{"id":169,"date":"2023-05-05T14:33:42","date_gmt":"2023-05-05T18:33:42","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/chapter\/__unknown__\/"},"modified":"2024-01-12T18:52:39","modified_gmt":"2024-01-12T23:52:39","slug":"probabilities","status":"web-only","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/chapter\/probabilities\/","title":{"raw":"Probabilities","rendered":"Probabilities"},"content":{"raw":"<div class=\"__UNKNOWN__\">\r\n\r\n<em>Note:\u00a0 I'm posting this, but the formulae definitely need work!<\/em>\r\n<h1 class=\"import-Normal\">Why are Probabilities Important?<\/h1>\r\n<p class=\"import-Normal\">But first, some VERY FAST recap of the theory of probabilities:<\/p>\r\n\r\n<h2>Probability Distributions<\/h2>\r\n<p class=\"import-Normal\">A [pb_glossary id=\"177\"]<strong>Random Variable<\/strong> [\/pb_glossary]is variable (typically represented by X) that has a single numerical value (determined by chance) for each outcome of a procedure (or experiment).<\/p>\r\n\r\n<ul>\r\n \t<li class=\"import-Normal\">For example, X = the velocity of an unladen swallow<\/li>\r\n \t<li class=\"import-Normal\">X = number of tabs open on your browser as you read this book.<\/li>\r\n<\/ul>\r\n<p class=\"import-Normal\">There are two types of random variable:<\/p>\r\n<p class=\"import-Normal\">A [pb_glossary id=\"178\"]<strong>Discrete Random Variable<\/strong>[\/pb_glossary] has either a finite number of values or a countable number of values, that is, it results from a counting process.<\/p>\r\n\r\n<ul>\r\n \t<li class=\"import-Normal\">For example, the number of BCIT grads among 50 newly hired employees<\/li>\r\n \t<li class=\"import-Normal\">The number of lemons in my fridge right now.<\/li>\r\n<\/ul>\r\n<p class=\"import-Normal\">A <strong>[pb_glossary id=\"179\"]Continuous Random Variable[\/pb_glossary]<\/strong> has infinitely many values, which can be associated with measurements on a continuous scale in such a way that there are no gaps or interruptions.<\/p>\r\n\r\n<ul>\r\n \t<li class=\"import-Normal\">ie, the height of a randomly selected BCIT student<\/li>\r\n \t<li class=\"import-Normal\">The time at which you close this window and stop reviewing probabilities<\/li>\r\n<\/ul>\r\n[h5p id=\"1\"]\r\n<p class=\"import-Normal\"><em>Example<\/em><\/p>\r\n[h5p id=\"2\"]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">Definitions:<\/div>\r\nThe union of two sets is denoted by [latex]A\\cup B[\/latex] and this is the OR.\r\n\r\nThe [pb_glossary id=\"263\"]intersection [\/pb_glossary]of two sets is denoted by [latex]A\\cap B[\/latex] and this is the AND.\r\n\r\n&nbsp;\r\n<h3>[pb_glossary id=\"180\"]Mutually Exclusive[\/pb_glossary]:<\/h3>\r\nA and B are mutually exclusive if P(A and B) = 0.\r\n\r\nFor example, being a heart and a spade.\r\n<div class=\"__UNKNOWN__\">\r\n<h3>[pb_glossary id=\"181\"]Compliments[\/pb_glossary]:<\/h3>\r\nA and B are compliments if they are mutually exclusive and mutually exhaustive.\u00a0 That is:\r\n\r\n[latex]P(A \\, and \\, B) = 0[\/latex]\u00a0 and [latex]P(A \\, or \\, B) = 1[\/latex]\r\n<h3>Conditional Probability of an Event<\/h3>\r\n<p class=\"import-Normal\">Given A and B are events in an experiment and<\/p>\r\n\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(A)\u22600, <\/span>then the <strong>conditional <\/strong><strong>probability<\/strong> that event <span style=\"font-size: NaNpt;color: #;text-decoration: none\">B <\/span>will occur given that the event <span style=\"font-size: NaNpt;color: #;text-decoration: none\">A <\/span><strong>has<\/strong> <strong>already occurred<\/strong> is:<\/div>\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[P(B|A)=\\frac{P(A \\cap B)}{P(A)}\\]<\/span><\/p>\r\n<p class=\"import-Normal\">Rearranging the above formula gives:<\/p>\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[P(A \\cap B)=P(A) \\times P(B|A)\\]<\/span><\/p>\r\n\r\n<h3 style=\"text-align: left\">Independence vs Dependence<\/h3>\r\n<p class=\"import-Normal\">We say two events are <strong>independent<\/strong> if the occurrence of one does not affect the other.<\/p>\r\n<p class=\"import-Normal\">We say two events are <span style=\"font-size: 18.6667px\"><b>dependent<\/b><\/span>, if they are not independent, that is the occurrence of one affects the occurrence of the other.<\/p>\r\n\r\n<h3>Probability Tables and Contingency Tables (also called joint distribution tables)<\/h3>\r\n<p class=\"import-Normal\">Put one <em lang=\"en-US\" xml:lang=\"en-US\">mutually exclusive<\/em> event and its complement along the top.<\/p>\r\n<p class=\"import-Normal\">Put the other <em lang=\"en-US\" xml:lang=\"en-US\">mutually exclusive<\/em> event and its complement along the left side.<\/p>\r\n<p class=\"import-Normal\">Inside the table put the \u201cAND\u201d probabilities.<\/p>\r\n\r\n<table>\r\n<tbody>\r\n<tr class=\"TableNormal-R\" style=\"height: 13.4pt\">\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">A<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">A\u0305<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Total<\/strong><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableNormal-R\" style=\"height: 13.4pt\">\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">B<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(A and B)<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(A\u0305 and B)<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(B)<\/span><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableNormal-R\" style=\"height: 13.4pt\">\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">B\u0305<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(A and B\u0305)<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(A\u0305 and B\u0305)<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(B\u0305)<\/span><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableNormal-R\" style=\"height: 2.5pt\">\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Total<\/strong><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(A)<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(A\u0305)<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">1<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercises<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p class=\"import-Normal\"><em>Example<\/em><\/p>\r\n<p class=\"import-Normal\">In your stats computer lab, you decide to conduct of study of the ages of 300 business students and whether or not they live with their parents. You obtain the following results:<\/p>\r\n\r\n<table style=\"width: 306pt\">\r\n<tbody>\r\n<tr class=\"TableNormal-R\" style=\"height: 11.35pt\">\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;padding: 0pt 5.4pt 0pt 5.4pt;border: none windowtext 0pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;padding: 0pt 5.4pt 0pt 5.4pt;border: none windowtext 0pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;padding: 0pt 5.4pt 0pt 5.4pt;border: none windowtext 0pt\" colspan=\"3\">\r\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">Ages<\/em><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;padding: 0pt 5.4pt 0pt 5.4pt;border: none windowtext 0pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableNormal-R\" style=\"height: 17pt\">\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;padding: 0pt 5.4pt 0pt 5.4pt;border: none windowtext 0pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: none windowtext 0pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;border-left: none windowtext 0pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: none windowtext 0pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">18 \u2013 27<\/strong><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: none windowtext 0pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">28 \u2013 37<\/strong><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: none windowtext 0pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">38 \u2013 47<\/strong><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: none windowtext 0pt;border-right: none windowtext 0pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Total<\/strong><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableNormal-R\" style=\"height: 0\">\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;padding: 0pt 5.4pt 0pt 5.4pt;border: none windowtext 0pt\" rowspan=\"2\">\r\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">Live with<\/em><em lang=\"en-US\" xml:lang=\"en-US\"><br style=\"clear: both\" \/>Parents?<\/em><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;border-bottom: none windowtext 0pt;border-left: none windowtext 0pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Yes<\/strong><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">185<\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">5<\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">0<\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: none windowtext 0pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">190<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableNormal-R\" style=\"height: 0\">\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: none windowtext 0pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;border-left: none windowtext 0pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">No<\/strong><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: none windowtext 0pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">95<\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: none windowtext 0pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">11<\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: none windowtext 0pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">4<\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: none windowtext 0pt;border-right: none windowtext 0pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">110<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableNormal-R\">\r\n<td class=\"TableNormal-C\" style=\"padding: 0pt 5.4pt 0pt 5.4pt;border: none windowtext 0pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;border-bottom: none windowtext 0pt;border-left: none windowtext 0pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Total<\/strong><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">280<\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">16<\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">4<\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: none windowtext 0pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">300<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol type=\"a\">\r\n \t<li class=\"import-Normal\">Give a <em lang=\"en-US\" xml:lang=\"en-US\">non-trivial<\/em> example of two mutually exclusive events in this study.<\/li>\r\n \t<li class=\"import-Normal\">If a student is at least 28 years old, what is the probability they do <strong lang=\"en-US\" xml:lang=\"en-US\">not<\/strong> live with their parents?