{"id":3643,"date":"2023-05-25T20:03:22","date_gmt":"2023-05-25T20:03:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=3643"},"modified":"2024-10-18T20:58:32","modified_gmt":"2024-10-18T20:58:32","slug":"math-in-politics-background-youll-need-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/math-in-politics-background-youll-need-2\/","title":{"raw":"Math in Politics: Background You'll Need 2","rendered":"Math in Politics: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Apply basic probability concepts<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Probability<\/h2>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\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>A <strong>compound event <\/strong>is a combination of two or more simple events.<\/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<\/section>\r\n<section class=\"textbox seeExample\">If we roll a standard [latex]6[\/latex]-sided die, describe the sample space and some simple events.[reveal-answer q=\"997648\"]Show Solution[\/reveal-answer] [hidden-answer a=\"997648\"]The sample space is the set of all possible simple events: [latex]{1,2,3,4,5,6}[\/latex]. Some examples of simple events:\r\n\r\n<ul>\r\n\t<li>We roll a [latex]1[\/latex]<\/li>\r\n\t<li>We roll a [latex]5[\/latex]<\/li>\r\n<\/ul>\r\n\r\nSome compound events:\r\n\r\n<ul>\r\n\t<li>We roll a number bigger than [latex]4[\/latex]<\/li>\r\n\t<li>We roll an even number<\/li>\r\n<\/ul>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2807[\/ohm2_question]<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>basic probability<\/h3>\r\n<p>Given that all outcomes are equally likely, we can compute the probability of an event [latex]E[\/latex] using this formula:<\/p>\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<p>&nbsp;<\/p>\r\n<p>Probabilities can be expressed as decimals, fractions, or percentages.<\/p>\r\n<p class=\"student12ptnumberlist\"><strong>Notation:<\/strong> The probability of an event is notated as [latex]P(\\text{event})[\/latex]<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox proTip\">Probability, likelihood, and chance are related concepts that are often used interchangeably, but they have distinct meanings. Probability is a mathematical concept used to quantify the likelihood of an event occurring, likelihood is a measure of how well a hypothesis or model fits the data, and chance is an informal way of expressing the uncertainty of an event. For this lesson, we will be focusing on probability.<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>certain and impossible events<\/h3>\r\n<ul>\r\n\t<li>An <strong>impossible event<\/strong> has a probability of [latex]0[\/latex].<\/li>\r\n\t<li>A <strong>certain event<\/strong> has a probability of [latex]1[\/latex].<\/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<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2808[\/ohm2_question]<\/section>\r\n<h2>Probability of Two Independent Events<\/h2>\r\n<p>Suppose we flipped a coin and rolled a die, and wanted to know the probability of getting a head on the coin and a [latex]6[\/latex] on the die. Take a moment to think about how you could approach finding the probability. You could start by listing all the possible outcomes: [latex]{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}[\/latex] and determining the total number of outcomes. Notice, out of the twelve outcomes only one is the desired outcome, so the probability is [latex]\\frac{1}{12}[\/latex]. This example contains two <strong>independent<\/strong> <strong>events <\/strong>since getting a certain outcome from rolling a die had no influence on the outcome from flipping the coin. When two events are independent, the probability of both occurring is the product of the probabilities of the individual events.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>independent events<\/h3>\r\n<p>Events [latex]A[\/latex] and [latex]B[\/latex] are <strong>independent events<\/strong> if the probability of event [latex]B[\/latex] occurring is the same whether or not event [latex]A[\/latex] occurs.<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>[latex]P(A \\text{ and } B)[\/latex] for independent events<\/h3>\r\n<p>If events [latex]A[\/latex] and [latex]B[\/latex] are independent, then the probability of both [latex]A[\/latex] and [latex]B[\/latex]\u00a0 occurring is:<\/p>\r\n<p>&nbsp;<\/p>\r\n<img class=\"wp-image-177 alignright\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/02182816\/Intersection_of_sets_A_and_B.svg_-300x230.png\" alt=\"A venn diagram with the intersection of sets A and B is shown. The intersection is labeled as A and B\" width=\"187\" height=\"143\" \/><center><\/center><center>[latex]P\\left(A\\text{ and }B\\right)=P\\left(A\\right)\\cdot{P}\\left(B\\right)[\/latex]<\/center>\r\n<p>&nbsp;<\/p>\r\n<br \/>\r\n<p><br \/>\r\nwhere [latex]P(A \\text{ and } B)[\/latex] is the probability of events [latex]A[\/latex] and [latex]B[\/latex] both occurring, [latex]P(A)[\/latex] is the probability of event [latex]A[\/latex] occurring, and [latex]P(B)[\/latex] is the probability of event [latex]B[\/latex] occurring.