{"id":1500,"date":"2025-07-25T02:15:49","date_gmt":"2025-07-25T02:15:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1500"},"modified":"2026-03-24T07:06:45","modified_gmt":"2026-03-24T07:06:45","slug":"probability-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/probability-fresh-take\/","title":{"raw":"Probability: Fresh Take","rendered":"Probability: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Compute probabilities of equally likely outcomes.<\/li>\r\n \t<li>Compute probabilities of the union of two events.<\/li>\r\n \t<li>Use the complement rule to find probabilities.<\/li>\r\n \t<li>Compute probability using counting theory.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Probability<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Experiment<\/strong>: An activity with an observable result.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Outcome<\/strong>: A possible result of an experiment.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Sample Space<\/strong>: The set of all possible outcomes of an experiment.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Event<\/strong>: Any subset of a sample space.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Probability<\/strong>: The likelihood of an event occurring.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Probability Model<\/strong>: A mathematical description of an experiment listing all possible outcomes and their associated probabilities.<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Probability Fundamentals<\/strong><\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Probability is always between [latex]0[\/latex] and [latex]1[\/latex] (or [latex]0\\%[\/latex] and [latex]100\\%[\/latex]).\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]0 \\leq p \\leq 1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]0[\/latex] ([latex]0\\%[\/latex]) indicates an impossible event.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]1[\/latex] ([latex]100\\%[\/latex]) indicates a certain event.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The sum of all probabilities in a probability model must equal [latex]1[\/latex] ([latex]100\\%[\/latex]).<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Calculating Probability<\/strong><\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For equally likely outcomes:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]P(\\text{outcome}) = \\frac{\\text{Number of ways the outcome can occur}}{\\text{Total number of possible outcomes}}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Important: This formula only applies when all outcomes are equally likely to occur.<\/p>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Constructing a Probability Model<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify all possible outcomes (sample space).<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine the total number of possible outcomes.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate the probability of each outcome.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">List each outcome with its associated probability.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Construct a probability model for tossing a fair coin.[reveal-answer q=\"613018\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"613018\"]\r\n<table>\r\n<thead>\r\n<tr>\r\n<th style=\"width: 113px;\">Outcome<\/th>\r\n<th style=\"width: 251.5px;\">Probability<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 113px;\">Heads<\/td>\r\n<td style=\"width: 251.5px;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 113px;\">Tails<\/td>\r\n<td style=\"width: 251.5px;\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ddcggbbb-YWt_u5l_jHs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/YWt_u5l_jHs?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ddcggbbb-YWt_u5l_jHs\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851416&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ddcggbbb-YWt_u5l_jHs&amp;vembed=0&amp;video_id=YWt_u5l_jHs&amp;video_target=tpm-plugin-ddcggbbb-YWt_u5l_jHs\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Introduction+to+Probability_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Probability\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Computing Probabilities of Equally Likely Outcomes<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\r\n<\/strong>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Sample Space (S)<\/strong>: The set of all possible outcomes in an experiment.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Event (E)<\/strong>: Any subset of the sample space.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Equally Likely Outcomes<\/strong>: When all outcomes in the sample space have an equal chance of occurring.<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Fundamental Formula<\/strong><\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For an event [latex]E[\/latex] in a sample space [latex]S[\/latex] with equally likely outcomes:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]P(E) = \\frac{\\text{number of elements in E}}{\\text{number of elements in S}} = \\frac{n(E)}{n(S)}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Important: [latex]0 \\leq P(E) \\leq 1[\/latex] always holds true, as [latex]E[\/latex] is a subset of [latex]S[\/latex].<\/p>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Steps to Compute Probability<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the sample space [latex]S[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Count the total number of outcomes in [latex]S[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Identify the outcomes that make up the event [latex]E[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Count the number of outcomes in [latex]E[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Divide the number of outcomes in [latex]E[\/latex] by the total number of outcomes in [latex]S[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">A number cube (a fair six-sided die) is rolled. Find the probability of rolling a number greater than [latex]2[\/latex].