to BCIT follows a uniform distribution and is between 30 and 55 minutes. You slept in this morning and woke up 45 minutes before your class is about to begin.\r\n\r\n<strong><span style=\"color: #003366\">Question<\/span><\/strong>: If it only takes you 5 minutes to get ready, what are the odds that you will be late for class?\r\n\r\n<span style=\"color: #003366\"><strong>You Try<\/strong><\/span>: First, let's fill in the values for [latex]a[\/latex], [latex]b[\/latex], [latex]x_1[\/latex], and [latex]x_2[\/latex]:\r\n\r\n[h5p id=\"94\"]\r\n\r\n<span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: Next, let's plug them into the equation:\r\n\r\n\\[ P(\\text{late for class}) = P(x \\ge 40) = \\frac{x_2-x_1}{b-a}=\\frac{55-40}{55-30} = 0.6 \\]\r\n\r\n<span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: There is a 60% chance that you will be late for class this morning.\r\n\r\n<span style=\"color: #003366\"><strong>Hints for Exercise<\/strong><\/span>: Click below to reveal explanations for the above exercise.\r\n\r\n[h5p id=\"95\"]\r\n<h1>Travel Times Example Continued (Less Than Video)<\/h1>\r\nLet us now change up the problem slightly and show the solution in a video (see below).\r\n<h2>Example 35.1.2<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Let us continue on with the previous example:\r\n<ul>\r\n \t<li>You have 40 minutes to get to class.<\/li>\r\n \t<li>Your commute times are between 30 and 55 minutes.<\/li>\r\n \t<li>The times follow a uniform distribution.<\/li>\r\n<\/ul>\r\n<strong><span style=\"color: #003366\">Question<\/span><\/strong>: What is the probability of being on time or early for class?\r\n\r\n<strong><span style=\"color: #003366\">Solution<\/span><\/strong>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/Continuous-Uniform-Probability-of-At-Most-X-Value.pdf\">Click here<\/a> to download the written solutions. Also, see the video below:\r\n\r\nhttps:\/\/youtu.be\/tNSs4gkHU9Y\r\n<h1>Key Takeaways (EXERCISE)<\/h1>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Takeaways: Calculating Probabilities Above\/Below Values<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[h5p id=\"96\"]\r\n\r\n[h5p id=\"97\"]\r\n\r\n<\/div>\r\n<\/div>\r\n<h1>Your Own Notes (EXERCISE)<\/h1>\r\n<ul>\r\n \t<li>Are there any notes you want to take from this section? Is there anything you'd like to copy and paste below?<\/li>\r\n \t<li>These notes are for you only (they will not be stored anywhere)<\/li>\r\n \t<li>Make sure to download them at the end to use as a reference<\/li>\r\n<\/ul>\r\n[h5p id=\"16\"]","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Calculate the probability of [latex]x[\/latex] being at least, more than, at most or less than a value.<\/p>\n<\/div>\n<\/div>\n<h2>Probability of Exactly One Value<\/h2>\n<p>For continuous distributions:<\/p>\n<ul>\n<li>The probability of being exactly at one value is zero.<\/li>\n<li>Ie: [latex]P(x= X)=0[\/latex]<\/li>\n<\/ul>\n<p>Because of this:<\/p>\n<ul>\n<li>[latex]P(x\\ge X)=P(x= X)+P(x\\gt X) = 0+P(x\\gt X) = P(x\\gt X)[\/latex]<\/li>\n<li>[latex]P(x\\le X)= P(x= X)+P(x\\lt X) = 0+P(x\\lt X) = P(x\\lt X)[\/latex]<\/li>\n<\/ul>\n<h2>At Least or More Than<\/h2>\n<p>For the probability of at least or more than [latex]X[\/latex], ie: [latex]P(x\\ge X)[\/latex] or [latex]P(x\\gt X)[\/latex]<\/p>\n<ul>\n<li>[latex]x_1 = X[\/latex] (the lowest value in the [latex]x[\/latex]-range)<\/li>\n<li>[latex]x_2 = b[\/latex] (the highest possible [latex]x[\/latex]-value)<\/li>\n<\/ul>\n<p>This gives [latex]P(x_1 \\le x \\le x_2) =\u00a0 \\frac{x_2 - x_1}{b-a} = \\frac{b-X}{b-a}[\/latex].<\/p>\n<h2>At Most or Less Than<\/h2>\n<p>For the probability of at most or less than [latex]X[\/latex], ie: [latex]P(x\\le X)[\/latex] or [latex]P(x\\lt X)[\/latex]:<\/p>\n<ul>\n<li>[latex]x_1 = a[\/latex] (the lowest possible [latex]x[\/latex]-value)<\/li>\n<li>[latex]x_2 = X[\/latex] (the highest value in the [latex]x[\/latex]-range)<\/li>\n<\/ul>\n<p>This gives [latex]P(x_1 \\le x \\le x_2) =\u00a0 \\frac{x_2 - x_1}{b-a} = \\frac{X-a}{b-a}[\/latex]<\/p>\n<h1>Travel Times (At Least Exercise)<\/h1>\n<p>Let us look at calculating the probabilities of at least or more than a certain [latex]x[\/latex] value.<\/p>\n<h2>Example 35.1.1<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: During the morning commute, the time it takes to drive to BCIT follows a uniform distribution and is between 30 and 55 minutes. You slept in this morning and woke up 45 minutes before your class is about to begin.<\/p>\n<p><strong><span style=\"color: #003366\">Question<\/span><\/strong>: If it only takes you 5 minutes to get ready, what are the odds that you will be late for class?