<\/li>\r\n \t<li class=\"import-Normal\">What is the probability a student lives with their parents <strong lang=\"en-US\" xml:lang=\"en-US\">or<\/strong> is from 28 to 37 years old?<\/li>\r\n \t<li class=\"import-Normal\">What is the probability of a student <em lang=\"en-US\" xml:lang=\"en-US\">neither<\/em> living with their parents <em lang=\"en-US\" xml:lang=\"en-US\">nor<\/em> being from <br style=\"clear: both\" \/>28 to 37 years old?<\/li>\r\n \t<li class=\"import-Normal\">Are <em lang=\"en-US\" xml:lang=\"en-US\">ages<\/em> and <em lang=\"en-US\" xml:lang=\"en-US\">living with <\/em><em lang=\"en-US\" xml:lang=\"en-US\">parents<\/em> independent events?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n<h3>What\u2019s a Probability Distribution?<\/h3>\r\n<p class=\"import-Normal\">A <strong>Probability Distribution<\/strong> is a graph, table, or formula that gives the probability of each value of the random variable.<\/p>\r\n<p class=\"import-Normal\">There are two requirements for a probability distribution:<\/p>\r\n\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[\\Sigma P(x)=1\\]<\/span><\/div>\r\nwhere x assumes all possible values (\u2211 refers to the sum)\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[0\\leq P(x)\\leq 1\\]<\/span><\/div>\r\nfor every value of [latex]x[\/latex]\r\n<h3>Expected Value or Mean of a Probability Distribution<\/h3>\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[\\mu=\\Sigma x P(x)=E(x)\\]<\/span><\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Examples<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p class=\"import-Normal\"><em>Example<\/em><\/p>\r\n<p class=\"import-Normal\">A rock concert producer has scheduled an outdoor concert. If it is warm that day, she expects to make a $20,000 profit. If it is cool that day, she expects to make a $5,000 profit. If it is very cold, she expects to suffer a $12,000 loss. Based upon historical data, the weather office has estimated the chances of a warm day to be 0.60; the chances of a cool day to be 0.25.<\/p>\r\n<p class=\"import-Normal\">Construct a probability distribution. <em lang=\"en-US\" xml:lang=\"en-US\">P(x)<\/em> = probability that <strong lang=\"en-US\" xml:lang=\"en-US\"><em>x<\/em><\/strong> occurs<\/p>\r\n\r\n<table style=\"width: 347.4pt\">\r\n<tbody>\r\n<tr class=\"TableNormal-R\">\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 1pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">Weather<\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">X<\/em> = Profit<\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">P(x)<\/em><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">x\u00d7 P(x) <\/em><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableNormal-R\" style=\"height: 19.6pt\">\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 1pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">Warm<\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 1pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableNormal-R\" style=\"height: 19.6pt\">\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 1pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">Cool<\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 1pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"padding: 0pt 5.4pt 0pt 5.4pt;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableNormal-R\" style=\"height: 19.6pt\">\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 1pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">Very Cold<\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 1pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 0.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableNormal-R\" style=\"height: 19.45pt\">\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 1pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\">1.00<\/p>\r\n<\/td>\r\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol type=\"a\">\r\n \t<li>What is the producer\u2019s expected profit (or loss)?<\/li>\r\n \t<li>What is the probability that she will make a profit?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n<h2>Different Probability Distributions<\/h2>\r\n<h3>The Discrete Uniform Distribution<\/h3>\r\n<p class=\"import-Normal\">Every outcome from <em>a<\/em> to <em>b<\/em> is equally likely.<\/p>\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[P(X)=\\frac{1}{b-a+1}\\]<\/span><\/p>\r\n<p class=\"import-Normal\">For all <span style=\"font-size: NaNpt;color: #;text-decoration: none\">[latex]a\u2264X\u2264b[\/latex].<\/span><\/p>\r\n&nbsp;\r\n<p class=\"import-Normal\"><strong><em>Mean and Standard Deviation of a Discrete Uniform Distribution<\/em><\/strong><\/p>\r\n\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[\\mu=E(X)=\\frac{a+b}{2}\\]<\/span><\/div>\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[\\sigma=\\frac{b-a+12}{-112}\\] <\/span><\/div>\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u00a0<\/span><\/div>\r\n<div><\/div>\r\n[h5p id=\"3\"]\r\n<h3>The Binomial Distribution<\/h3>\r\n<ul>\r\n \t<li class=\"import-Normal\">The experiment must have a <strong><em>fixed number<\/em><\/strong> of <em>identical<\/em> trials.<\/li>\r\n \t<li class=\"import-Normal\">Each trial must have all outcomes classified into two categories (a <strong><em>success<\/em><\/strong> or <strong><em>failure<\/em><\/strong>)<\/li>\r\n \t<li class=\"import-Normal\">The trials must be <strong><em>independent<\/em><\/strong> (the outcome of any individual trial doesn't influence the outcome of another trial).<\/li>\r\n \t<li class=\"import-Normal\">The probabilities must remain <strong><em>constant\/same<\/em><\/strong> for each trial.<\/li>\r\n \t<li class=\"import-Normal\">The variable of interest must be the <strong><em>number of successes<\/em><\/strong> in <em>n<\/em> trials.<\/li>\r\n<\/ul>\r\nTerminology:\r\n<ul>\r\n \t<li class=\"import-Normal\">[latex]n[\/latex] = fixed number of trials<\/li>\r\n \t<li class=\"import-Normal\">[latex]x[\/latex] = specific number of successes in n trials<\/li>\r\n \t<li class=\"import-Normal\">[latex]p[\/latex]= probability of success in each one of the n trials<\/li>\r\n \t<li class=\"import-Normal\">[latex]1-p[\/latex]= probability of failure in each one of the n trials[latex]p[\\latex]<\/li>\r\n \t<li class=\"import-Normal\">[latex]P(x)[\/latex]= probability of getting exactly <em>x<\/em> successes among <em>n<\/em> trials<\/li>\r\n<\/ul>\r\n<h2 class=\"import-Normal\"><strong><em>Combinations<\/em><\/strong><\/h2>\r\n<p class=\"import-Normal\"><em>How many ways can you choose x things out of a list of n things?<\/em><\/p>\r\n\\[_nC_x= \\binom{n}{x} = \\frac{n!}{x!(n-x)!}\\]\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p class=\"import-Normal\">Where<\/p>\r\n\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[n!=n\\times (n-1)\\times (n-2)\\times \\cdots \\times 2\\times 1\\]<\/span><\/div>\r\n<p class=\"import-Normal\"><strong><em>Binomial Probability Formula<\/em><\/strong><\/p>\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[P(x) = _nC_{x} p^{x} (1-p)^{n-x}\\]\r\n<\/span><\/p>\r\n<p class=\"import-Normal\">If [latex]n = 5, p = 0.2, x = 3[\/latex]:<\/p>\r\n\\[P(x=3) = _5C_{3} 0.2^{3} (0.8)^{2}=0.0512\\]\r\n<p class=\"import-Normal\">Excel: <code>=BINOM.DIST(3, 5, 0.2,0)<\/code><\/p>\r\n<p class=\"import-Normal\">R: <code>dbinom(x = 3, size = 5, prob = 0.2)<\/code><\/p>\r\nCUMULATIVE\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[P(X\u22643)=P(x = 0)+P(x=1)+P(x=2)+P(x=3)=0.99328\\]<\/span><\/p>\r\n<p class=\"import-Normal\">Excel: <code>=BINOM.DIST(3, 5, 0.2,1)<\/code><\/p>\r\n<p class=\"import-Normal\">R: <code>pbinom(q = 3, size = 5, prob = 0.2)<\/code><\/p>\r\n<p class=\"import-Normal\"><strong><em>Mean and Standard Deviation of a Binomial Distribution<\/em><\/strong><\/p>\r\n\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[\\mu=np=E(x)\\]<\/span><\/div>\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[\\sigma=\\sqrt{n p(1-p)}\\]<\/span><\/div>\r\n<p class=\"import-Normal\">A quiz consists of 5 multiple-choice questions with 4 possible answers for each question. The student is unprepared for the exam and randomly selects answers.<\/p>\r\n\r\n<div style=\"margin: auto\">\r\n<table>\r\n<tbody>\r\n<tr class=\"TableGrid-R\" style=\"height: 0\">\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\"># of successes<\/strong><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\"># of failures<\/strong><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Probability<\/strong><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 0\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">0<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 0\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">1<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 0\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">2<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 0\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">3<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 0\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">4<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 0\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">5<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<p class=\"import-Normal\">Find the probability that the student gets:<\/p>\r\n<p class=\"import-Normal\" style=\"margin-left: 14.2pt;text-indent: 14.2pt\">all five correct<\/p>\r\n<p class=\"import-Normal\" style=\"margin-left: 14.2pt;text-indent: 14.2pt\">none correct<\/p>\r\n<p class=\"import-Normal\" style=\"margin-left: 14.2pt;text-indent: 14.2pt\">exactly two correct<\/p>\r\n<p class=\"import-Normal\" style=\"margin-left: 14.2pt;text-indent: 14.2pt\">(d) at least one correct<\/p>\r\n\r\n<h3>The Poisson Distributions<\/h3>\r\n<p class=\"import-Normal\">the random variable <em>x<\/em> is the number of occurrences of an event over some <strong>time interval<\/strong><\/p>\r\n<p class=\"import-Normal\">the occurrences must be random<\/p>\r\n<p class=\"import-Normal\">the occurrences must be independent of each other<\/p>\r\n<p class=\"import-Normal\">the occurrences must be <strong><em>uniformly distributed<\/em><\/strong> over the interval being used<\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\"><em>Examples where it applies<\/em><\/strong><strong lang=\"en-US\" xml:lang=\"en-US\">:<\/strong><\/p>\r\n<p class=\"import-Normal\">The <strong lang=\"en-US\" xml:lang=\"en-US\">number of<\/strong> car accidents in the lower mainland <strong lang=\"en-US\" xml:lang=\"en-US\">per day<\/strong>.<\/p>\r\n<p class=\"import-Normal\">The <strong lang=\"en-US\" xml:lang=\"en-US\">number of<\/strong> spam emails that a computer user will receive <strong lang=\"en-US\" xml:lang=\"en-US\">per week<\/strong>.<\/p>\r\n<p class=\"import-Normal\">The <strong lang=\"en-US\" xml:lang=\"en-US\">number of<\/strong> phone calls to a call center <strong lang=\"en-US\" xml:lang=\"en-US\">per minute<\/strong>.