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p><strong>Notation: <\/strong>[latex]P\\left(A\\text{ and }B\\right)[\/latex] can also be notated as [latex]P(A \\cap B)[\/latex]<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2816[\/ohm2_question]<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>[latex]P(A \\text{ or } B)[\/latex] for independent events<\/h3>\r\n<p><img class=\"wp-image-177 alignright\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/02183504\/Union_of_sets_A_and_B.svg_-300x230.png\" alt=\"A venn diagram of the union of sets A and B is shown\" width=\"187\" height=\"143\" \/><br \/>\r\nThe probability of either [latex]A[\/latex] or [latex]B[\/latex] occurring (or both) is:<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]P(A\\text{ or }B)=P(A)+P(B)\u2013P(A\\text{ and }B)[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<p><strong>Notation: <\/strong>[latex]P\\left(A\\text{ or }B\\right)[\/latex] can also be notated as [latex]P(A \\cup B)[\/latex]<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2824[\/ohm2_question]<\/section>\r\n<h2>Conditional Probability<\/h2>\r\n<p>So far we have computed the probabilities of events that were independent of each other. We saw that getting a certain outcome from rolling a die had no influence on the outcome from flipping a coin, even though we were computing a probability based on doing them at the same time. In this section, we will consider events that are dependent on each other, called <strong>conditional probabilities<\/strong>.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>conditional probability<\/h3>\r\n<p>The probability the event [latex]B[\/latex] occurs, given that event [latex]A[\/latex] has happened, is represented as:<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]P(B | A)[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>This is read as \u201cthe probability of [latex]B[\/latex] given [latex]A[\/latex].\u201d<\/p>\r\n<p>&nbsp;<\/p>\r\n<p><strong>Conditional Probability Formula:<\/strong><\/p>\r\n<p>If Events [latex]A[\/latex] and [latex]B[\/latex] are not independent, then [latex]P(A \\text{ and } B) = P(A) \u00b7 P(B | A)[\/latex]<\/p>\r\n<\/div>\r\n<\/section>\r\n<p>It's important to remember the conditional probability formula can also be written as [latex]P(A \\text{ and } B) = P(B) \u00b7 P(A|B)[\/latex].<\/p>\r\n<section class=\"textbox seeExample\">What is the probability that two cards drawn at random from a deck of playing cards will both be aces?<br \/>\r\n[reveal-answer q=\"284277\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"284277\"]It might seem that you could use the formula for the probability of two independent events and simply multiply [latex]\\frac{4}{52}\\cdot\\frac{4}{52}=\\frac{1}{169}[\/latex]. This would be incorrect, however, because the two events are not independent.If the first card drawn is an ace, then the probability that the second card is also an ace would be lower because there would only be three aces left in the deck. <br \/>\r\n<br \/>\r\nOnce the first card chosen is an ace, the probability that the second card chosen is also an ace is called the <strong>conditional probability<\/strong> of drawing an ace. In this case, the \"condition\" is that the first card is an ace. Symbolically, we write this as: [latex]P(\\text{ace on second draw } | \\text{ an ace on the first draw})[\/latex]. The vertical bar \"|\" is read as \"given,\" so the above expression is short for \"The probability that an ace is drawn on the second draw given that an ace was drawn on the first draw.\" What is this probability? <br \/>\r\n<br \/>\r\nAfter an ace is drawn on the first draw, there are [latex]3[\/latex] aces out of [latex]51[\/latex] total cards left. This means that the conditional probability of drawing an ace after one ace has already been drawn is [latex]\\frac{3}{51}=\\frac{1}{17}[\/latex]. Thus, the probability of both cards being aces is [latex]\\frac{4}{52}\\cdot\\frac{3}{51}=\\frac{12}{2652}=\\frac{1}{221}[\/latex].[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2825[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2826 [\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Apply basic probability concepts<\/li>\n<\/ul>\n<\/section>\n<h2>Probability<\/h2>\n<section class=\"textbox keyTakeaway\">\n<div>\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>A <strong>compound event <\/strong>is a combination of two or more simple events.