{"id":400,"date":"2023-06-05T15:31:03","date_gmt":"2023-06-05T15:31:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/chapter\/basic-concepts\/"},"modified":"2025-12-08T00:23:25","modified_gmt":"2025-12-08T00:23:25","slug":"basic-concepts","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/chapter\/basic-concepts\/","title":{"raw":"Basic Concepts","rendered":"Basic Concepts"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Describe a sample space and 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<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, write 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 deck of playing cards, or randomly select a person and observe their hair color, we are executing an experiment or procedure. 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 class=\"aligncenter size-full wp-image-987\" 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>Events and Outcomes<\/h3>\r\n<ul>\r\n \t<li>The result of an experiment is called an <strong>outcome<\/strong>.<\/li>\r\n \t<li>An <strong>event<\/strong> is any particular outcome or group of outcomes.<\/li>\r\n \t<li>A <strong>simple event <\/strong>is an event that cannot be broken down further<\/li>\r\n \t<li>The <strong>sample space<\/strong> is the set of all possible simple events.<\/li>\r\n<\/ul>\r\n<\/div>\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\n\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&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Basic Probability<\/h3>\r\nGiven that all outcomes are equally likely, we can compute the probability of an event <em>E<\/em> using this formula:\r\n<p style=\"text-align: center;\">[latex]P(E)=\\frac{\\text{Number of outcomes corresponding to the event E}}{\\text{Total number of equally-likely outcomes}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>examples<\/h3>\r\nIf we roll a 6-sided die, calculate\r\n<ol>\r\n \t<li>P(rolling a 1)<\/li>\r\n \t<li>P(rolling a number bigger than 4)<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"417705\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"417705\"]\r\n\r\nRecall that the sample space is {1,2,3,4,5,6}\r\n<ol>\r\n \t<li>There is one outcome corresponding to \u201crolling a 1,\u201d so the probability is [latex]\\frac{1}{6}[\/latex]<\/li>\r\n \t<li>There are two outcomes bigger than a 4, so the probability is [latex]\\frac{2}{6}=\\frac{1}{3}[\/latex]<\/li>\r\n<\/ol>\r\nProbabilities are essentially fractions, and can be reduced to lower terms like fractions.\r\n\r\n[\/hidden-answer]\r\n\r\nThis video describes this example and the previous one in detail.\r\n\r\nhttps:\/\/youtu.be\/37P01dt0zsE\r\n\r\nLet's say you have a bag with 20 cherries, 14 sweet and 6 sour. If you pick a cherry at random, what is the probability that it will be sweet?\r\n[reveal-answer q=\"9680\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"9680\"]\r\n\r\nThere are 20 possible cherries that could be picked, so the number of possible outcomes is 20. Of these 20 possible outcomes, 14 are favorable (sweet), so the probability that the cherry will be sweet is [latex]\\frac{14}{20}=\\frac{7}{10}[\/latex].\r\nThere is one potential complication to this example, however. It must be assumed that the probability of picking any of the cherries is the same as the probability of picking any other. This wouldn't be true if (let us imagine) the sweet cherries are smaller than the sour ones. (The sour cherries would come to hand more readily when you sampled from the bag.) Let us keep in mind, therefore, that when we assess probabilities in terms of the ratio of favorable to all potential cases, we rely heavily on the assumption of equal probability for all outcomes.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nAt some random moment, you look at your clock and note the minutes reading.\r\n\r\na. What is probability the minutes reading is 15?\r\n\r\nb. What is the probability the minutes reading is 15 or less?\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Cards<\/h3>\r\nA standard deck of 52 playing cards consists of four <strong>suits<\/strong> (hearts, spades, diamonds and clubs). Spades and clubs are black while hearts and diamonds are red. Each suit contains 13 cards, each of a different <strong>rank<\/strong>: an Ace (which in many games functions as both a low card and a high card), cards numbered 2 through 10, a Jack, a Queen and a King.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nCompute the probability of randomly drawing one card from a deck and getting an Ace.\r\n[reveal-answer q=\"652517\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"652517\"]\r\n\r\nThere are 52 cards in the deck and 4 Aces so\u00a0[latex]P(Ace)=\\frac{4}{52}=\\frac{1}{13}\\approx 0.0769[\/latex]\r\n\r\nWe can also think of probabilities as percents: There is a 7.69% chance that a randomly selected card will be an Ace.\r\n\r\nNotice that the smallest possible probability is 0 \u2013 if there are no outcomes that correspond with the event. The largest possible probability is 1 \u2013 if all possible outcomes correspond with the event.\r\n\r\n[\/hidden-answer]\r\n\r\nThis video demonstrates both this example and the previous cherry example\u00a0on the page.\r\n\r\nhttps:\/\/youtu.be\/EBqj_R3dzd4\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>SURE and Impossible events<\/h3>\r\n<ul>\r\n \t<li>An impossible event has a probability of 0.<\/li>\r\n \t<li>A sure event has a probability of 1.<\/li>\r\n \t<li>The probability of any event must be [latex]0\\le P(E)\\le 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]7130[\/ohm_question]\r\n\r\n<\/div>\r\nIn the course of this section, <strong>if you compute a probability and get an answer that is negative or greater than 1, you have made a mistake and should check your work<\/strong>.