<\/p>\r\n<p class=\"import-Normal\"><strong><em>Notation<\/em><\/strong><\/p>\r\n<p class=\"import-Normal\">x = specific number occurrences in the time interval<\/p>\r\n\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03bc<\/span><\/div>\r\n=\u03bb = the average number of occurrences in the time interval\r\n<p class=\"import-Normal\"><strong><em>Poisson Probability Formula<\/em><\/strong><\/p>\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[P(x)=\u03bcxe-\u03bcx!=\u03bbxe-\u03bbx!\\]<\/span><\/p>\r\n<p class=\"import-Normal\">If \u03bb = 6, x = 3<\/p>\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">PX=3=63e-63!=0.089235<\/span><\/p>\r\n<p class=\"import-Normal\">Excel: =POISSON.DIST(3, 6, 0)<\/p>\r\n<p class=\"import-Normal\">R: dpois(x = 3, lambda =6)<\/p>\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">PX\u22643=P0+P1+P2+P3=0.1512039<\/span><\/p>\r\n<p class=\"import-Normal\">Excel: =POISSON.DIST(3, 6, 1)<\/p>\r\n<p class=\"import-Normal\">R: ppois(q = 3, lambda =6)<\/p>\r\n<p class=\"import-Normal\"><strong><em>Mean and Standard Deviation of a Poisson Distribution<\/em><\/strong><\/p>\r\n\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03bc=\u03bb=E(x)<\/span><\/div>\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03c3=\u03bb=\u03bc<\/span><\/div>\r\n<p class=\"import-Normal\">An internet provider receives an average of 5 calls per half hour for technical support. The calls follow a Poisson distribution.<\/p>\r\n<p class=\"import-Normal\">What is the probability there will be three calls in 30 minutes?<\/p>\r\n<p class=\"import-Normal\">What is the probability there will be at least one call in 15 minutes?<\/p>\r\n<p class=\"import-Normal\">What is the probability there will be at least 2 calls in half an hour?<\/p>\r\n<p class=\"import-Normal\"><strong><em>The Continuous Uniform Distribution<\/em><\/strong><\/p>\r\n<p class=\"import-Normal\">Every outcome from <em>a<\/em> to <em>b<\/em> is equally likely.<\/p>\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">Px1\u2264X\u2264x2=x2-x1b-a,<\/span><\/p>\r\n<p class=\"import-Normal\">For all<\/p>\r\n\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">a\u2264x1\u2264x2\u2264b<\/span><\/div>\r\n.\r\n<p class=\"import-Normal\"><strong><em>Mean and Standard Deviation of a continuous uniform Distribution<\/em><\/strong><\/p>\r\n\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03bc=Ex=a+b2<\/span><\/div>\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03c3=b-a12<\/span><\/div>\r\n<p class=\"import-Normal\">The fill volume of a regular can of pop ranges uniformly from 354 mL to 358 mL. What is the mean fill volume?<\/p>\r\n<p class=\"import-Normal\">b. What is the probability that the fill volume is between 354.3 mL and 355.2 mL?<\/p>\r\n\r\n<h3>The Normal Distribution<\/h3>\r\n<p class=\"import-Normal\">Used for continuous variables only. (e.g. Age, time, income, temperature, house prices).<\/p>\r\n<p class=\"import-Normal\">Symmetrical about the mean (mean = median = mode) and bell-shaped.<\/p>\r\n<p class=\"import-Normal\">Since there are many different normal curves, we convert all normal curves into one single curve called the standard normal (mean \u03bc = 0, standard deviation \u03c3 = 1) by finding the <em lang=\"en-US\" xml:lang=\"en-US\">z<\/em>-score. (subtract the mean and divide by the standard deviation to calculate the <em lang=\"en-US\" xml:lang=\"en-US\">z<\/em>-score).<\/p>\r\n<p class=\"import-Normal\">The area under the normal curve represents probability. The total area under the curve = 1.<\/p>\r\n<p class=\"import-Normal\">Continuous random variables can take on any value within an interval since there are no gaps between the numbers. We no longer talk about the probability of the random variable assuming a particular value. Instead, we talk about the probability of the variable assuming a value within some given interval. The probability of the random variable assuming a particular value = 0.<\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\"><em>Empirical Rule: <\/em><\/strong><em lang=\"en-US\" xml:lang=\"en-US\">Applies only to bell-shaped (normally distributed) populations<\/em><\/p>\r\n<p class=\"import-Normal\">About 68% of the values will fall within <strong lang=\"en-US\" xml:lang=\"en-US\">one <\/strong>standard deviation of the mean.<\/p>\r\n<p class=\"import-Normal\">About 95% of the values will fall within <strong lang=\"en-US\" xml:lang=\"en-US\">two <\/strong>standard deviations of the mean.<\/p>\r\n<p class=\"import-Normal\">About 99.7% (<em lang=\"en-US\" xml:lang=\"en-US\">almost<\/em> 100% of values will fall within <strong lang=\"en-US\" xml:lang=\"en-US\">three <\/strong>standard deviations of the mean).<\/p>\r\n<p class=\"import-Normal\">The selling prices of all stocks listed on the <em lang=\"en-US\" xml:lang=\"en-US\">CDNX<\/em> stock exchange are known to be normally distributed with a mean of $20 and a standard deviation of $4.<\/p>\r\n<p class=\"import-Normal\" style=\"margin-left: 28.35pt;text-indent: 21.25pt\">What percentage of stocks have a selling price between $20 and $40?<\/p>\r\n<p class=\"import-Normal\" style=\"margin-left: 28.35pt;text-indent: 21.25pt\">What is the minimum selling price of the most expensive 5% of stocks?<\/p>\r\n<p class=\"import-Normal\" style=\"margin-left: 28.35pt;text-indent: 21.25pt\">The prices of the middle 80% of stocks are between what two values?<\/p>\r\n\r\n<h3>The Exponential Distribution<\/h3>\r\n<p class=\"import-Normal\">How long between occurrences?<\/p>\r\n<p class=\"import-Normal\">Measures the time between two things in a Poisson distribution: for example, what is the probability that there is at least a 15 minute gap between customers at your coffee shop?<\/p>\r\n<p class=\"import-Normal\">Here, \u03bb (the Greek letter lambda) is the average number of occurrences in a time interval.<\/p>\r\n<p class=\"import-Normal\"><strong><em>Exponential Formula<\/em><\/strong><\/p>\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">PX\u2265a=1e\u03bba<\/span><\/p>\r\n<p class=\"import-Normal\"><strong><em>Mean and Standard Deviation of a continuous uniform Distribution<\/em><\/strong><\/p>\r\n\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03bc=Ex=1\u03bb<\/span><\/div>\r\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03c3=\u03bc=1\u03bb<\/span><\/div>\r\n<p class=\"import-Normal\">Telemarketers at ARG make an average of 3 calls per minute. Assuming that the calls follow a Poisson Distribution, what is the mean time between calls?<\/p>\r\n<p class=\"import-Normal\">What is the probability that there will be at least a 30 second gap between incoming calls?<\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\"><em>Try these at home. Which Distribution should you use?<\/em><\/strong><\/p>\r\n<p class=\"import-Normal\">A hotel\u2019s records show that 70% of its guests are from the United States. In a random sample of 10 guests, what is the probability that exactly half are from the U.S.?<\/p>\r\n<p class=\"import-Normal\">At the airport a plane takes off, on average, every 2 minutes. In any 10-minute interval, what is the probability that there are more than 2 departures?<\/p>\r\n<p class=\"import-Normal\">What is the probability that there is at least a minute between successive departures?<\/p>\r\n<p class=\"import-Normal\"><strong><em>Fun with Excel <\/em><\/strong><\/p>\r\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">0 or 1 is ALWAYS the last thing: do you want cumulative or not?<\/em><\/p>\r\n<p class=\"import-Normal\">=BINOM.DIST(x, n, p, 0 or 1) and =BINOM.INV(n,p,P)<\/p>\r\n<p class=\"import-Normal\">=POISSON.DIST(x, \u03bc, 0 or 1)<\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Excel Commands for Normal Distribution:<\/strong><\/p>\r\n<p class=\"import-Normal\">If we know \u00b5, \u03c3 and X and are trying to find the probability that x&lt;X: =norm.dist(x,\u00b5,\u03c3,1).<\/p>\r\n<p class=\"import-Normal\"><img src=\"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-content\/uploads\/sites\/1653\/2023\/05\/image1.png\" alt=\"image\" width=\"225px\" height=\"127.466666666667px\" \/>Example SEQ Example_1 \\* ARABIC 1<\/p>\r\n<p class=\"import-Normal\">(<em lang=\"en-US\" xml:lang=\"en-US\">The 1 is for cumulative, in Normal, we will use it in every command.)<\/em><\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 1:<\/strong> What is the probability that a person\u2019s IQ is less than 125? (note: \u00b5=100, \u03c3=15)<\/p>\r\n<p class=\"import-Normal\"><img src=\"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-content\/uploads\/sites\/1653\/2023\/05\/image3.png\" alt=\"image\" width=\"240px\" height=\"112px\" \/><strong lang=\"en-US\" xml:lang=\"en-US\">Example 2:<\/strong> What is the probability that an IQ is greater than 130?<\/p>\r\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">To get the right side, just subtract your answer from 1: this gives the answer on the table.<\/em><\/p>\r\n<p class=\"import-Normal\">If you have z-scores, you may use the norm.s.dist command which cuts down on the typing:<\/p>\r\n<p class=\"import-Normal\"><em><img src=\"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-content\/uploads\/sites\/1653\/2023\/05\/image4.png\" alt=\"image\" width=\"237px\" height=\"114.466666666667px\" \/>Example3\r\n<\/em><\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 3:<\/strong> What is the probability that z is less than .38?<\/p>\r\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">Again, the 1 stands for cumulative. This is the same as typing =<\/em><em lang=\"en-US\" xml:lang=\"en-US\">norm.dist<\/em><em lang=\"en-US\" xml:lang=\"en-US\">(.38,1,0,1)<\/em><\/p>\r\n<p class=\"import-Normal\"><img src=\"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-content\/uploads\/sites\/1653\/2023\/05\/image6.png\" alt=\"image\" width=\"284px\" height=\"121.466666666667px\" \/>Example 4<\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 4:<\/strong> What is the area (or probability) that Z is between -1 and 1?<\/p>\r\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">You can subtract in 2 cells or in one\u2026 Notice that the larger z score goes first!<\/em><\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Inverses:<\/strong><\/p>\r\n<p class=\"import-Normal\">If you know that the area below X is P, and you are solving for X use =norm.inv(P,\u00b5,\u03c3):<\/p>\r\n<p class=\"import-Normal\"><img src=\"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-content\/uploads\/sites\/1653\/2023\/05\/image8.png\" alt=\"image\" width=\"223px\" height=\"124.466666666667px\" \/>Example 5<\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 5:<\/strong> 75% of the population have an IQ below what score?<\/p>\r\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">Make sure your area is in decimal! It needs to be greater than 0 and less than 1. Putting in 0 or 1 will give an error message.