{"id":464,"date":"2019-07-15T22:45:19","date_gmt":"2019-07-15T22:45:19","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/construct-probability-models\/"},"modified":"2019-07-15T22:45:19","modified_gmt":"2019-07-15T22:45:19","slug":"construct-probability-models","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/construct-probability-models\/","title":{"raw":"Construct Probability Models","rendered":"Construct Probability Models"},"content":{"raw":"\n<div class=\"textbox learning-objectives\"><h3>Learning Outcomes<\/h3><ul><li>Construct a probability model that assigns the probability of each outcome in a sample space.<\/li><li>Compute the probability of an event with equally likely outcomes.<\/li><\/ul><\/div>\n\nSuppose we roll a six-sided number cube. Rolling a number cube is an example of an <strong>experiment<\/strong>, or an activity with an observable result. The numbers on the cube are possible results, or <strong>outcomes<\/strong>, of this experiment. The set of all possible outcomes of an experiment is called the <strong>sample space<\/strong> of the experiment. The sample space for this experiment is [latex]\\left\\{1,2,3,4,5,6\\right\\}[\/latex]. An <strong>event<\/strong> is any subset of a sample space.\n\nThe likelihood of an event is known as <strong>probability<\/strong>. The probability of an event [latex]p[\/latex] is a number that always satisfies [latex]0\\le p\\le 1[\/latex], where 0 indicates an impossible event and 1 indicates a certain event. A <strong>probability model<\/strong> is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. For instance, if there is a 1% chance of winning a raffle and a 99% chance of losing the raffle, a probability model would look much like the table below.\n\n<table><thead><tr><th>Outcome<\/th><th>Probability<\/th><\/tr><\/thead><tbody><tr><td>Winning the raffle<\/td><td>1%<\/td><\/tr><tr><td>Losing the raffle<\/td><td>99%<\/td><\/tr><\/tbody><\/table>The sum of the probabilities listed in a probability model must equal 1, or 100%.\n\n<div class=\"textbox\"><h3>How To: Given a probability event where each event is equally likely, construct a probability model.<\/h3><ol><li>Identify every outcome.<\/li><li>Determine the total number of possible outcomes.<\/li><li>Compare each outcome to the total number of possible outcomes.<\/li><\/ol><\/div><div class=\"textbox examples\"><h3>tip for success<\/h3>Be sure to try the examples and practice problems in this section on paper to build up your intuition for the next section's material.\n\n<\/div><div class=\"textbox exercises\"><h3>Example: Constructing a Probability Model<\/h3>Construct a probability model for rolling a single, fair die, with the event being the number shown on the die.\n\n[reveal-answer q=\"640659\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"640659\"]\n\nBegin by making a list of all possible outcomes for the experiment. The possible outcomes are the numbers that can be rolled: 1, 2, 3, 4, 5, and 6. There are six possible outcomes that make up the sample space.\n\nAssign probabilities to each outcome in the sample space by determining a ratio of the outcome to the number of possible outcomes. There is one of each of the six numbers on the cube, and there is no reason to think that any particular face is more likely to show up than any other one, so the probability of rolling any number is [latex]\\frac{1}{6}[\/latex].\n\n<table><tbody><tr><td><strong>Outcome<\/strong><\/td><td>Roll of 1<\/td><td>Roll of 2<\/td><td>Roll of 3<\/td><td>Roll of 4<\/td><td>Roll of 5<\/td><td>Roll of 6<\/td><\/tr><tr><td><strong>Probability<\/strong><\/td><td>[latex]\\frac{1}{6}[\/latex]<\/td><td>[latex]\\frac{1}{6}[\/latex]<\/td><td>[latex]\\frac{1}{6}[\/latex]<\/td><td>[latex]\\frac{1}{6}[\/latex]<\/td><td>[latex]\\frac{1}{6}[\/latex]<\/td><td>[latex]\\frac{1}{6}[\/latex]<\/td><\/tr><\/tbody><\/table>[\/hidden-answer]\n\n<\/div><div class=\"textbox\"><h3>Q &amp; A<\/h3><h4>Do probabilities always have to be expressed as fractions?<\/h4><em>No. Probabilities can be expressed as fractions, decimals, or percents. Probability must always be a number between 0 and 1, inclusive of 0 and 1.<\/em>\n\n<\/div><div class=\"textbox key-takeaways\"><h3>Try It<\/h3>Construct a probability model for tossing a fair coin.\n\n[reveal-answer q=\"613018\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"613018\"]\n\n<table><thead><tr><th style=\"width: 113px\">Outcome<\/th><th style=\"width: 251.5px\">Probability<\/th><\/tr><\/thead><tbody><tr><td style=\"width: 113px\">Heads<\/td><td style=\"width: 251.5px\">[latex]\\frac{1}{2}[\/latex]<\/td><\/tr><tr><td style=\"width: 113px\">Tails<\/td><td style=\"width: 251.5px\">[latex]\\frac{1}{2}[\/latex]<\/td><\/tr><\/tbody><\/table>[\/hidden-answer]\n\n<\/div><h2>Computing Probabilities of Equally Likely Outcomes<\/h2>Let [latex]S[\/latex] be a sample space for an experiment. When investigating probability, an event is any subset of [latex]S[\/latex]. When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in [latex]S[\/latex]. Suppose a number cube is rolled, and we are interested in finding the probability of the event \"rolling a number less than or equal to 4.