{"id":332,"date":"2016-10-11T15:35:03","date_gmt":"2016-10-11T15:35:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/math4libarts\/?post_type=chapter&#038;p=332"},"modified":"2022-01-07T01:44:22","modified_gmt":"2022-01-07T01:44:22","slug":"4-1-basic-probability-concepts","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/chapter\/4-1-basic-probability-concepts\/","title":{"raw":"4.1 Basic Probability Concepts","rendered":"4.1 Basic Probability Concepts"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning OUTCOMES<\/h3>\r\n<ul>\r\n \t<li>Theoretical Probability<\/li>\r\n \t<li>Empirical Probability<\/li>\r\n \t<li>Describe a sample space, outcome, simple and compound events in it using standard notation<\/li>\r\n \t<li>Calculate the probability of an event using standard notation<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Learning new Math vocabulary and notation<\/h3>\r\nYou've seen before that learning mathematics is similar to learning a new language -- it takes repetition and practice to obtain new vocabulary and symbols. Probability is no different. The set of vocabulary and symbols used in probability will likely be completely unfamiliar to you unless you've studied probability before. Remember to read the text with your pencil, writing out the terms, definitions, and practice problems multiple times in order to learn them. You'll need to spend time with these new ways of thinking to make them your own.\r\n\r\n<\/div>\r\nIf you roll a die, pick a card from a deck of playing cards, or randomly select a person and observe their hair color, we are executing an experiment. In probability, we look at the likelihood of different outcomes.\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/28180953\/dice-219263_640.jpg\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/28180953\/dice-219263_640.jpg\" alt=\"Five dice, red and white, on a marble surface\" width=\"640\" height=\"480\" \/><\/a>\r\n\r\nWe begin with some terminology.\r\n<div class=\"textbox\">\r\n<h3>Theoretical Probability<\/h3>\r\n<ul>\r\n \t<li style=\"text-align: center;\">a probability that is expected based on <strong>theory<\/strong>\u00a0\u00a0<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2016\/10\/11042344\/theoretical-probability.png\"><img class=\"alignnone wp-image-5737\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2016\/10\/11042344\/theoretical-probability-300x72.png\" alt=\"\" width=\"307\" height=\"74\" \/><\/a><\/li>\r\n<\/ul>\r\n<h3>Empirical Probability<\/h3>\r\n<ul>\r\n \t<li>a probability that is based on the result of an <strong>experiment<\/strong><\/li>\r\n<\/ul>\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2016\/10\/11042653\/experimental-probability.png\"><img class=\"size-medium wp-image-5738 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2016\/10\/11042653\/experimental-probability-300x54.png\" alt=\"\" width=\"300\" height=\"54\" \/><\/a>\r\n<h3>Event, Outcome, and Sample SPace<\/h3>\r\n<ul>\r\n \t<li>The result of an experiment is called an <strong>outcome<\/strong>.<\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 90px;\">If you flip one fair coin,\u00a0<em>H<\/em>\u00a0= heads and\u00a0<em>T<\/em>\u00a0= tails are the outcomes<\/p>\r\n\r\n<ul>\r\n \t<li>An <strong>event<\/strong> is any particular outcome or group of outcomes.<\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 90px;\">If you roll one fair six-sided die, the event might be getting an even number on the die.\u00a0 The probability of this occurring would be written as P(an even number).<\/p>\r\n\r\n<ul>\r\n \t<li>The <strong>sample space<\/strong> is the set of all possible outcomes.<\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 90px;\">If you roll one fair six-sided die, the sample space is S = {1, 2, 3, 4, 5, 6}.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example - Theoretical Probability<\/h3>\r\nA bag contains three <a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2016\/10\/11043004\/bag-of-marbles.png\"><img class=\" wp-image-5739 alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2016\/10\/11043004\/bag-of-marbles.png\" alt=\"\" width=\"168\" height=\"176\" \/><\/a>red marbles and three blue marbles.\r\nP(red) = 3\/6 =1\/2\r\n\r\n(The result is based on the possible outcomes therefore this is Theoretical Probability)\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example - Empirical<\/h3>\r\nOne hundred consumers were asked the reason for choosing a particular car.