<\/em><\/p>\r\n<p class=\"import-Normal\"><img src=\"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-content\/uploads\/sites\/1653\/2023\/05\/image10.png\" alt=\"image\" width=\"215px\" height=\"141.466666666667px\" \/>Example 6<\/p>\r\n<p class=\"import-Normal\">Or you may recover z scores:<\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 6:<\/strong> Above what Z score do you find 5% of the population?<\/p>\r\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">Since we are always working with areas to the left, we need to use .95, as 5% <\/em><em lang=\"en-US\" xml:lang=\"en-US\">above<\/em><em lang=\"en-US\" xml:lang=\"en-US\"> is 95% <\/em><em lang=\"en-US\" xml:lang=\"en-US\">below.<\/em><\/p>\r\n\r\n<h2>Fun with R<\/h2>\r\n<p class=\"import-Normal\">R is a great language for probabilities! Let\u2019s do the same Normal problems, but in R:<\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 1:<\/strong> What is the probability that a person\u2019s IQ is less than 125? (note: \u00b5=100, \u03c3=15)<\/p>\r\n<p class=\"import-Normal\">Answer: `pnorm(q = 125, mean = 100, sd = 15)`<\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 2:<\/strong> What is the probability that an IQ is greater than 130?<\/p>\r\n<p class=\"import-Normal\">Answer: pnorm(q = 130, mean = 100, sd = 15, lower.tail = FALSE)<\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 3:<\/strong> What is the probability that z is less than .38?<\/p>\r\n<p class=\"import-Normal\">pnorm(q = 0.38, mean = 0, sd = 1)<\/p>\r\n<p class=\"import-Normal\">or<\/p>\r\n<p class=\"import-Normal\">pnorm(q = 0.38)<\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 4:<\/strong> What is the area (or probability) that Z is between -1 and 1?<\/p>\r\n<p class=\"import-Normal\">pnorm(q = 1)-pnorm(q = -1)<\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 5:<\/strong> 75% of the population have an IQ below what score?<\/p>\r\n<p class=\"import-Normal\">qnorm(p = 0.75, mean = 100, sd = 15)<\/p>\r\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 6:<\/strong> Above what Z score do you find 5% of the population?<\/p>\r\n\r\n<ul>\r\n \t<li>qnorm(p = 0.05, lower.tail = FALSE)<\/li>\r\n \t<li><\/li>\r\n \t<li><\/li>\r\n<\/ul>\r\n<h2><\/h2>\r\n<p class=\"import-Normal\"><strong>Distributions: <\/strong>NORM.DIST(1.96, 1, 0, 1) = 0.975 means that NORM.INV(0.975, 1, 0) = 1.96<\/p>\r\n\r\n<table style=\"width: 638pt\">\r\n<tbody>\r\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Binomial Distribution<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">Cnxpxqn-x<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=BINOM.DIST(X, n, p,0)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Finds P(x)<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(x\u2264a)<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=BINOM.DIST(a, n, p,1)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Cumulative probability, includes all values smaller!<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(a\u2264x\u2264b)<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=BINOM.DIST.RANGE(n, p, a, b)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">The probability it\u2019s between 2 numbers<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(x\u2264?)=P<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=BINOM.INV(n, p, P)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Inverse or reverse cumulative probability.<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Poisson Distribution<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03bbxe-\u03bcx!<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=POISSON.DIST(x, \u03bb, 0)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Exactly x<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(x\u2264a)<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=POISSON.DIST(a, \u03bb, 1)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">a or smaller<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(x\u2265a)<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=1 - POISSON.DIST(a-1, \u03bb, 1)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">a or bigger<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(a\u2264x\u2264b)<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">= POISSON.DIST(b, \u03bb, 1) - POISSON.DIST(a-1, \u03bb, 1)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Normal Distribution<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(x\u2264a)<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=NORM.DIST(a, \u03bc, \u03c3, 1)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Always use cumulative!<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(x\u2265a)<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=1 - NORM.DIST(a, \u03bc, \u03c3, 1)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(a\u2264x\u2264b)<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">= NORM.DIST(b, \u03bc, \u03c3, 1) - NORM.DIST(a, \u03bc, \u03c3, , 1)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Subtract 2 things!<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(x\u2264?)=P<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=NORM.INV(P, \u03bc, \u03c3)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Reverse lookup!<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Exponential Distribution<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">Px\u2264a=1-e-\u03bba<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=EXPON.DIST(x, \u03bb, 1)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Excel always counts x and below<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">Px\u2265a=e-\u03bba<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=1 - EXPON.DIST(x, \u03bb, 1)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Use (1 \u2013) to get x and above<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(a\u2264x\u2264b)<\/span><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">= EXPON.DIST(b, \u03bb, 1) - EXPON.DIST(a, \u03bb, 1)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Subtract 2 things<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p class=\"import-Normal\">Randomizing<\/p>\r\n\r\n<table style=\"width: 673.1pt\">\r\n<tbody>\r\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><strong>Name<\/strong><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><strong>Command<\/strong><\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\"><strong>Notes &amp; Examples<\/strong><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Uniform Discrete<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=RANDBETWEEN(a,b)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Gives a number between 1 and 10 inclusive<\/p>\r\n<p class=\"import-Normal\">=RANDBETWEEN(1,10)<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Binomial<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=BINOM.INV(n,p,RAND())<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">There are n = 20 people, the liklihood of one showing up is p = 0.30<\/p>\r\n<p class=\"import-Normal\"># show up =BINOM.INV(20,0.3,RAND())<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Bernoulli Trial<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=IF(RAND()&lt;p,ans1,ans_2)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Choose between two things, a 20% chance of being \u201chappy\u201d, an 80% chance of being \u201csad\u201d<\/p>\r\n<p class=\"import-Normal\">=IF(RAND()&lt;.2,\u201dhappy\u201d,\u201dsad\u201d)<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">General Discrete<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=LOOKUP(RAND(),lower_limits, answers)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Pull a value from a table with given probabilities<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Uniform<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=RAND()<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Give a random number between 0 and 1<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Uniform Continuous<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=a+(b-a)*RAND()<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Give a number between 10 and 11<\/p>\r\n<p class=\"import-Normal\">=11+(12 - 11)*RAND()<\/p>\r\n<p class=\"import-Normal\"><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Normal<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=NORM.INV(RAND(),\u03bc ,\u03c3)<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Pull a random IQ score if the mean is 100 and standard deviation 15<\/p>\r\n<p class=\"import-Normal\">=NORM.INV(RAND(),100 ,15)<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">Exponential<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">=-(1\/\u03bb )*LN(RAND())<\/p>\r\n<\/td>\r\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\r\n<p class=\"import-Normal\">5 customers an hour come into my coffee shop. Pick a random time between two customers<\/p>\r\n<p class=\"import-Normal\">=-(1\/5)*LN(RAND())<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p class=\"import-Normal\"><\/p>\r\n\r\n<\/div>","rendered":"<div class=\"__UNKNOWN__\">\n<p><em>Note:\u00a0 I&#8217;m posting this, but the formulae definitely need work!<\/em><\/p>\n<h1 class=\"import-Normal\">Why are Probabilities Important?<\/h1>\n<p class=\"import-Normal\">But first, some VERY FAST recap of the theory of probabilities:<\/p>\n<h2>Probability Distributions<\/h2>\n<p class=\"import-Normal\">A <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_169_177\"><strong>Random Variable<\/strong> <\/a>is variable (typically represented by X) that has a single numerical value (determined by chance) for each outcome of a procedure (or experiment).<\/p>\n<ul>\n<li class=\"import-Normal\">For example, X = the velocity of an unladen swallow<\/li>\n<li class=\"import-Normal\">X = number of tabs open on your browser as you read this book.<\/li>\n<\/ul>\n<p class=\"import-Normal\">There are two types of random variable:<\/p>\n<p class=\"import-Normal\">A <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_169_178\"><strong>Discrete Random Variable<\/strong><\/a> has either a finite number of values or a countable number of values, that is, it results from a counting process.<\/p>\n<ul>\n<li class=\"import-Normal\">For example, the number of BCIT grads among 50 newly hired employees<\/li>\n<li class=\"import-Normal\">The number of lemons in my fridge right now.<\/li>\n<\/ul>\n<p class=\"import-Normal\">A <strong><a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_169_179\">Continuous Random Variable<\/a><\/strong> has infinitely many values, which can be associated with measurements on a continuous scale in such a way that there are no gaps or interruptions.