\" There are 4 possible outcomes in the event and 6 possible outcomes in [latex]S[\/latex], so the probability of the event is [latex]\\frac{4}{6}=\\frac{2}{3}[\/latex].\n\n<div class=\"textbox\"><h3>A General Note: Computing the Probability of an Event with Equally Likely Outcomes<\/h3>The probability of an event [latex]E[\/latex] in an experiment with sample space [latex]S[\/latex] with equally likely outcomes is given by\n\n<p style=\"text-align: center\">[latex]P\\left(E\\right)=\\dfrac{\\text{number of elements in }E}{\\text{number of elements in }S}=\\dfrac{n\\left(E\\right)}{n\\left(S\\right)}[\/latex]\n\n[latex]E[\/latex] is a subset of [latex]S[\/latex], so it is always true that [latex]0\\le P\\left(E\\right)\\le 1[\/latex].\n\n<\/p><\/div><div class=\"textbox exercises\"><h3>Example: Computing the Probability of an Event with Equally Likely Outcomes<\/h3>A number cube is rolled. Find the probability of rolling an odd number.\n\n[reveal-answer q=\"320469\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"320469\"]\n\nThe event \"rolling an odd number\" contains three outcomes. There are 6 equally likely outcomes in the sample space. Divide to find the probability of the event.\n\n<p style=\"text-align: center\">[latex]P\\left(E\\right)=\\dfrac{3}{6}=\\dfrac{1}{2}[\/latex]\n\n[\/hidden-answer]\n\n<\/p><\/div><div class=\"textbox key-takeaways\"><h3>Try It<\/h3>A number cube is rolled. Find the probability of rolling a number greater than 2.\n\n[reveal-answer q=\"521967\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"521967\"]\n\n[latex]\\dfrac{2}{3}[\/latex]\n\n[\/hidden-answer]\n[embed]https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=838&amp;theme=oea&amp;iframe_resize_id=mom5[\/embed]\n\n\n\n<\/div>&nbsp;\n\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Construct a probability model that assigns the probability of each outcome in a sample space.<\/li>\n<li>Compute the probability of an event with equally likely outcomes.<\/li>\n<\/ul>\n<\/div>\n<p>Suppose we roll a six-sided number cube. Rolling a number cube is an example of an <strong>experiment<\/strong>, or an activity with an observable result. The numbers on the cube are possible results, or <strong>outcomes<\/strong>, of this experiment. The set of all possible outcomes of an experiment is called the <strong>sample space<\/strong> of the experiment. The sample space for this experiment is [latex]\\left\\{1,2,3,4,5,6\\right\\}[\/latex]. An <strong>event<\/strong> is any subset of a sample space.<\/p>\n<p>The likelihood of an event is known as <strong>probability<\/strong>. The probability of an event [latex]p[\/latex] is a number that always satisfies [latex]0\\le p\\le 1[\/latex], where 0 indicates an impossible event and 1 indicates a certain event. A <strong>probability model<\/strong> is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. For instance, if there is a 1% chance of winning a raffle and a 99% chance of losing the raffle, a probability model would look much like the table below.<\/p>\n<table>\n<thead>\n<tr>\n<th>Outcome<\/th>\n<th>Probability<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Winning the raffle<\/td>\n<td>1%<\/td>\n<\/tr>\n<tr>\n<td>Losing the raffle<\/td>\n<td>99%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The sum of the probabilities listed in a probability model must equal 1, or 100%.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a probability event where each event is equally likely, construct a probability model.<\/h3>\n<ol>\n<li>Identify every outcome.<\/li>\n<li>Determine the total number of possible outcomes.<\/li>\n<li>Compare each outcome to the total number of possible outcomes.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>Be sure to try the examples and practice problems in this section on paper to build up your intuition for the next section&#8217;s material.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Constructing a Probability Model<\/h3>\n<p>Construct a probability model for rolling a single, fair die, with the event being the number shown on the die.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q640659\">Show Solution<\/span><\/p>\n<div id=\"q640659\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by making a list of all possible outcomes for the experiment. The possible outcomes are the numbers that can be rolled: 1, 2, 3, 4, 5, and 6. There are six possible outcomes that make up the sample space.