[reveal-answer q=\"521967\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"521967\"][latex]\\dfrac{2}{3}[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cddebffh-IZAMLgS5x6w\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/IZAMLgS5x6w?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cddebffh-IZAMLgS5x6w\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851417&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cddebffh-IZAMLgS5x6w&amp;vembed=0&amp;video_id=IZAMLgS5x6w&amp;video_target=tpm-plugin-cddebffh-IZAMLgS5x6w\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Determining+Probability+Using+Combinations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermining Probability Using Combinations\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Computing the Probability of the Union of Two Events<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Union of Events<\/strong>: The event that occurs if either or both events occur.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Intersection of Events<\/strong>: The event that occurs when both events occur simultaneously.<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Fundamental Formula<\/strong><\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For two events [latex]E[\/latex] and [latex]F[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]P(E \\cup F) = P(E) + P(F) - P(E \\cap F)[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Where:<\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]P(E \\cup F)[\/latex] is the probability of [latex]E[\/latex] or [latex]F[\/latex] occurring<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]P(E \\cap F)[\/latex] is the probability of both [latex]E[\/latex] and [latex]F[\/latex] occurring<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Important Notation<\/strong><\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Union: [latex]\\cup[\/latex] (represents \"or\")<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Intersection: [latex]\\cap[\/latex] (represents \"and\")<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Steps to Compute Probability of Union<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]P(E)[\/latex] and [latex]P(F)[\/latex] individually<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Determine [latex]P(E \\cap F)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the formula: Add [latex]P(E)[\/latex] and [latex]P(F)[\/latex], then subtract [latex]P(E \\cap F)[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">A card is drawn from a standard deck. Find the probability of drawing a red card or an ace.[reveal-answer q=\"594379\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"594379\"][latex]\\dfrac{7}{13}[\/latex][\/hidden-answer]<\/section>\r\n<h2>Computing the Probability of Mutually Exclusive Events<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p class=\"whitespace-pre-wrap break-words\"><strong>Mutually Exclusive Events<\/strong>: Events that cannot occur at the same time, i.e., they have no outcomes in common.<\/p>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Simplified Formula<\/strong><\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For mutually exclusive events [latex]E[\/latex] and [latex]F[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]P(E \\cup F) = P(E) + P(F)[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Note: This is a simplification of the general union formula because [latex]P(E \\cap F) = 0[\/latex] for mutually exclusive events.<\/p>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Steps to Compute Probability of Mutually Exclusive Events<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Verify that the events are mutually exclusive<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]P(E)[\/latex] for the first event<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate [latex]P(F)[\/latex] for the second event<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Add the individual probabilities<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">A card is drawn from a standard deck. Find the probability of drawing an ace or a king.[reveal-answer q=\"672890\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"672890\"][latex]\\dfrac{2}{13}[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aehfcdhb-MRwKsuEMWZk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/MRwKsuEMWZk?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-aehfcdhb-MRwKsuEMWZk\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780924&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-aehfcdhb-MRwKsuEMWZk&amp;vembed=0&amp;video_id=MRwKsuEMWZk&amp;video_target=tpm-plugin-aehfcdhb-MRwKsuEMWZk\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Unit+11+Mutually+Exclusive1_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cUnit 11 Mutually Exclusive1\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Find the Probability That an Event Will Not Happen<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Complement of an Event<\/strong>: The set of all outcomes in the sample space that are not in the event.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Notation<\/strong>: The complement of event E is denoted as [latex]E'[\/latex].<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>The Complement Rule<\/strong><\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For any event [latex]E[\/latex]:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">[latex]P(E') = 1 - P(E)[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Where:<\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]P(E')[\/latex] is the probability that event [latex]E[\/latex] will not occur<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]P(E)[\/latex] is the probability that event [latex]E[\/latex] will occur<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Why It Works<\/strong><\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">The sum of the probabilities of all possible outcomes in a sample space must equal [latex]1[\/latex]. Therefore, the probability of an event not occurring is the remainder after subtracting the probability of it occurring from [latex]1[\/latex].