<\/p>\n<ul>\n<li class=\"import-Normal\">ie, the height of a randomly selected BCIT student<\/li>\n<li class=\"import-Normal\">The time at which you close this window and stop reviewing probabilities<\/li>\n<\/ul>\n<div id=\"h5p-1\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-1\" class=\"h5p-iframe\" data-content-id=\"1\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Fill in probabilities\"><\/iframe><\/div>\n<\/div>\n<p class=\"import-Normal\"><em>Example<\/em><\/p>\n<div id=\"h5p-2\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-2\" class=\"h5p-iframe\" data-content-id=\"2\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Card Probabilities\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">Definitions:<\/div>\n<p>The union of two sets is denoted by [latex]A\\cup B[\/latex] and this is the OR.<\/p>\n<p>The <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_169_263\">intersection <\/a>of two sets is denoted by [latex]A\\cap B[\/latex] and this is the AND.<\/p>\n<p>&nbsp;<\/p>\n<h3><a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_169_180\">Mutually Exclusive<\/a>:<\/h3>\n<p>A and B are mutually exclusive if P(A and B) = 0.<\/p>\n<p>For example, being a heart and a spade.<\/p>\n<div class=\"__UNKNOWN__\">\n<h3><a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_169_181\">Compliments<\/a>:<\/h3>\n<p>A and B are compliments if they are mutually exclusive and mutually exhaustive.\u00a0 That is:<\/p>\n<p>[latex]P(A \\, and \\, B) = 0[\/latex]\u00a0 and [latex]P(A \\, or \\, B) = 1[\/latex]<\/p>\n<h3>Conditional Probability of an Event<\/h3>\n<p class=\"import-Normal\">Given A and B are events in an experiment and<\/p>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(A)\u22600, <\/span>then the <strong>conditional <\/strong><strong>probability<\/strong> that event <span style=\"font-size: NaNpt;color: #;text-decoration: none\">B <\/span>will occur given that the event <span style=\"font-size: NaNpt;color: #;text-decoration: none\">A <\/span><strong>has<\/strong> <strong>already occurred<\/strong> is:<\/div>\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[P(B|A)=\\frac{P(A \\cap B)}{P(A)}\\]<\/span><\/p>\n<p class=\"import-Normal\">Rearranging the above formula gives:<\/p>\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[P(A \\cap B)=P(A) \\times P(B|A)\\]<\/span><\/p>\n<h3 style=\"text-align: left\">Independence vs Dependence<\/h3>\n<p class=\"import-Normal\">We say two events are <strong>independent<\/strong> if the occurrence of one does not affect the other.<\/p>\n<p class=\"import-Normal\">We say two events are <span style=\"font-size: 18.6667px\"><b>dependent<\/b><\/span>, if they are not independent, that is the occurrence of one affects the occurrence of the other.<\/p>\n<h3>Probability Tables and Contingency Tables (also called joint distribution tables)<\/h3>\n<p class=\"import-Normal\">Put one <em lang=\"en-US\" xml:lang=\"en-US\">mutually exclusive<\/em> event and its complement along the top.<\/p>\n<p class=\"import-Normal\">Put the other <em lang=\"en-US\" xml:lang=\"en-US\">mutually exclusive<\/em> event and its complement along the left side.<\/p>\n<p class=\"import-Normal\">Inside the table put the \u201cAND\u201d probabilities.<\/p>\n<table>\n<tbody>\n<tr class=\"TableNormal-R\" style=\"height: 13.4pt\">\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">A<\/span><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">A\u0305<\/span><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Total<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableNormal-R\" style=\"height: 13.4pt\">\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">B<\/span><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(A and B)<\/span><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(A\u0305 and B)<\/span><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(B)<\/span><\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableNormal-R\" style=\"height: 13.4pt\">\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">B\u0305<\/span><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(A and B\u0305)<\/span><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(A\u0305 and B\u0305)<\/span><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(B\u0305)<\/span><\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableNormal-R\" style=\"height: 2.5pt\">\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Total<\/strong><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(A)<\/span><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(A\u0305)<\/span><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">1<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercises<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p class=\"import-Normal\"><em>Example<\/em><\/p>\n<p class=\"import-Normal\">In your stats computer lab, you decide to conduct of study of the ages of 300 business students and whether or not they live with their parents. You obtain the following results:<\/p>\n<table style=\"width: 306pt\">\n<tbody>\n<tr class=\"TableNormal-R\" style=\"height: 11.35pt\">\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;padding: 0pt 5.4pt 0pt 5.4pt;border: none windowtext 0pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;padding: 0pt 5.4pt 0pt 5.4pt;border: none windowtext 0pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;padding: 0pt 5.4pt 0pt 5.4pt;border: none windowtext 0pt\" colspan=\"3\">\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">Ages<\/em><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;padding: 0pt 5.4pt 0pt 5.4pt;border: none windowtext 0pt\">\n<p class=\"import-Normal\">\n<\/td>\n<\/tr>\n<tr class=\"TableNormal-R\" style=\"height: 17pt\">\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;padding: 0pt 5.4pt 0pt 5.4pt;border: none windowtext 0pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: none windowtext 0pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;border-left: none windowtext 0pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: none windowtext 0pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">18 \u2013 27<\/strong><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: none windowtext 0pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">28 \u2013 37<\/strong><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: none windowtext 0pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">38 \u2013 47<\/strong><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: none windowtext 0pt;border-right: none windowtext 0pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Total<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableNormal-R\" style=\"height: 0\">\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;padding: 0pt 5.4pt 0pt 5.4pt;border: none windowtext 0pt\" rowspan=\"2\">\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">Live with<\/em><em lang=\"en-US\" xml:lang=\"en-US\"><br style=\"clear: both\" \/>Parents?<\/em><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;border-bottom: none windowtext 0pt;border-left: none windowtext 0pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Yes<\/strong><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">185<\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">5<\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">0<\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: none windowtext 0pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">190<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableNormal-R\" style=\"height: 0\">\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: none windowtext 0pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;border-left: none windowtext 0pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">No<\/strong><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: none windowtext 0pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">95<\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: none windowtext 0pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">11<\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: none windowtext 0pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">4<\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: none windowtext 0pt;border-right: none windowtext 0pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">110<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableNormal-R\">\n<td class=\"TableNormal-C\" style=\"padding: 0pt 5.4pt 0pt 5.4pt;border: none windowtext 0pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: bottom;border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;border-bottom: none windowtext 0pt;border-left: none windowtext 0pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Total<\/strong><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">280<\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">16<\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">4<\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: none windowtext 0pt;border-bottom: none windowtext 0pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">300<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol type=\"a\">\n<li class=\"import-Normal\">Give a <em lang=\"en-US\" xml:lang=\"en-US\">non-trivial<\/em> example of two mutually exclusive events in this study.<\/li>\n<li class=\"import-Normal\">If a student is at least 28 years old, what is the probability they do <strong lang=\"en-US\" xml:lang=\"en-US\">not<\/strong> live with their parents?<\/li>\n<li class=\"import-Normal\">What is the probability a student lives with their parents <strong lang=\"en-US\" xml:lang=\"en-US\">or<\/strong> is from 28 to 37 years old?<\/li>\n<li class=\"import-Normal\">What is the probability of a student <em lang=\"en-US\" xml:lang=\"en-US\">neither<\/em> living with their parents <em lang=\"en-US\" xml:lang=\"en-US\">nor<\/em> being from <br style=\"clear: both\" \/>28 to 37 years old?<\/li>\n<li class=\"import-Normal\">Are <em lang=\"en-US\" xml:lang=\"en-US\">ages<\/em> and <em lang=\"en-US\" xml:lang=\"en-US\">living with <\/em><em lang=\"en-US\" xml:lang=\"en-US\">parents<\/em> independent events?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>What\u2019s a Probability Distribution?<\/h3>\n<p class=\"import-Normal\">A <strong>Probability Distribution<\/strong> is a graph, table, or formula that gives the probability of each value of the random variable.<\/p>\n<p class=\"import-Normal\">There are two requirements for a probability distribution:<\/p>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[\\Sigma P(x)=1\\]<\/span><\/div>\n<p>where x assumes all possible values (\u2211 refers to the sum)<\/p>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[0\\leq P(x)\\leq 1\\]<\/span><\/div>\n<p>for every value of [latex]x[\/latex]<\/p>\n<h3>Expected Value or Mean of a Probability Distribution<\/h3>\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[\\mu=\\Sigma x P(x)=E(x)\\]<\/span><\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Examples<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p class=\"import-Normal\"><em>Example<\/em><\/p>\n<p class=\"import-Normal\">A rock concert producer has scheduled an outdoor concert. If it is warm that day, she expects to make a $20,000 profit. If it is cool that day, she expects to make a $5,000 profit. If it is very cold, she expects to suffer a $12,000 loss. Based upon historical data, the weather office has estimated the chances of a warm day to be 0.60; the chances of a cool day to be 0.25.