<\/p>\n<p>Assign probabilities to each outcome in the sample space by determining a ratio of the outcome to the number of possible outcomes. There is one of each of the six numbers on the cube, and there is no reason to think that any particular face is more likely to show up than any other one, so the probability of rolling any number is [latex]\\frac{1}{6}[\/latex].<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>Outcome<\/strong><\/td>\n<td>Roll of 1<\/td>\n<td>Roll of 2<\/td>\n<td>Roll of 3<\/td>\n<td>Roll of 4<\/td>\n<td>Roll of 5<\/td>\n<td>Roll of 6<\/td>\n<\/tr>\n<tr>\n<td><strong>Probability<\/strong><\/td>\n<td>[latex]\\frac{1}{6}[\/latex]<\/td>\n<td>[latex]\\frac{1}{6}[\/latex]<\/td>\n<td>[latex]\\frac{1}{6}[\/latex]<\/td>\n<td>[latex]\\frac{1}{6}[\/latex]<\/td>\n<td>[latex]\\frac{1}{6}[\/latex]<\/td>\n<td>[latex]\\frac{1}{6}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>Do probabilities always have to be expressed as fractions?<\/h4>\n<p><em>No. Probabilities can be expressed as fractions, decimals, or percents. Probability must always be a number between 0 and 1, inclusive of 0 and 1.<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Construct a probability model for tossing a fair coin.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q613018\">Show Solution<\/span><\/p>\n<div id=\"q613018\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<thead>\n<tr>\n<th style=\"width: 113px\">Outcome<\/th>\n<th style=\"width: 251.5px\">Probability<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 113px\">Heads<\/td>\n<td style=\"width: 251.5px\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 113px\">Tails<\/td>\n<td style=\"width: 251.5px\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<h2>Computing Probabilities of Equally Likely Outcomes<\/h2>\n<p>Let [latex]S[\/latex] be a sample space for an experiment. When investigating probability, an event is any subset of [latex]S[\/latex]. When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in [latex]S[\/latex]. Suppose a number cube is rolled, and we are interested in finding the probability of the event &#8220;rolling a number less than or equal to 4.&#8221; There are 4 possible outcomes in the event and 6 possible outcomes in [latex]S[\/latex], so the probability of the event is [latex]\\frac{4}{6}=\\frac{2}{3}[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Computing the Probability of an Event with Equally Likely Outcomes<\/h3>\n<p>The probability of an event [latex]E[\/latex] in an experiment with sample space [latex]S[\/latex] with equally likely outcomes is given by<\/p>\n<p style=\"text-align: center\">[latex]P\\left(E\\right)=\\dfrac{\\text{number of elements in }E}{\\text{number of elements in }S}=\\dfrac{n\\left(E\\right)}{n\\left(S\\right)}[\/latex]<\/p>\n<p>[latex]E[\/latex] is a subset of [latex]S[\/latex], so it is always true that [latex]0\\le P\\left(E\\right)\\le 1[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Computing the Probability of an Event with Equally Likely Outcomes<\/h3>\n<p>A number cube is rolled. Find the probability of rolling an odd number.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q320469\">Show Solution<\/span><\/p>\n<div id=\"q320469\" class=\"hidden-answer\" style=\"display: none\">\n<p>The event &#8220;rolling an odd number&#8221; contains three outcomes. There are 6 equally likely outcomes in the sample space. Divide to find the probability of the event.<\/p>\n<p style=\"text-align: center\">[latex]P\\left(E\\right)=\\dfrac{3}{6}=\\dfrac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A number cube is rolled. Find the probability of rolling a number greater than 2.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q521967\">Show Solution<\/span><\/p>\n<div id=\"q521967\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\dfrac{2}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm838\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=838&#38;theme=oea&#38;iframe_resize_id=ohm838&#38;show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/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-464\">\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>Question ID 838. <strong>Authored by<\/strong>: Eldridge,Jeff. <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><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <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>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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":17533,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 838\",\"author\":\"Eldridge,Jeff\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\" IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et 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