\u00a0 The responses were recorded below:\r\n<table style=\"border-collapse: collapse; width: 49.9227%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 21.0425%;\"><span style=\"text-decoration: underline; background-color: #ccffff; color: #000080;\">Reason<\/span><\/td>\r\n<td style=\"width: 21.0424%;\"><span style=\"text-decoration: underline; background-color: #ccffff; color: #000080;\">Frequency<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 21.0425%;\">Looks<\/td>\r\n<td style=\"width: 21.0424%;\">24<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 21.0425%;\">Reliability<\/td>\r\n<td style=\"width: 21.0424%;\">17<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 21.0425%;\">Cost<\/td>\r\n<td style=\"width: 21.0424%;\">49<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 21.0425%;\">Safety<\/td>\r\n<td style=\"width: 21.0424%;\">10<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ul>\r\n \t<li>Find the probability that a person chooses a car for cost reasons<\/li>\r\n<\/ul>\r\nP(cost) = 49\/100 = 0.490\r\n\r\n(The result is based on an experiment, therefore this is Empirical Probability)\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nIf we roll a standard 6-sided die, describe the sample space and some simple events.\r\n\r\n[reveal-answer q=\"997648\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"997648\"]\r\n\r\nThe sample space is the set of all possible simple events: {1,2,3,4,5,6}\r\nSome examples of simple events:\r\n<ul>\r\n \t<li>We roll a 1<\/li>\r\n \t<li>We roll a 5<\/li>\r\n<\/ul>\r\nSome compound events:\r\n<ul>\r\n \t<li>We roll a number bigger than 4<\/li>\r\n \t<li>We roll an even number<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning OUTCOMES<\/h3>\n<ul>\n<li>Theoretical Probability<\/li>\n<li>Empirical Probability<\/li>\n<li>Describe a sample space, outcome, simple and compound events in it using standard notation<\/li>\n<li>Calculate the probability of an event using standard notation<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Learning new Math vocabulary and notation<\/h3>\n<p>You&#8217;ve seen before that learning mathematics is similar to learning a new language &#8212; it takes repetition and practice to obtain new vocabulary and symbols. Probability is no different. The set of vocabulary and symbols used in probability will likely be completely unfamiliar to you unless you&#8217;ve studied probability before. Remember to read the text with your pencil, writing out the terms, definitions, and practice problems multiple times in order to learn them. You&#8217;ll need to spend time with these new ways of thinking to make them your own.<\/p>\n<\/div>\n<p>If you roll a die, pick a card from a deck of playing cards, or randomly select a person and observe their hair color, we are executing an experiment. In probability, we look at the likelihood of different outcomes.<br \/>\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/28180953\/dice-219263_640.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/28180953\/dice-219263_640.jpg\" alt=\"Five dice, red and white, on a marble surface\" width=\"640\" height=\"480\" \/><\/a><\/p>\n<p>We begin with some terminology.<\/p>\n<div class=\"textbox\">\n<h3>Theoretical Probability<\/h3>\n<ul>\n<li style=\"text-align: center;\">a probability that is expected based on <strong>theory<\/strong>\u00a0\u00a0<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2016\/10\/11042344\/theoretical-probability.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5737\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2016\/10\/11042344\/theoretical-probability-300x72.png\" alt=\"\" width=\"307\" height=\"74\" \/><\/a><\/li>\n<\/ul>\n<h3>Empirical Probability<\/h3>\n<ul>\n<li>a probability that is based on the result of an <strong>experiment<\/strong><\/li>\n<\/ul>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2016\/10\/11042653\/experimental-probability.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-5738 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2016\/10\/11042653\/experimental-probability-300x54.png\" alt=\"\" width=\"300\" height=\"54\" \/><\/a><\/p>\n<h3>Event, Outcome, and Sample SPace<\/h3>\n<ul>\n<li>The result of an experiment is called an <strong>outcome<\/strong>.<\/li>\n<\/ul>\n<p style=\"padding-left: 90px;\">If you flip one fair coin,\u00a0<em>H<\/em>\u00a0= heads and\u00a0<em>T<\/em>\u00a0= tails are the outcomes<\/p>\n<ul>\n<li>An <strong>event<\/strong> is any particular outcome or group of outcomes.<\/li>\n<\/ul>\n<p style=\"padding-left: 90px;\">If you roll one fair six-sided die, the event might be getting an even number on the die.