<\/p>\n<p class=\"import-Normal\">Construct a probability distribution. <em lang=\"en-US\" xml:lang=\"en-US\">P(x)<\/em> = probability that <strong lang=\"en-US\" xml:lang=\"en-US\"><em>x<\/em><\/strong> occurs<\/p>\n<table style=\"width: 347.4pt\">\n<tbody>\n<tr class=\"TableNormal-R\">\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 1pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">Weather<\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">X<\/em> = Profit<\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">P(x)<\/em><\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">x\u00d7 P(x) <\/em><\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableNormal-R\" style=\"height: 19.6pt\">\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 1pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">Warm<\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 1pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">\n<\/td>\n<\/tr>\n<tr class=\"TableNormal-R\" style=\"height: 19.6pt\">\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 1pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">Cool<\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 1pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableNormal-C\" style=\"padding: 0pt 5.4pt 0pt 5.4pt;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">\n<\/td>\n<\/tr>\n<tr class=\"TableNormal-R\" style=\"height: 19.6pt\">\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 1pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">Very Cold<\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 0.5pt;border-left: solid windowtext 1pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 0.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">\n<\/td>\n<\/tr>\n<tr class=\"TableNormal-R\" style=\"height: 19.45pt\">\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 1pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 0.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 1pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableNormal-C\" style=\"vertical-align: middle;border-top: solid windowtext 1.5pt;border-right: solid windowtext 0.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">1.00<\/p>\n<\/td>\n<td class=\"TableNormal-C\" style=\"border-top: solid windowtext 1.5pt;border-right: solid windowtext 1.5pt;border-bottom: solid windowtext 1.5pt;border-left: solid windowtext 0.5pt;padding: 0pt 5.4pt 0pt 5.4pt\">\n<p class=\"import-Normal\">\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol type=\"a\">\n<li>What is the producer\u2019s expected profit (or loss)?<\/li>\n<li>What is the probability that she will make a profit?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Different Probability Distributions<\/h2>\n<h3>The Discrete Uniform Distribution<\/h3>\n<p class=\"import-Normal\">Every outcome from <em>a<\/em> to <em>b<\/em> is equally likely.<\/p>\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[P(X)=\\frac{1}{b-a+1}\\]<\/span><\/p>\n<p class=\"import-Normal\">For all <span style=\"font-size: NaNpt;color: #;text-decoration: none\">[latex]a\u2264X\u2264b[\/latex].<\/span><\/p>\n<p>&nbsp;<\/p>\n<p class=\"import-Normal\"><strong><em>Mean and Standard Deviation of a Discrete Uniform Distribution<\/em><\/strong><\/p>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[\\mu=E(X)=\\frac{a+b}{2}\\]<\/span><\/div>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[\\sigma=\\frac{b-a+12}{-112}\\] <\/span><\/div>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u00a0<\/span><\/div>\n<div><\/div>\n<div id=\"h5p-3\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-3\" class=\"h5p-iframe\" data-content-id=\"3\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"20 sided die probability\"><\/iframe><\/div>\n<\/div>\n<h3>The Binomial Distribution<\/h3>\n<ul>\n<li class=\"import-Normal\">The experiment must have a <strong><em>fixed number<\/em><\/strong> of <em>identical<\/em> trials.<\/li>\n<li class=\"import-Normal\">Each trial must have all outcomes classified into two categories (a <strong><em>success<\/em><\/strong> or <strong><em>failure<\/em><\/strong>)<\/li>\n<li class=\"import-Normal\">The trials must be <strong><em>independent<\/em><\/strong> (the outcome of any individual trial doesn&#8217;t influence the outcome of another trial).<\/li>\n<li class=\"import-Normal\">The probabilities must remain <strong><em>constant\/same<\/em><\/strong> for each trial.<\/li>\n<li class=\"import-Normal\">The variable of interest must be the <strong><em>number of successes<\/em><\/strong> in <em>n<\/em> trials.<\/li>\n<\/ul>\n<p>Terminology:<\/p>\n<ul>\n<li class=\"import-Normal\">[latex]n[\/latex] = fixed number of trials<\/li>\n<li class=\"import-Normal\">[latex]x[\/latex] = specific number of successes in n trials<\/li>\n<li class=\"import-Normal\">[latex]p[\/latex]= probability of success in each one of the n trials<\/li>\n<li class=\"import-Normal\">[latex]1-p[\/latex]= probability of failure in each one of the n trials[latex]p[\\latex]<\/li>\n<li class=\"import-Normal\">[latex]P(x)[\/latex]= probability of getting exactly <em>x<\/em> successes among <em>n<\/em> trials<\/li>\n<\/ul>\n<h2 class=\"import-Normal\"><strong><em>Combinations<\/em><\/strong><\/h2>\n<p class=\"import-Normal\"><em>How many ways can you choose x things out of a list of n things?<\/em><\/p>\n<p>\\[_nC_x= \\binom{n}{x} = \\frac{n!}{x!(n-x)!}\\]<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p class=\"import-Normal\">Where<\/p>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[n!=n\\times (n-1)\\times (n-2)\\times \\cdots \\times 2\\times 1\\]<\/span><\/div>\n<p class=\"import-Normal\"><strong><em>Binomial Probability Formula<\/em><\/strong><\/p>\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[P(x) = _nC_{x} p^{x} (1-p)^{n-x}\\]<br \/>\n<\/span><\/p>\n<p class=\"import-Normal\">If [latex]n = 5, p = 0.2, x = 3[\/latex]:<\/p>\n<p>\\[P(x=3) = _5C_{3} 0.2^{3} (0.8)^{2}=0.0512\\]<\/p>\n<p class=\"import-Normal\">Excel: <code>=BINOM.DIST(3, 5, 0.2,0)<\/code><\/p>\n<p class=\"import-Normal\">R: <code>dbinom(x = 3, size = 5, prob = 0.2)<\/code><\/p>\n<p>CUMULATIVE<\/p>\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[P(X\u22643)=P(x = 0)+P(x=1)+P(x=2)+P(x=3)=0.99328\\]<\/span><\/p>\n<p class=\"import-Normal\">Excel: <code>=BINOM.DIST(3, 5, 0.2,1)<\/code><\/p>\n<p class=\"import-Normal\">R: <code>pbinom(q = 3, size = 5, prob = 0.2)<\/code><\/p>\n<p class=\"import-Normal\"><strong><em>Mean and Standard Deviation of a Binomial Distribution<\/em><\/strong><\/p>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[\\mu=np=E(x)\\]<\/span><\/div>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[\\sigma=\\sqrt{n p(1-p)}\\]<\/span><\/div>\n<p class=\"import-Normal\">A quiz consists of 5 multiple-choice questions with 4 possible answers for each question. The student is unprepared for the exam and randomly selects answers.<\/p>\n<div style=\"margin: auto\">\n<table>\n<tbody>\n<tr class=\"TableGrid-R\" style=\"height: 0\">\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\"># of successes<\/strong><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\"># of failures<\/strong><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Probability<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 0\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">0<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 0\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">1<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 0\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">2<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 0\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">3<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 0\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">4<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 0\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">5<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p class=\"import-Normal\">Find the probability that the student gets:<\/p>\n<p class=\"import-Normal\" style=\"margin-left: 14.2pt;text-indent: 14.2pt\">all five correct<\/p>\n<p class=\"import-Normal\" style=\"margin-left: 14.2pt;text-indent: 14.2pt\">none correct<\/p>\n<p class=\"import-Normal\" style=\"margin-left: 14.2pt;text-indent: 14.2pt\">exactly two correct<\/p>\n<p class=\"import-Normal\" style=\"margin-left: 14.2pt;text-indent: 14.2pt\">(d) at least one correct<\/p>\n<h3>The Poisson Distributions<\/h3>\n<p class=\"import-Normal\">the random variable <em>x<\/em> is the number of occurrences of an event over some <strong>time interval<\/strong><\/p>\n<p class=\"import-Normal\">the occurrences must be random<\/p>\n<p class=\"import-Normal\">the occurrences must be independent of each other<\/p>\n<p class=\"import-Normal\">the occurrences must be <strong><em>uniformly distributed<\/em><\/strong> over the interval being used<\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\"><em>Examples where it applies<\/em><\/strong><strong lang=\"en-US\" xml:lang=\"en-US\">:<\/strong><\/p>\n<p class=\"import-Normal\">The <strong lang=\"en-US\" xml:lang=\"en-US\">number of<\/strong> car accidents in the lower mainland <strong lang=\"en-US\" xml:lang=\"en-US\">per day<\/strong>.<\/p>\n<p class=\"import-Normal\">The <strong lang=\"en-US\" xml:lang=\"en-US\">number of<\/strong> spam emails that a computer user will receive <strong lang=\"en-US\" xml:lang=\"en-US\">per week<\/strong>.<\/p>\n<p class=\"import-Normal\">The <strong lang=\"en-US\" xml:lang=\"en-US\">number of<\/strong> phone calls to a call center <strong lang=\"en-US\" xml:lang=\"en-US\">per minute<\/strong>.<\/p>\n<p class=\"import-Normal\"><strong><em>Notation<\/em><\/strong><\/p>\n<p class=\"import-Normal\">x = specific number occurrences in the time interval<\/p>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03bc<\/span><\/div>\n<p>=\u03bb = the average number of occurrences in the time interval<\/p>\n<p class=\"import-Normal\"><strong><em>Poisson Probability Formula<\/em><\/strong><\/p>\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\\[P(x)=\u03bcxe-\u03bcx!=\u03bbxe-\u03bbx!\\]<\/span><\/p>\n<p class=\"import-Normal\">If \u03bb = 6, x = 3<\/p>\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">PX=3=63e-63!