{"id":163,"date":"2025-02-13T22:44:30","date_gmt":"2025-02-13T22:44:30","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/probability\/"},"modified":"2026-03-26T19:30:57","modified_gmt":"2026-03-26T19:30:57","slug":"probability","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/probability\/","title":{"raw":"Probability: Learn It 1","rendered":"Probability: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li style=\"font-weight: 400;\">Compute probabilities of equally likely outcomes.<\/li>\r\n \t<li style=\"font-weight: 400;\">Compute probabilities of the union of two events.<\/li>\r\n \t<li style=\"font-weight: 400;\">Use the complement rule to find probabilities.<\/li>\r\n \t<li style=\"font-weight: 400;\">Compute probability using counting theory.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Probability<\/h2>\r\nMany events in life are inherently uncertain: will it snow tomorrow? Am I going to get an \u2018A\u2019 in this course? None of these questions can be answered with certainty, however, we might say that some are unlikely, and others are more likely.\r\n\r\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.\r\n\r\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 [latex]0[\/latex] indicates an impossible event and [latex]1[\/latex] 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 [latex]1\\%[\/latex] chance of winning a raffle and a [latex]99\\%[\/latex] chance of losing the raffle, a probability model would look much like the table below.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Outcome<\/th>\r\n<th>Probability<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Winning the raffle<\/td>\r\n<td>[latex]1\\%[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Losing the raffle<\/td>\r\n<td>[latex]99\\%[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe sum of the probabilities listed in a probability model must equal [latex]1[\/latex], or [latex]100\\%[\/latex].\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>probability<\/h3>\r\nThe <strong>probability of an event<\/strong> is a description of how likely it is that an event will happen.\r\n<p style=\"padding-left: 40px;\">A probability is a number between [latex]0[\/latex] and [latex]1[\/latex] (that is, between [latex]0\\%[\/latex] and [latex]100\\%[\/latex]), where probabilities closer to [latex]100\\%[\/latex] are very likely to occur, and probabilities closer to [latex]0\\%[\/latex] are very unlikely to occur. A probability of [latex]0\\%[\/latex] means the event is impossible, and a probability of [latex]100\\%[\/latex] means the event will certainly occur.<\/p>\r\n&nbsp;\r\n\r\nTo calculate the probability of an event, we divide the number of possible outcomes of the event by the number of possible outcomes of the sample space.\r\n<p style=\"text-align: center;\">[latex]P(\\text{outcome}) = \\dfrac{\\text{Number of ways that outcome can occur}}{\\text{Total number of outcomes}}[\/latex]<\/p>\r\n\r\n<ul>\r\n \t<li>It is important to note that in order to use this formula, <strong>all outcomes must be equally likely to happen<\/strong>.<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\nA <strong>probability model<\/strong> is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. It is defined by its sample space, events within the sample space, and probabilities associated with each event.\r\n<ul>\r\n \t<li>The <strong>sample space<\/strong> [latex]S[\/latex] for a probability model is the set of all possible outcomes.<\/li>\r\n \t<li class=\"page\" title=\"Page 1142\">An <strong>event<\/strong> [latex]A[\/latex] is a subset of the sample space [latex]S[\/latex].<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a probability event where each event is equally likely, construct a probability model.<\/strong>\r\n<ol>\r\n \t<li>Identify every outcome.<\/li>\r\n \t<li>Determine the total number of possible outcomes.<\/li>\r\n \t<li>Compare each outcome to the total number of possible outcomes.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Construct a probability model for rolling a single, fair die, with the event being the number shown on the die.[reveal-answer q=\"640659\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"640659\"]Begin by making a list of all possible outcomes for the experiment. The possible outcomes are the numbers that can be rolled: [latex]1, 2, 3, 4, 5,[\/latex] and [latex]6[\/latex]. There are six possible outcomes that make up the <strong>sample space<\/strong>.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].\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Outcome<\/strong><\/td>\r\n<td>Roll of [latex]1[\/latex]<\/td>\r\n<td>Roll of [latex]2[\/latex]<\/td>\r\n<td>Roll of [latex]3[\/latex]<\/td>\r\n<td>Roll of [latex]4[\/latex]<\/td>\r\n<td>Roll of [latex]5[\/latex]<\/td>\r\n<td>Roll of [latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Probability<\/strong><\/td>\r\n<td>[latex]\\frac{1}{6}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{6}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{6}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{6}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{6}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{6}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322239[\/ohm_question]<\/section><section aria-label=\"Try It\"><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]322240[\/ohm_question]<\/section><\/section><\/div>\r\n<dl id=\"fs-id1657370\" class=\"definition\">\r\n \t<dd id=\"fs-id1569949\"><\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li style=\"font-weight: 400;\">Compute probabilities of equally likely outcomes.