<\/p>\n<p><span style=\"color: #003366\"><strong>You Try<\/strong><\/span>: First, let&#8217;s fill in the values for [latex]a[\/latex], [latex]b[\/latex], [latex]x_1[\/latex], and [latex]x_2[\/latex]:<\/p>\n<div id=\"h5p-94\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-94\" class=\"h5p-iframe\" data-content-id=\"94\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 35.1.1 - Uniform - values of a, b, x1 and x2\"><\/iframe><\/div>\n<\/div>\n<p><span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: Next, let&#8217;s plug them into the equation:<\/p>\n<p>\\[ P(\\text{late for class}) = P(x \\ge 40) = \\frac{x_2-x_1}{b-a}=\\frac{55-40}{55-30} = 0.6 \\]<\/p>\n<p><span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: There is a 60% chance that you will be late for class this morning.<\/p>\n<p><span style=\"color: #003366\"><strong>Hints for Exercise<\/strong><\/span>: Click below to reveal explanations for the above exercise.<\/p>\n<div id=\"h5p-95\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-95\" class=\"h5p-iframe\" data-content-id=\"95\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 35.1.1 \u2013 Uniform Example Solution\"><\/iframe><\/div>\n<\/div>\n<h1>Travel Times Example Continued (Less Than Video)<\/h1>\n<p>Let us now change up the problem slightly and show the solution in a video (see below).<\/p>\n<h2>Example 35.1.2<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Let us continue on with the previous example:<\/p>\n<ul>\n<li>You have 40 minutes to get to class.<\/li>\n<li>Your commute times are between 30 and 55 minutes.<\/li>\n<li>The times follow a uniform distribution.<\/li>\n<\/ul>\n<p><strong><span style=\"color: #003366\">Question<\/span><\/strong>: What is the probability of being on time or early for class?<\/p>\n<p><strong><span style=\"color: #003366\">Solution<\/span><\/strong>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/Continuous-Uniform-Probability-of-At-Most-X-Value.pdf\">Click here<\/a> to download the written solutions. Also, see the video below:<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Probability of At Most X for a Continuous Uniform Distribution\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/tNSs4gkHU9Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h1>Key Takeaways (EXERCISE)<\/h1>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways: Calculating Probabilities Above\/Below Values<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"h5p-96\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-96\" class=\"h5p-iframe\" data-content-id=\"96\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Continuous Uniform - Calculating Probabilities Above\/Below Values Key Takeaways\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-97\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-97\" class=\"h5p-iframe\" data-content-id=\"97\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Continuous Uniform - Calculating Probabilities Above\/Below Values Key Takeaways Solutions\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Your Own Notes (EXERCISE)<\/h1>\n<ul>\n<li>Are there any notes you want to take from this section? Is there anything you&#8217;d like to copy and paste below?<\/li>\n<li>These notes are for you only (they will not be stored anywhere)<\/li>\n<li>Make sure to download them at the end to use as a reference<\/li>\n<\/ul>\n<div id=\"h5p-16\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-16\" class=\"h5p-iframe\" data-content-id=\"16\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key takeaways, notes and comments from this section document tool.\"><\/iframe><\/div>\n<\/div>\n","protected":false},"author":865,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1747","chapter","type-chapter","status-publish","hentry"],"part":1701,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1747","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/users\/865"}],"version-history":[{"count":25,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1747\/revisions"}],"predecessor-version":[{"id":1962,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1747\/revisions\/1962"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/1701"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1747\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=1747"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1747"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=1747"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=1747"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}