<\/p>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Steps to Find the Probability of an Event Not Happening<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the event [latex]E[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate or determine [latex]P(E)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Subtract [latex]P(E)[\/latex] from [latex]1[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Two number cubes are rolled. Use the Complement Rule to find the probability that the sum is less than [latex]10[\/latex].[reveal-answer q=\"431695\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"431695\"][latex]\\dfrac{5}{6}[\/latex][\/hidden-answer]<\/section>\r\n<h2>Computing Probability Using Counting Theory<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Probability problems often involve counting principles, permutations, and combinations.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Break down complex problems into smaller counting problems.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use combinations [latex](C(n,r))[\/latex] to count ways of selecting items when order doesn't matter.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the Multiplication Principle when dealing with multiple independent selections.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use the Complement Rule for \"at least\" problems.<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-xl font-bold\"><strong>General Approach<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the event you're looking for.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Count the number of ways this event can occur using appropriate counting techniques.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Count the total number of possible outcomes (sample space).<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Divide the number of favorable outcomes by the total number of possible outcomes.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">A child randomly selects [latex]3[\/latex] gumballs from a container holding [latex]4[\/latex] purple gumballs, [latex]8[\/latex] yellow gumballs, and [latex]2[\/latex] green gumballs.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>Find the probability that all [latex]3[\/latex] gumballs selected are purple.<\/li>\r\n \t<li>Find the probability that no yellow gumballs are selected.<\/li>\r\n \t<li>Find the probability that at least [latex]1[\/latex] yellow gumball is selected.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"212100\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"212100\"]\r\n\r\n[latex]\\begin{align} &amp;\\text{a}\\text{. }\\frac{1}{91} \\\\[1mm] &amp; \\text{b}\\text{. }\\frac{\\text{5}}{\\text{91}} \\\\[1mm] &amp; \\text{c}\\text{. }\\frac{86}{91} \\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Compute probabilities of equally likely outcomes.<\/li>\n<li>Compute probabilities of the union of two events.<\/li>\n<li>Use the complement rule to find probabilities.<\/li>\n<li>Compute probability using counting theory.<\/li>\n<\/ul>\n<\/section>\n<h2>Probability<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\"><strong>Experiment<\/strong>: An activity with an observable result.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Outcome<\/strong>: A possible result of an experiment.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Sample Space<\/strong>: The set of all possible outcomes of an experiment.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Event<\/strong>: Any subset of a sample space.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Probability<\/strong>: The likelihood of an event occurring.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Probability Model<\/strong>: A mathematical description of an experiment listing all possible outcomes and their associated probabilities.<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Probability Fundamentals<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Probability is always between [latex]0[\/latex] and [latex]1[\/latex] (or [latex]0\\%[\/latex] and [latex]100\\%[\/latex]).\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]0 \\leq p \\leq 1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">[latex]0[\/latex] ([latex]0\\%[\/latex]) indicates an impossible event.<\/li>\n<li class=\"whitespace-normal break-words\">[latex]1[\/latex] ([latex]100\\%[\/latex]) indicates a certain event.<\/li>\n<li class=\"whitespace-normal break-words\">The sum of all probabilities in a probability model must equal [latex]1[\/latex] ([latex]100\\%[\/latex]).<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Calculating Probability<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">For equally likely outcomes:<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]P(\\text{outcome}) = \\frac{\\text{Number of ways the outcome can occur}}{\\text{Total number of possible outcomes}}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Important: This formula only applies when all outcomes are equally likely to occur.<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Constructing a Probability Model<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify all possible outcomes (sample space).<\/li>\n<li class=\"whitespace-normal break-words\">Determine the total number of possible outcomes.<\/li>\n<li class=\"whitespace-normal break-words\">Calculate the probability of each outcome.<\/li>\n<li class=\"whitespace-normal break-words\">List each outcome with its associated probability.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Construct a probability model for tossing a fair coin.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q613018\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ddcggbbb-YWt_u5l_jHs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/YWt_u5l_jHs?