=0.089235<\/span><\/p>\n<p class=\"import-Normal\">Excel: =POISSON.DIST(3, 6, 0)<\/p>\n<p class=\"import-Normal\">R: dpois(x = 3, lambda =6)<\/p>\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">PX\u22643=P0+P1+P2+P3=0.1512039<\/span><\/p>\n<p class=\"import-Normal\">Excel: =POISSON.DIST(3, 6, 1)<\/p>\n<p class=\"import-Normal\">R: ppois(q = 3, lambda =6)<\/p>\n<p class=\"import-Normal\"><strong><em>Mean and Standard Deviation of a Poisson Distribution<\/em><\/strong><\/p>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03bc=\u03bb=E(x)<\/span><\/div>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03c3=\u03bb=\u03bc<\/span><\/div>\n<p class=\"import-Normal\">An internet provider receives an average of 5 calls per half hour for technical support. The calls follow a Poisson distribution.<\/p>\n<p class=\"import-Normal\">What is the probability there will be three calls in 30 minutes?<\/p>\n<p class=\"import-Normal\">What is the probability there will be at least one call in 15 minutes?<\/p>\n<p class=\"import-Normal\">What is the probability there will be at least 2 calls in half an hour?<\/p>\n<p class=\"import-Normal\"><strong><em>The Continuous Uniform Distribution<\/em><\/strong><\/p>\n<p class=\"import-Normal\">Every outcome from <em>a<\/em> to <em>b<\/em> is equally likely.<\/p>\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">Px1\u2264X\u2264x2=x2-x1b-a,<\/span><\/p>\n<p class=\"import-Normal\">For all<\/p>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">a\u2264x1\u2264x2\u2264b<\/span><\/div>\n<p>.<\/p>\n<p class=\"import-Normal\"><strong><em>Mean and Standard Deviation of a continuous uniform Distribution<\/em><\/strong><\/p>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03bc=Ex=a+b2<\/span><\/div>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03c3=b-a12<\/span><\/div>\n<p class=\"import-Normal\">The fill volume of a regular can of pop ranges uniformly from 354 mL to 358 mL. What is the mean fill volume?<\/p>\n<p class=\"import-Normal\">b. What is the probability that the fill volume is between 354.3 mL and 355.2 mL?<\/p>\n<h3>The Normal Distribution<\/h3>\n<p class=\"import-Normal\">Used for continuous variables only. (e.g. Age, time, income, temperature, house prices).<\/p>\n<p class=\"import-Normal\">Symmetrical about the mean (mean = median = mode) and bell-shaped.<\/p>\n<p class=\"import-Normal\">Since there are many different normal curves, we convert all normal curves into one single curve called the standard normal (mean \u03bc = 0, standard deviation \u03c3 = 1) by finding the <em lang=\"en-US\" xml:lang=\"en-US\">z<\/em>-score. (subtract the mean and divide by the standard deviation to calculate the <em lang=\"en-US\" xml:lang=\"en-US\">z<\/em>-score).<\/p>\n<p class=\"import-Normal\">The area under the normal curve represents probability. The total area under the curve = 1.<\/p>\n<p class=\"import-Normal\">Continuous random variables can take on any value within an interval since there are no gaps between the numbers. We no longer talk about the probability of the random variable assuming a particular value. Instead, we talk about the probability of the variable assuming a value within some given interval. The probability of the random variable assuming a particular value = 0.<\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\"><em>Empirical Rule: <\/em><\/strong><em lang=\"en-US\" xml:lang=\"en-US\">Applies only to bell-shaped (normally distributed) populations<\/em><\/p>\n<p class=\"import-Normal\">About 68% of the values will fall within <strong lang=\"en-US\" xml:lang=\"en-US\">one <\/strong>standard deviation of the mean.<\/p>\n<p class=\"import-Normal\">About 95% of the values will fall within <strong lang=\"en-US\" xml:lang=\"en-US\">two <\/strong>standard deviations of the mean.<\/p>\n<p class=\"import-Normal\">About 99.7% (<em lang=\"en-US\" xml:lang=\"en-US\">almost<\/em> 100% of values will fall within <strong lang=\"en-US\" xml:lang=\"en-US\">three <\/strong>standard deviations of the mean).<\/p>\n<p class=\"import-Normal\">The selling prices of all stocks listed on the <em lang=\"en-US\" xml:lang=\"en-US\">CDNX<\/em> stock exchange are known to be normally distributed with a mean of $20 and a standard deviation of $4.<\/p>\n<p class=\"import-Normal\" style=\"margin-left: 28.35pt;text-indent: 21.25pt\">What percentage of stocks have a selling price between $20 and $40?<\/p>\n<p class=\"import-Normal\" style=\"margin-left: 28.35pt;text-indent: 21.25pt\">What is the minimum selling price of the most expensive 5% of stocks?<\/p>\n<p class=\"import-Normal\" style=\"margin-left: 28.35pt;text-indent: 21.25pt\">The prices of the middle 80% of stocks are between what two values?<\/p>\n<h3>The Exponential Distribution<\/h3>\n<p class=\"import-Normal\">How long between occurrences?<\/p>\n<p class=\"import-Normal\">Measures the time between two things in a Poisson distribution: for example, what is the probability that there is at least a 15 minute gap between customers at your coffee shop?<\/p>\n<p class=\"import-Normal\">Here, \u03bb (the Greek letter lambda) is the average number of occurrences in a time interval.<\/p>\n<p class=\"import-Normal\"><strong><em>Exponential Formula<\/em><\/strong><\/p>\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">PX\u2265a=1e\u03bba<\/span><\/p>\n<p class=\"import-Normal\"><strong><em>Mean and Standard Deviation of a continuous uniform Distribution<\/em><\/strong><\/p>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03bc=Ex=1\u03bb<\/span><\/div>\n<div><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03c3=\u03bc=1\u03bb<\/span><\/div>\n<p class=\"import-Normal\">Telemarketers at ARG make an average of 3 calls per minute. Assuming that the calls follow a Poisson Distribution, what is the mean time between calls?<\/p>\n<p class=\"import-Normal\">What is the probability that there will be at least a 30 second gap between incoming calls?<\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\"><em>Try these at home. Which Distribution should you use?<\/em><\/strong><\/p>\n<p class=\"import-Normal\">A hotel\u2019s records show that 70% of its guests are from the United States. In a random sample of 10 guests, what is the probability that exactly half are from the U.S.?<\/p>\n<p class=\"import-Normal\">At the airport a plane takes off, on average, every 2 minutes. In any 10-minute interval, what is the probability that there are more than 2 departures?<\/p>\n<p class=\"import-Normal\">What is the probability that there is at least a minute between successive departures?<\/p>\n<p class=\"import-Normal\"><strong><em>Fun with Excel <\/em><\/strong><\/p>\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">0 or 1 is ALWAYS the last thing: do you want cumulative or not?<\/em><\/p>\n<p class=\"import-Normal\">=BINOM.DIST(x, n, p, 0 or 1) and =BINOM.INV(n,p,P)<\/p>\n<p class=\"import-Normal\">=POISSON.DIST(x, \u03bc, 0 or 1)<\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Excel Commands for Normal Distribution:<\/strong><\/p>\n<p class=\"import-Normal\">If we know \u00b5, \u03c3 and X and are trying to find the probability that x&lt;X: =norm.dist(x,\u00b5,\u03c3,1).<\/p>\n<p class=\"import-Normal\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-content\/uploads\/sites\/1653\/2023\/05\/image1.png\" alt=\"image\" width=\"225px\" height=\"127.466666666667px\" \/>Example SEQ Example_1 \\* ARABIC 1<\/p>\n<p class=\"import-Normal\">(<em lang=\"en-US\" xml:lang=\"en-US\">The 1 is for cumulative, in Normal, we will use it in every command.)<\/em><\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 1:<\/strong> What is the probability that a person\u2019s IQ is less than 125? (note: \u00b5=100, \u03c3=15)<\/p>\n<p class=\"import-Normal\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-content\/uploads\/sites\/1653\/2023\/05\/image3.png\" alt=\"image\" width=\"240px\" height=\"112px\" \/><strong lang=\"en-US\" xml:lang=\"en-US\">Example 2:<\/strong> What is the probability that an IQ is greater than 130?<\/p>\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">To get the right side, just subtract your answer from 1: this gives the answer on the table.<\/em><\/p>\n<p class=\"import-Normal\">If you have z-scores, you may use the norm.s.dist command which cuts down on the typing:<\/p>\n<p class=\"import-Normal\"><em><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-content\/uploads\/sites\/1653\/2023\/05\/image4.png\" alt=\"image\" width=\"237px\" height=\"114.466666666667px\" \/>Example3<br \/>\n<\/em><\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 3:<\/strong> What is the probability that z is less than .38?<\/p>\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">Again, the 1 stands for cumulative. This is the same as typing =<\/em><em lang=\"en-US\" xml:lang=\"en-US\">norm.dist<\/em><em lang=\"en-US\" xml:lang=\"en-US\">(.38,1,0,1)<\/em><\/p>\n<p class=\"import-Normal\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-content\/uploads\/sites\/1653\/2023\/05\/image6.png\" alt=\"image\" width=\"284px\" height=\"121.466666666667px\" \/>Example 4<\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 4:<\/strong> What is the area (or probability) that Z is between -1 and 1?<\/p>\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">You can subtract in 2 cells or in one\u2026 Notice that the larger z score goes first!<\/em><\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Inverses:<\/strong><\/p>\n<p class=\"import-Normal\">If you know that the area below X is P, and you are solving for X use =norm.inv(P,\u00b5,\u03c3):<\/p>\n<p class=\"import-Normal\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-content\/uploads\/sites\/1653\/2023\/05\/image8.png\" alt=\"image\" width=\"223px\" height=\"124.466666666667px\" \/>Example 5<\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 5:<\/strong> 75% of the population have an IQ below what score?<\/p>\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">Make sure your area is in decimal! It needs to be greater than 0 and less than 1. Putting in 0 or 1 will give an error message.<\/em><\/p>\n<p class=\"import-Normal\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-content\/uploads\/sites\/1653\/2023\/05\/image10.png\" alt=\"image\" width=\"215px\" height=\"141.466666666667px\" \/>Example 6<\/p>\n<p class=\"import-Normal\">Or you may recover z scores:<\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 6:<\/strong> Above what Z score do you find 5% of the population?<\/p>\n<p class=\"import-Normal\"><em lang=\"en-US\" xml:lang=\"en-US\">Since we are always working with areas to the left, we need to use .95, as 5% <\/em><em lang=\"en-US\" xml:lang=\"en-US\">above<\/em><em lang=\"en-US\" xml:lang=\"en-US\"> is 95% <\/em><em lang=\"en-US\" xml:lang=\"en-US\">below.<\/em><\/p>\n<h2>Fun with R<\/h2>\n<p class=\"import-Normal\">R is a great language for probabilities! Let\u2019s do the same Normal problems, but in R:<\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 1:<\/strong> What is the probability that a person\u2019s IQ is less than 125? (note: \u00b5=100, \u03c3=15)<\/p>\n<p class=\"import-Normal\">Answer: `pnorm(q = 125, mean = 100, sd = 15)`<\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 2:<\/strong> What is the probability that an IQ is greater than 130?