\u00a0 The probability of this occurring would be written as P(an even number).<\/p>\n<ul>\n<li>The <strong>sample space<\/strong> is the set of all possible outcomes.<\/li>\n<\/ul>\n<p style=\"padding-left: 90px;\">If you roll one fair six-sided die, the sample space is S = {1, 2, 3, 4, 5, 6}.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example &#8211; Theoretical Probability<\/h3>\n<p>A bag contains three <a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2016\/10\/11043004\/bag-of-marbles.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5739 alignright\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2016\/10\/11043004\/bag-of-marbles.png\" alt=\"\" width=\"168\" height=\"176\" \/><\/a>red marbles and three blue marbles.<br \/>\nP(red) = 3\/6 =1\/2<\/p>\n<p>(The result is based on the possible outcomes therefore this is Theoretical Probability)<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example &#8211; Empirical<\/h3>\n<p>One hundred consumers were asked the reason for choosing a particular car.\u00a0 The responses were recorded below:<\/p>\n<table style=\"border-collapse: collapse; width: 49.9227%;\">\n<tbody>\n<tr>\n<td style=\"width: 21.0425%;\"><span style=\"text-decoration: underline; background-color: #ccffff; color: #000080;\">Reason<\/span><\/td>\n<td style=\"width: 21.0424%;\"><span style=\"text-decoration: underline; background-color: #ccffff; color: #000080;\">Frequency<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 21.0425%;\">Looks<\/td>\n<td style=\"width: 21.0424%;\">24<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 21.0425%;\">Reliability<\/td>\n<td style=\"width: 21.0424%;\">17<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 21.0425%;\">Cost<\/td>\n<td style=\"width: 21.0424%;\">49<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 21.0425%;\">Safety<\/td>\n<td style=\"width: 21.0424%;\">10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ul>\n<li>Find the probability that a person chooses a car for cost reasons<\/li>\n<\/ul>\n<p>P(cost) = 49\/100 = 0.490<\/p>\n<p>(The result is based on an experiment, therefore this is Empirical Probability)<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>If we roll a standard 6-sided die, describe the sample space and some simple events.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q997648\">Show Solution<\/span><\/p>\n<div id=\"q997648\" class=\"hidden-answer\" style=\"display: none\">\n<p>The sample space is the set of all possible simple events: {1,2,3,4,5,6}<br \/>\nSome examples of simple events:<\/p>\n<ul>\n<li>We roll a 1<\/li>\n<li>We roll a 5<\/li>\n<\/ul>\n<p>Some compound events:<\/p>\n<ul>\n<li>We roll a number bigger than 4<\/li>\n<li>We roll an even number<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-332\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Math in Society. <strong>Authored by<\/strong>: Open Textbook Store, Transition Math Project, and the Open Course Library. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\">http:\/\/www.opentextbookstore.com\/mathinsociety\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>dice-die-probability-fortune-luck. <strong>Authored by<\/strong>: jodylehigh. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/pixabay.com\/en\/dice-die-probability-fortune-luck-219263\/\">https:\/\/pixabay.com\/en\/dice-die-probability-fortune-luck-219263\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/cc0\">CC0: No Rights Reserved<\/a><\/em><\/li><li>Basics of Probability - events and outcomes. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/37P01dt0zsE\">https:\/\/youtu.be\/37P01dt0zsE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Basic Probabilities. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/EBqj_R3dzd4\">https:\/\/youtu.be\/EBqj_R3dzd4<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 7130. <strong>Authored by<\/strong>: WebWork-Rochester, mb Lippman,David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":20,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Math in Society\",\"author\":\"Open Textbook Store, Transition Math Project, and the Open Course Library\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"dice-die-probability-fortune-luck\",\"author\":\"jodylehigh\",\"organization\":\"\",\"url\":\"https:\/\/pixabay.com\/en\/dice-die-probability-fortune-luck-219263\/\",\"project\":\"\",\"license\":\"cc0\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Basics of Probability - 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