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Describe a sample space and 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<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, write 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 deck of playing cards, or randomly select a person and observe their hair color, we are executing an experiment or procedure. 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\" class=\"aligncenter size-full wp-image-987\" 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>Events and Outcomes<\/h3>\n<ul>\n<li>The result of an experiment is called an <strong>outcome<\/strong>.<\/li>\n<li>An <strong>event<\/strong> is any particular outcome or group of outcomes.<\/li>\n<li>A <strong>simple event <\/strong>is an event that cannot be broken down further<\/li>\n<li>The <strong>sample space<\/strong> is the set of all possible simple events.<\/li>\n<\/ul>\n<\/div>\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}<\/p>\n<p>Some 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<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Basic Probability<\/h3>\n<p>Given that all outcomes are equally likely, we can compute the probability of an event <em>E<\/em> using this formula:<\/p>\n<p style=\"text-align: center;\">[latex]P(E)=\\frac{\\text{Number of outcomes corresponding to the event E}}{\\text{Total number of equally-likely outcomes}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>examples<\/h3>\n<p>If we roll a 6-sided die, calculate<\/p>\n<ol>\n<li>P(rolling a 1)<\/li>\n<li>P(rolling a number bigger than 4)<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q417705\">Show Solution<\/span><\/p>\n<div id=\"q417705\" class=\"hidden-answer\" style=\"display: none\">\n<p>Recall that the sample space is {1,2,3,4,5,6}<\/p>\n<ol>\n<li>There is one outcome corresponding to \u201crolling a 1,\u201d so the probability is [latex]\\frac{1}{6}[\/latex]<\/li>\n<li>There are two outcomes bigger than a 4, so the probability is [latex]\\frac{2}{6}=\\frac{1}{3}[\/latex]<\/li>\n<\/ol>\n<p>Probabilities are essentially fractions, and can be reduced to lower terms like fractions.<\/p>\n<\/div>\n<\/div>\n<p>This video describes this example and the previous one in detail.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Basics of Probability - events and outcomes\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/37P01dt0zsE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Let&#8217;s say you have a bag with 20 cherries, 14 sweet and 6 sour. If you pick a cherry at random, what is the probability that it will be sweet?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q9680\">Show Solution<\/span><\/p>\n<div id=\"q9680\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are 20 possible cherries that could be picked, so the number of possible outcomes is 20. Of these 20 possible outcomes, 14 are favorable (sweet), so the probability that the cherry will be sweet is [latex]\\frac{14}{20}=\\frac{7}{10}[\/latex].<br \/>\nThere is one potential complication to this example, however. It must be assumed that the probability of picking any of the cherries is the same as the probability of picking any other. This wouldn&#8217;t be true if (let us imagine) the sweet cherries are smaller than the sour ones. (The sour cherries would come to hand more readily when you sampled from the bag.) Let us keep in mind, therefore, that when we assess probabilities in terms of the ratio of favorable to all potential cases, we rely heavily on the assumption of equal probability for all outcomes.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>At some random moment, you look at your clock and note the minutes reading.<\/p>\n<p>a. What is probability the minutes reading is 15?<\/p>\n<p>b. What is the probability the minutes reading is 15 or less?<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Cards<\/h3>\n<p>A standard deck of 52 playing cards consists of four <strong>suits<\/strong> (hearts, spades, diamonds and clubs). Spades and clubs are black while hearts and diamonds are red. Each suit contains 13 cards, each of a different <strong>rank<\/strong>: an Ace (which in many games functions as both a low card and a high card), cards numbered 2 through 10, a Jack, a Queen and a King.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Compute the probability of randomly drawing one card from a deck and getting an Ace.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q652517\">Show Solution<\/span><\/p>\n<div id=\"q652517\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are 52 cards in the deck and 4 Aces so\u00a0[latex]P(Ace)=\\frac{4}{52}=\\frac{1}{13}\\approx 0.0769[\/latex]<\/p>\n<p>We can also think of probabilities as percents: There is a 7.69% chance that a randomly selected card will be an Ace.<\/p>\n<p>Notice that the smallest possible probability is 0 \u2013 if there are no outcomes that correspond with the event. The largest possible probability is 1 \u2013 if all possible outcomes correspond with the event.<\/p>\n<\/div>\n<\/div>\n<p>This video demonstrates both this example and the previous cherry example\u00a0on the page.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Basic Probabilities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/EBqj_R3dzd4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>SURE and Impossible events<\/h3>\n<ul>\n<li>An impossible event has a probability of 0.<\/li>\n<li>A sure event has a probability of 1.<\/li>\n<li>The probability of any event must be [latex]0\\le P(E)\\le 1[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm7130\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=7130&theme=oea&iframe_resize_id=ohm7130&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the course of this section, <strong>if you compute a probability and get an answer that is negative or greater than 1, you have made a mistake and should check your work<\/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-400\">\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":503070,"menu_order":16,"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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