<\/li>\n<li>The <strong>sample space<\/strong> is the set of all possible simple events.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox seeExample\">If we roll a standard [latex]6[\/latex]-sided die, describe the sample space and some simple events.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q997648\">Show Solution<\/button> <\/p>\n<div id=\"q997648\" class=\"hidden-answer\" style=\"display: none\">The sample space is the set of all possible simple events: [latex]{1,2,3,4,5,6}[\/latex]. Some examples of simple events:<\/p>\n<ul>\n<li>We roll a [latex]1[\/latex]<\/li>\n<li>We roll a [latex]5[\/latex]<\/li>\n<\/ul>\n<p>Some compound events:<\/p>\n<ul>\n<li>We roll a number bigger than [latex]4[\/latex]<\/li>\n<li>We roll an even number<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2807\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2807&theme=lumen&iframe_resize_id=ohm2807&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>basic probability<\/h3>\n<p>Given that all outcomes are equally likely, we can compute the probability of an event [latex]E[\/latex] 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<p>&nbsp;<\/p>\n<p>Probabilities can be expressed as decimals, fractions, or percentages.<\/p>\n<p class=\"student12ptnumberlist\"><strong>Notation:<\/strong> The probability of an event is notated as [latex]P(\\text{event})[\/latex]<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\">Probability, likelihood, and chance are related concepts that are often used interchangeably, but they have distinct meanings. Probability is a mathematical concept used to quantify the likelihood of an event occurring, likelihood is a measure of how well a hypothesis or model fits the data, and chance is an informal way of expressing the uncertainty of an event. For this lesson, we will be focusing on probability.<\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>certain and impossible events<\/h3>\n<ul>\n<li>An <strong>impossible event<\/strong> has a probability of [latex]0[\/latex].<\/li>\n<li>A <strong>certain event<\/strong> has a probability of [latex]1[\/latex].<\/li>\n<li>The probability of any event must be [latex]0\\le P(E)\\le 1[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2808\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2808&theme=lumen&iframe_resize_id=ohm2808&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Probability of Two Independent Events<\/h2>\n<p>Suppose we flipped a coin and rolled a die, and wanted to know the probability of getting a head on the coin and a [latex]6[\/latex] on the die. Take a moment to think about how you could approach finding the probability. You could start by listing all the possible outcomes: [latex]{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}[\/latex] and determining the total number of outcomes. Notice, out of the twelve outcomes only one is the desired outcome, so the probability is [latex]\\frac{1}{12}[\/latex]. This example contains two <strong>independent<\/strong> <strong>events <\/strong>since getting a certain outcome from rolling a die had no influence on the outcome from flipping the coin. When two events are independent, the probability of both occurring is the product of the probabilities of the individual events.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>independent events<\/h3>\n<p>Events [latex]A[\/latex] and [latex]B[\/latex] are <strong>independent events<\/strong> if the probability of event [latex]B[\/latex] occurring is the same whether or not event [latex]A[\/latex] occurs.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>[latex]P(A \\text{ and } B)[\/latex] for independent events<\/h3>\n<p>If events [latex]A[\/latex] and [latex]B[\/latex] are independent, then the probability of both [latex]A[\/latex] and [latex]B[\/latex]\u00a0 occurring is:<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-177 alignright\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/02182816\/Intersection_of_sets_A_and_B.svg_-300x230.png\" alt=\"A venn diagram with the intersection of sets A and B is shown. The intersection is labeled as A and B\" width=\"187\" height=\"143\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/02182816\/Intersection_of_sets_A_and_B.svg_-300x230.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/02182816\/Intersection_of_sets_A_and_B.svg_-65x50.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/02182816\/Intersection_of_sets_A_and_B.svg_-225x172.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/02182816\/Intersection_of_sets_A_and_B.svg_-350x268.