<\/li>\n<li style=\"font-weight: 400;\">Compute probabilities of the union of two events.<\/li>\n<li style=\"font-weight: 400;\">Use the complement rule to find probabilities.<\/li>\n<li style=\"font-weight: 400;\">Compute probability using counting theory.<\/li>\n<\/ul>\n<\/section>\n<h2>Probability<\/h2>\n<p>Many events in life are inherently uncertain: will it snow tomorrow? Am I going to get an \u2018A\u2019 in this course? None of these questions can be answered with certainty, however, we might say that some are unlikely, and others are more likely.<\/p>\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 [latex]0[\/latex] indicates an impossible event and [latex]1[\/latex] 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 [latex]1\\%[\/latex] chance of winning a raffle and a [latex]99\\%[\/latex] 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>[latex]1\\%[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Losing the raffle<\/td>\n<td>[latex]99\\%[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The sum of the probabilities listed in a probability model must equal [latex]1[\/latex], or [latex]100\\%[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>probability<\/h3>\n<p>The <strong>probability of an event<\/strong> is a description of how likely it is that an event will happen.<\/p>\n<p style=\"padding-left: 40px;\">A probability is a number between [latex]0[\/latex] and [latex]1[\/latex] (that is, between [latex]0\\%[\/latex] and [latex]100\\%[\/latex]), where probabilities closer to [latex]100\\%[\/latex] are very likely to occur, and probabilities closer to [latex]0\\%[\/latex] are very unlikely to occur. A probability of [latex]0\\%[\/latex] means the event is impossible, and a probability of [latex]100\\%[\/latex] means the event will certainly occur.<\/p>\n<p>&nbsp;<\/p>\n<p>To calculate the probability of an event, we divide the number of possible outcomes of the event by the number of possible outcomes of the sample space.<\/p>\n<p style=\"text-align: center;\">[latex]P(\\text{outcome}) = \\dfrac{\\text{Number of ways that outcome can occur}}{\\text{Total number of outcomes}}[\/latex]<\/p>\n<ul>\n<li>It is important to note that in order to use this formula, <strong>all outcomes must be equally likely to happen<\/strong>.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>A <strong>probability model<\/strong> is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. It is defined by its sample space, events within the sample space, and probabilities associated with each event.<\/p>\n<ul>\n<li>The <strong>sample space<\/strong> [latex]S[\/latex] for a probability model is the set of all possible outcomes.<\/li>\n<li class=\"page\" title=\"Page 1142\">An <strong>event<\/strong> [latex]A[\/latex] is a subset of the sample space [latex]S[\/latex].<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a probability event where each event is equally likely, construct a probability model.<\/strong><\/p>\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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q640659\">Show Solution<\/button><\/p>\n<div id=\"q640659\" class=\"hidden-answer\" style=\"display: none\">Begin by making a list of all possible outcomes for the experiment. The possible outcomes are the numbers that can be rolled: [latex]1, 2, 3, 4, 5,[\/latex] and [latex]6[\/latex]. There are six possible outcomes that make up the <strong>sample space<\/strong>.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 [latex]1[\/latex]<\/td>\n<td>Roll of [latex]2[\/latex]<\/td>\n<td>Roll of [latex]3[\/latex]<\/td>\n<td>Roll of [latex]4[\/latex]<\/td>\n<td>Roll of [latex]5[\/latex]<\/td>\n<td>Roll of [latex]6[\/latex]<\/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<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322239\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322239&theme=lumen&iframe_resize_id=ohm322239&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section aria-label=\"Try It\">\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm322240\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=322240&theme=lumen&iframe_resize_id=ohm322240&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<\/div>\n<dl id=\"fs-id1657370\" class=\"definition\">\n<dd id=\"fs-id1569949\"><\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":18,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":513,"module-header":"learn_it","content_attributions":[{"type":"original","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/163"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/163\/revisions"}],"predecessor-version":[{"id":6057,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/163\/revisions\/6057"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/513"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/163\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=163"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=163"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=163"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=163"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}