<\/p>\n<p class=\"import-Normal\">Answer: pnorm(q = 130, mean = 100, sd = 15, lower.tail = FALSE)<\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 3:<\/strong> What is the probability that z is less than .38?<\/p>\n<p class=\"import-Normal\">pnorm(q = 0.38, mean = 0, sd = 1)<\/p>\n<p class=\"import-Normal\">or<\/p>\n<p class=\"import-Normal\">pnorm(q = 0.38)<\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 4:<\/strong> What is the area (or probability) that Z is between -1 and 1?<\/p>\n<p class=\"import-Normal\">pnorm(q = 1)-pnorm(q = -1)<\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 5:<\/strong> 75% of the population have an IQ below what score?<\/p>\n<p class=\"import-Normal\">qnorm(p = 0.75, mean = 100, sd = 15)<\/p>\n<p class=\"import-Normal\"><strong lang=\"en-US\" xml:lang=\"en-US\">Example 6:<\/strong> Above what Z score do you find 5% of the population?<\/p>\n<ul>\n<li>qnorm(p = 0.05, lower.tail = FALSE)<\/li>\n<li><\/li>\n<li><\/li>\n<\/ul>\n<h2><\/h2>\n<p class=\"import-Normal\"><strong>Distributions: <\/strong>NORM.DIST(1.96, 1, 0, 1) = 0.975 means that NORM.INV(0.975, 1, 0) = 1.96<\/p>\n<table style=\"width: 638pt\">\n<tbody>\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Binomial Distribution<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">Cnxpxqn-x<\/span><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=BINOM.DIST(X, n, p,0)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Finds P(x)<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(x\u2264a)<\/span><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=BINOM.DIST(a, n, p,1)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Cumulative probability, includes all values smaller!<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(a\u2264x\u2264b)<\/span><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=BINOM.DIST.RANGE(n, p, a, b)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">The probability it\u2019s between 2 numbers<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(x\u2264?)=P<\/span><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=BINOM.INV(n, p, P)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Inverse or reverse cumulative probability.<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Poisson Distribution<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">\u03bbxe-\u03bcx!<\/span><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=POISSON.DIST(x, \u03bb, 0)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Exactly x<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(x\u2264a)<\/span><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=POISSON.DIST(a, \u03bb, 1)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">a or smaller<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(x\u2265a)<\/span><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=1 - POISSON.DIST(a-1, \u03bb, 1)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">a or bigger<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(a\u2264x\u2264b)<\/span><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">= POISSON.DIST(b, \u03bb, 1) - POISSON.DIST(a-1, \u03bb, 1)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Normal Distribution<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(x\u2264a)<\/span><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=NORM.DIST(a, \u03bc, \u03c3, 1)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Always use cumulative!<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(x\u2265a)<\/span><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=1 - NORM.DIST(a, \u03bc, \u03c3, 1)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(a\u2264x\u2264b)<\/span><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">= NORM.DIST(b, \u03bc, \u03c3, 1) - NORM.DIST(a, \u03bc, \u03c3, , 1)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Subtract 2 things!<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(x\u2264?)=P<\/span><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=NORM.INV(P, \u03bc, \u03c3)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Reverse lookup!<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Exponential Distribution<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">Px\u2264a=1-e-\u03bba<\/span><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=EXPON.DIST(x, \u03bb, 1)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Excel always counts x and below<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">Px\u2265a=e-\u03bba<\/span><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=1 - EXPON.DIST(x, \u03bb, 1)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Use (1 \u2013) to get x and above<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 27.4pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><span style=\"font-size: NaNpt;color: #;text-decoration: none\">P(a\u2264x\u2264b)<\/span><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">= EXPON.DIST(b, \u03bb, 1) - EXPON.DIST(a, \u03bb, 1)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Subtract 2 things<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"import-Normal\">Randomizing<\/p>\n<table style=\"width: 673.1pt\">\n<tbody>\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><strong>Name<\/strong><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><strong>Command<\/strong><\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\"><strong>Notes &amp; Examples<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Uniform Discrete<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=RANDBETWEEN(a,b)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Gives a number between 1 and 10 inclusive<\/p>\n<p class=\"import-Normal\">=RANDBETWEEN(1,10)<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Binomial<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=BINOM.INV(n,p,RAND())<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">There are n = 20 people, the liklihood of one showing up is p = 0.30<\/p>\n<p class=\"import-Normal\"># show up =BINOM.INV(20,0.3,RAND())<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Bernoulli Trial<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=IF(RAND()&lt;p,ans1,ans_2)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Choose between two things, a 20% chance of being \u201chappy\u201d, an 80% chance of being \u201csad\u201d<\/p>\n<p class=\"import-Normal\">=IF(RAND()&lt;.2,\u201dhappy\u201d,\u201dsad\u201d)<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">General Discrete<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=LOOKUP(RAND(),lower_limits, answers)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Pull a value from a table with given probabilities<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Uniform<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=RAND()<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Give a random number between 0 and 1<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Uniform Continuous<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=a+(b-a)*RAND()<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Give a number between 10 and 11<\/p>\n<p class=\"import-Normal\">=11+(12 - 11)*RAND()<\/p>\n<p class=\"import-Normal\">\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Normal<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=NORM.INV(RAND(),\u03bc ,\u03c3)<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Pull a random IQ score if the mean is 100 and standard deviation 15<\/p>\n<p class=\"import-Normal\">=NORM.INV(RAND(),100 ,15)<\/p>\n<\/td>\n<\/tr>\n<tr class=\"TableGrid-R\" style=\"height: 25.3pt\">\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">Exponential<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">=-(1\/\u03bb )*LN(RAND())<\/p>\n<\/td>\n<td class=\"TableGrid-C\" style=\"vertical-align: middle;border: solid windowtext 0.5pt\">\n<p class=\"import-Normal\">5 customers an hour come into my coffee shop. Pick a random time between two customers<\/p>\n<p class=\"import-Normal\">=-(1\/5)*LN(RAND())<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"import-Normal\">\n<\/div>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_169_177\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_169_177\"><div tabindex=\"-1\"><p>A Random Variable is variable (typically represented by X) that has a single numerical value (determined by chance) for each outcome of a procedure (or experiment).<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_169_178\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_169_178\"><div tabindex=\"-1\"><p>has either a finite number of values or a countable number of values, that is, it results from a counting process.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_169_179\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_169_179\"><div tabindex=\"-1\"><p>has infinitely many values, which can be associated with measurements on a continuous scale in such a way that there are no gaps or interruptions.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_169_263\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_169_263\"><div tabindex=\"-1\"><\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_169_180\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_169_180\"><div tabindex=\"-1\"><p>A and B are mutually exclusive if P(A and B) = 0.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_169_181\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_169_181\"><div tabindex=\"-1\"><p>A and B are compliments if they are mutually exclusive and mutually exhaustive.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":883,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[48],"contributor":[],"license":[],"class_list":["post-169","chapter","type-chapter","status-web-only","hentry","chapter-type-standard"],"part":102,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-json\/pressbooks\/v2\/chapters\/169","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-json\/wp\/v2\/users\/883"}],"version-history":[{"count":22,"href":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-json\/pressbooks\/v2\/chapters\/169\/revisions"}],"predecessor-version":[{"id":390,"href":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-json\/pressbooks\/v2\/chapters\/169\/revisions\/390"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-json\/pressbooks\/v2\/parts\/102"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-json\/pressbooks\/v2\/chapters\/169\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-json\/wp\/v2\/media?parent=169"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-json\/pressbooks\/v2\/chapter-type?post=169"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-json\/wp\/v2\/contributor?post=169"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/businessanalytics\/wp-json\/wp\/v2\/license?post=169"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}