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/02182816\/Intersection_of_sets_A_and_B.svg_.png 371w\" sizes=\"(max-width: 187px) 100vw, 187px\" \/><\/p>\n<div style=\"text-align: center;\"><\/div>\n<div style=\"text-align: center;\">[latex]P\\left(A\\text{ and }B\\right)=P\\left(A\\right)\\cdot{P}\\left(B\\right)[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<p>\nwhere [latex]P(A \\text{ and } B)[\/latex] is the probability of events [latex]A[\/latex] and [latex]B[\/latex] both occurring, [latex]P(A)[\/latex] is the probability of event [latex]A[\/latex] occurring, and [latex]P(B)[\/latex] is the probability of event [latex]B[\/latex] occurring.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Notation: <\/strong>[latex]P\\left(A\\text{ and }B\\right)[\/latex] can also be notated as [latex]P(A \\cap B)[\/latex]<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2816\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2816&theme=lumen&iframe_resize_id=ohm2816&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>[latex]P(A \\text{ or } B)[\/latex] for independent events<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-177 alignright\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/01\/02183504\/Union_of_sets_A_and_B.svg_-300x230.png\" alt=\"A venn diagram of the union of sets A and B is shown\" width=\"187\" height=\"143\" \/><br \/>\nThe probability of either [latex]A[\/latex] or [latex]B[\/latex] occurring (or both) is:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]P(A\\text{ or }B)=P(A)+P(B)\u2013P(A\\text{ and }B)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Notation: <\/strong>[latex]P\\left(A\\text{ or }B\\right)[\/latex] can also be notated as [latex]P(A \\cup B)[\/latex]<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2824\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2824&theme=lumen&iframe_resize_id=ohm2824&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Conditional Probability<\/h2>\n<p>So far we have computed the probabilities of events that were independent of each other. We saw that getting a certain outcome from rolling a die had no influence on the outcome from flipping a coin, even though we were computing a probability based on doing them at the same time. In this section, we will consider events that are dependent on each other, called <strong>conditional probabilities<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>conditional probability<\/h3>\n<p>The probability the event [latex]B[\/latex] occurs, given that event [latex]A[\/latex] has happened, is represented as:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]P(B | A)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>This is read as \u201cthe probability of [latex]B[\/latex] given [latex]A[\/latex].\u201d<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Conditional Probability Formula:<\/strong><\/p>\n<p>If Events [latex]A[\/latex] and [latex]B[\/latex] are not independent, then [latex]P(A \\text{ and } B) = P(A) \u00b7 P(B | A)[\/latex]<\/p>\n<\/div>\n<\/section>\n<p>It&#8217;s important to remember the conditional probability formula can also be written as [latex]P(A \\text{ and } B) = P(B) \u00b7 P(A|B)[\/latex].<\/p>\n<section class=\"textbox seeExample\">What is the probability that two cards drawn at random from a deck of playing cards will both be aces?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q284277\">Show Solution<\/button><\/p>\n<div id=\"q284277\" class=\"hidden-answer\" style=\"display: none\">It might seem that you could use the formula for the probability of two independent events and simply multiply [latex]\\frac{4}{52}\\cdot\\frac{4}{52}=\\frac{1}{169}[\/latex]. This would be incorrect, however, because the two events are not independent.If the first card drawn is an ace, then the probability that the second card is also an ace would be lower because there would only be three aces left in the deck. <\/p>\n<p>Once the first card chosen is an ace, the probability that the second card chosen is also an ace is called the <strong>conditional probability<\/strong> of drawing an ace. In this case, the &#8220;condition&#8221; is that the first card is an ace. Symbolically, we write this as: [latex]P(\\text{ace on second draw } | \\text{ an ace on the first draw})[\/latex]. The vertical bar &#8220;|&#8221; is read as &#8220;given,&#8221; so the above expression is short for &#8220;The probability that an ace is drawn on the second draw given that an ace was drawn on the first draw.&#8221; What is this probability? <\/p>\n<p>After an ace is drawn on the first draw, there are [latex]3[\/latex] aces out of [latex]51[\/latex] total cards left. This means that the conditional probability of drawing an ace after one ace has already been drawn is [latex]\\frac{3}{51}=\\frac{1}{17}[\/latex]. 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