{"id":1716,"date":"2024-05-24T11:14:21","date_gmt":"2024-05-24T15:14:21","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&#038;p=1716"},"modified":"2024-06-12T14:48:38","modified_gmt":"2024-06-12T18:48:38","slug":"calculating-the-area-between-x-values","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/calculating-the-area-between-x-values\/","title":{"raw":"Calculating Probabilities Between Values","rendered":"Calculating Probabilities Between 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 probabilities above or below [latex]x[\/latex]-values.\r\n\r\n<\/div>\r\n<\/div>\r\nWe know that the probability being between two values, [latex]x_1[\/latex] and [latex]x_2[\/latex] is:\r\n\\[P(x_1 \\le x \\le x_2) = (x_2 - x_1) \\times \\frac{1}{b-a} = \\frac{x_2 - x_1}{b-a} \\]\r\n\r\nWhere the following is true:\r\n<ul>\r\n \t<li>[latex]a[\/latex] = minimum (lowest) value<\/li>\r\n \t<li>[latex]b[\/latex] = maximum (highest) value<\/li>\r\n \t<li>[latex]x_1[\/latex] = lowest [latex]x[\/latex] value<\/li>\r\n \t<li>[latex]x_2[\/latex] = highest [latex]x[\/latex] value<\/li>\r\n<\/ul>\r\n<h1>Bus Arrival Times (Example)<\/h1>\r\nLet us look at some examples below that go through calculating the probabilities of being between [latex]x[\/latex]-values.\r\n<h2>Example 34.1.1<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: During peak periods, the times between <a href=\"https:\/\/www.translink.ca\/schedules-and-maps\/route\/130\/direction\/0\/schedule\">130<\/a> busses arriving at <a href=\"https:\/\/www.bcit.ca\/\">BCIT<\/a> follow a continuous uniform distribution with a minimum of 2 minutes and a maximum of 12 minutes.\r\n\r\n<strong><span style=\"color: #003366\">Question<\/span><\/strong>: What is the probability that it takes between 5 to 10 minutes for the next bus to arrive?\r\n\r\n<span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: This gives the following values:\r\n<ul>\r\n \t<li>[latex]a=2[\/latex]<\/li>\r\n \t<li>[latex]b=12[\/latex]<\/li>\r\n \t<li>[latex]x_1 = 5[\/latex]<\/li>\r\n \t<li>[latex]x_2 = 10[\/latex]<\/li>\r\n<\/ul>\r\nPlugging this all into the formula gives: \\[ P(5\\le x \\le 10) = \\frac{x_2-x_1}{b-a} = \\frac{10-5}{12-2} = 0.5 \\]\r\n\r\n<span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: There is a 50% chance that the 130 bus will take between 5 to 10 minutes to arrive.\r\n<h1>Bus Arrival Times Example Continued (Video)<\/h1>\r\nLet us now change up the problem slightly and show the solution in a video (see below).\r\n<h2>Example 34.1.2<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Your friend was waiting for you at the bus stop and texted that you JUST missed the bus. You are hoping to join them to watch a movie. If you can catch a bus in the next 10 minutes, you will make it the movie on time.\r\n<ul>\r\n \t<li>It takes you 2 minutes to pack up your stuff.<\/li>\r\n \t<li>It takes 1 minute to run to the bus stop.<\/li>\r\n \t<li>The times between busses are between 2 and 12 minutes<\/li>\r\n \t<li>The times follow a continuous uniform distribution<\/li>\r\n<\/ul>\r\n<strong><span style=\"color: #003366\">Question<\/span><\/strong>: What is the probability of making it to the movie on time?\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-Between-X-Values.pdf\">Click here<\/a>\u00a0to download the written solutions. Also, see the video below:\r\n\r\nhttps:\/\/youtu.be\/XBJa3GbWuw8\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 Between Values<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[h5p id=\"90\"]\r\n\r\n[h5p id=\"93\"]\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 probabilities above or below [latex]x[\/latex]-values.<\/p>\n<\/div>\n<\/div>\n<p>We know that the probability being between two values, [latex]x_1[\/latex] and [latex]x_2[\/latex] is:<br \/>\n\\[P(x_1 \\le x \\le x_2) = (x_2 &#8211; x_1) \\times \\frac{1}{b-a} = \\frac{x_2 &#8211; x_1}{b-a} \\]<\/p>\n<p>Where the following is true:<\/p>\n<ul>\n<li>[latex]a[\/latex] = minimum (lowest) value<\/li>\n<li>[latex]b[\/latex] = maximum (highest) value<\/li>\n<li>[latex]x_1[\/latex] = lowest [latex]x[\/latex] value<\/li>\n<li>[latex]x_2[\/latex] = highest [latex]x[\/latex] value<\/li>\n<\/ul>\n<h1>Bus Arrival Times (Example)<\/h1>\n<p>Let us look at some examples below that go through calculating the probabilities of being between [latex]x[\/latex]-values.<\/p>\n<h2>Example 34.1.1<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: During peak periods, the times between <a href=\"https:\/\/www.translink.ca\/schedules-and-maps\/route\/130\/direction\/0\/schedule\">130<\/a> busses arriving at <a href=\"https:\/\/www.bcit.ca\/\">BCIT<\/a> follow a continuous uniform distribution with a minimum of 2 minutes and a maximum of 12 minutes.<\/p>\n<p><strong><span style=\"color: #003366\">Question<\/span><\/strong>: What is the probability that it takes between 5 to 10 minutes for the next bus to arrive?<\/p>\n<p><span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: This gives the following values:<\/p>\n<ul>\n<li>[latex]a=2[\/latex]<\/li>\n<li>[latex]b=12[\/latex]<\/li>\n<li>[latex]x_1 = 5[\/latex]<\/li>\n<li>[latex]x_2 = 10[\/latex]<\/li>\n<\/ul>\n<p>Plugging this all into the formula gives: \\[ P(5\\le x \\le 10) = \\frac{x_2-x_1}{b-a} = \\frac{10-5}{12-2} = 0.5 \\]<\/p>\n<p><span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: There is a 50% chance that the 130 bus will take between 5 to 10 minutes to arrive.<\/p>\n<h1>Bus Arrival Times Example Continued (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 34.1.2<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Your friend was waiting for you at the bus stop and texted that you JUST missed the bus. You are hoping to join them to watch a movie. If you can catch a bus in the next 10 minutes, you will make it the movie on time.<\/p>\n<ul>\n<li>It takes you 2 minutes to pack up your stuff.<\/li>\n<li>It takes 1 minute to run to the bus stop.<\/li>\n<li>The times between busses are between 2 and 12 minutes<\/li>\n<li>The times follow a continuous uniform distribution<\/li>\n<\/ul>\n<p><strong><span style=\"color: #003366\">Question<\/span><\/strong>: What is the probability of making it to the movie on time?<\/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-Between-X-Values.pdf\">Click here<\/a>\u00a0to download the written solutions. Also, see the video below:<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"How to calculate the probability between two values for continuous uniform distributions\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/XBJa3GbWuw8?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 Between Values<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"h5p-90\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-90\" class=\"h5p-iframe\" data-content-id=\"90\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Continuous Uniform - Probability Between X-Values Key Takeaways\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-93\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-93\" class=\"h5p-iframe\" data-content-id=\"93\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Continuous Uniform - Probability Between X-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":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1716","chapter","type-chapter","status-publish","hentry"],"part":1701,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1716","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\/1716\/revisions"}],"predecessor-version":[{"id":1961,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1716\/revisions\/1961"}],"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\/1716\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=1716"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1716"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=1716"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=1716"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}