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ddcggbbb-YWt_u5l_jHs\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851416&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ddcggbbb-YWt_u5l_jHs&amp;vembed=0&amp;video_id=YWt_u5l_jHs&amp;video_target=tpm-plugin-ddcggbbb-YWt_u5l_jHs\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Introduction+to+Probability_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntroduction to Probability\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Computing Probabilities of Equally Likely Outcomes<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\"><strong>Sample Space (S)<\/strong>: The set of all possible outcomes in an experiment.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Event (E)<\/strong>: Any subset of the sample space.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Equally Likely Outcomes<\/strong>: When all outcomes in the sample space have an equal chance of occurring.<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Fundamental Formula<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">For an event [latex]E[\/latex] in a sample space [latex]S[\/latex] with equally likely outcomes:<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]P(E) = \\frac{\\text{number of elements in E}}{\\text{number of elements in S}} = \\frac{n(E)}{n(S)}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Important: [latex]0 \\leq P(E) \\leq 1[\/latex] always holds true, as [latex]E[\/latex] is a subset of [latex]S[\/latex].<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Steps to Compute Probability<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the sample space [latex]S[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Count the total number of outcomes in [latex]S[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Identify the outcomes that make up the event [latex]E[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Count the number of outcomes in [latex]E[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Divide the number of outcomes in [latex]E[\/latex] by the total number of outcomes in [latex]S[\/latex].<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">A number cube (a fair six-sided die) is rolled. Find the probability of rolling a number greater than [latex]2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q521967\">Show Solution<\/button><\/p>\n<div id=\"q521967\" class=\"hidden-answer\" style=\"display: none\">[latex]\\dfrac{2}{3}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cddebffh-IZAMLgS5x6w\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/IZAMLgS5x6w?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cddebffh-IZAMLgS5x6w\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12851417&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-cddebffh-IZAMLgS5x6w&amp;vembed=0&amp;video_id=IZAMLgS5x6w&amp;video_target=tpm-plugin-cddebffh-IZAMLgS5x6w\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Determining+Probability+Using+Combinations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermining Probability Using Combinations\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Computing the Probability of the Union of Two Events<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\"><strong>Union of Events<\/strong>: The event that occurs if either or both events occur.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Intersection of Events<\/strong>: The event that occurs when both events occur simultaneously.<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>Fundamental Formula<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">For two events [latex]E[\/latex] and [latex]F[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]P(E \\cup F) = P(E) + P(F) - P(E \\cap F)[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Where:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]P(E \\cup F)[\/latex] is the probability of [latex]E[\/latex] or [latex]F[\/latex] occurring<\/li>\n<li class=\"whitespace-normal break-words\">[latex]P(E \\cap F)[\/latex] is the probability of both [latex]E[\/latex] and [latex]F[\/latex] occurring<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Important Notation<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Union: [latex]\\cup[\/latex] (represents &#8220;or&#8221;)<\/li>\n<li class=\"whitespace-normal break-words\">Intersection: [latex]\\cap[\/latex] (represents &#8220;and&#8221;)<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Steps to Compute Probability of Union<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Calculate [latex]P(E)[\/latex] and [latex]P(F)[\/latex] individually<\/li>\n<li class=\"whitespace-normal break-words\">Determine [latex]P(E \\cap F)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Apply the formula: Add [latex]P(E)[\/latex] and [latex]P(F)[\/latex], then subtract [latex]P(E \\cap F)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">A card is drawn from a standard deck. Find the probability of drawing a red card or an ace.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q594379\">Show Solution<\/button><\/p>\n<div id=\"q594379\" class=\"hidden-answer\" style=\"display: none\">[latex]\\dfrac{7}{13}[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Computing the Probability of Mutually Exclusive Events<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\"><strong>Mutually Exclusive Events<\/strong>: Events that cannot occur at the same time, i.e., they have no outcomes in common.<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Simplified Formula<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">For mutually exclusive events [latex]E[\/latex] and [latex]F[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]P(E \\cup F) = P(E) + P(F)[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Note: This is a simplification of the general union formula because [latex]P(E \\cap F) = 0[\/latex] for mutually exclusive events.<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Steps to Compute Probability of Mutually Exclusive Events<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Verify that the events are mutually exclusive<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]P(E)[\/latex] for the first event<\/li>\n<li class=\"whitespace-normal break-words\">Calculate [latex]P(F)[\/latex] for the second event<\/li>\n<li class=\"whitespace-normal break-words\">Add the individual probabilities<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">A card is drawn from a standard deck. Find the probability of drawing an ace or a king.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q672890\">Show Solution<\/button><\/p>\n<div id=\"q672890\" class=\"hidden-answer\" style=\"display: none\">[latex]\\dfrac{2}{13}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aehfcdhb-MRwKsuEMWZk\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/MRwKsuEMWZk?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-aehfcdhb-MRwKsuEMWZk\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12780924&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-aehfcdhb-MRwKsuEMWZk&amp;vembed=0&amp;video_id=MRwKsuEMWZk&amp;video_target=tpm-plugin-aehfcdhb-MRwKsuEMWZk\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Unit+11+Mutually+Exclusive1_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cUnit 11 Mutually Exclusive1\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Find the Probability That an Event Will Not Happen<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\"><strong>Complement of an Event<\/strong>: The set of all outcomes in the sample space that are not in the event.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Notation<\/strong>: The complement of event E is denoted as [latex]E'[\/latex].<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>The Complement Rule<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">For any event [latex]E[\/latex]:<\/p>\n<p class=\"whitespace-pre-wrap break-words\">[latex]P(E') = 1 - P(E)[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Where:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]P(E')[\/latex] is the probability that event [latex]E[\/latex] will not occur<\/li>\n<li class=\"whitespace-normal break-words\">[latex]P(E)[\/latex] is the probability that event [latex]E[\/latex] will occur<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Why It Works<\/strong><\/p>\n<p class=\"whitespace-pre-wrap break-words\">The sum of the probabilities of all possible outcomes in a sample space must equal [latex]1[\/latex]. Therefore, the probability of an event not occurring is the remainder after subtracting the probability of it occurring from [latex]1[\/latex].<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Steps to Find the Probability of an Event Not Happening<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the event [latex]E[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Calculate or determine [latex]P(E)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Subtract [latex]P(E)[\/latex] from [latex]1[\/latex]<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Two number cubes are rolled. Use the Complement Rule to find the probability that the sum is less than [latex]10[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q431695\">Show Solution<\/button><\/p>\n<div id=\"q431695\" class=\"hidden-answer\" style=\"display: none\">[latex]\\dfrac{5}{6}[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Computing Probability Using Counting Theory<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Probability problems often involve counting principles, permutations, and combinations.<\/li>\n<li class=\"whitespace-normal break-words\">Break down complex problems into smaller counting problems.<\/li>\n<li class=\"whitespace-normal break-words\">Use combinations [latex](C(n,r))[\/latex] to count ways of selecting items when order doesn&#8217;t matter.<\/li>\n<li class=\"whitespace-normal break-words\">Apply the Multiplication Principle when dealing with multiple independent selections.<\/li>\n<li class=\"whitespace-normal break-words\">Use the Complement Rule for &#8220;at least&#8221; problems.<\/li>\n<\/ol>\n<p class=\"font-600 text-xl font-bold\"><strong>General Approach<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the event you&#8217;re looking for.<\/li>\n<li class=\"whitespace-normal break-words\">Count the number of ways this event can occur using appropriate counting techniques.<\/li>\n<li class=\"whitespace-normal break-words\">Count the total number of possible outcomes (sample space).<\/li>\n<li class=\"whitespace-normal break-words\">Divide the number of favorable outcomes by the total number of possible outcomes.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">A child randomly selects [latex]3[\/latex] gumballs from a container holding [latex]4[\/latex] purple gumballs, [latex]8[\/latex] yellow gumballs, and [latex]2[\/latex] green gumballs.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Find the probability that all [latex]3[\/latex] gumballs selected are purple.<\/li>\n<li>Find the probability that no yellow gumballs are selected.<\/li>\n<li>Find the probability that at least [latex]1[\/latex] yellow gumball is selected.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q212100\">Show Solution<\/button><\/p>\n<div id=\"q212100\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{align} &\\text{a}\\text{. }\\frac{1}{91} \\\\[1mm] & \\text{b}\\text{. }\\frac{\\text{5}}{\\text{91}} \\\\[1mm] & \\text{c}\\text{. }\\frac{86}{91} \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":67,"menu_order":25,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Introduction to Probability\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/YWt_u5l_jHs\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Determining Probability Using Combinations\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/IZAMLgS5x6w\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Unit 11 Mutually Exclusive1\",\"author\":\"Kris Frees\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/MRwKsuEMWZk\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube 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