{"id":1031,"date":"2023-06-22T01:45:11","date_gmt":"2023-06-22T01:45:11","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/binomial-distribution-background-youll-need-2\/"},"modified":"2024-02-10T00:02:38","modified_gmt":"2024-02-10T00:02:38","slug":"binomial-distribution-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/binomial-distribution-background-youll-need-2\/","title":{"raw":"Supplemental Module - Binomial Distribution: Background You\u2019ll Need 3","rendered":"Supplemental Module &#8211; Binomial Distribution: Background You\u2019ll Need 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Interpret statements of inequality<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Inequalities and Probabilities<\/h2>\r\n<p>Let's\u00a0consider probability questions where each trial of an experiment has exactly two outcomes (which we call \u201csuccess\u201d and \u201cfailure\u201d), and we want to count how many of the trials result in successes.<\/p>\r\n<p><img class=\"alignleft wp-image-1505\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/10\/10155303\/Picture14.png\" alt=\"A spinner with 4 quadrants, labeled 1, 2, 3, and 4.\" width=\"101\" height=\"99\" \/>Suppose that we have a spinner with [latex]4[\/latex] sections, each of equal size, and the sections are labeled [latex]1, 2, 3, \\text{and} 4[\/latex]. We might want to count how many times the spinner lands on the [latex]1[\/latex] section if we spin the spinner [latex]10[\/latex] times. We will also consider questions like, \u201cWhat is the probability that in at least [latex]6[\/latex] out of [latex]10[\/latex] spin attempts, the arrow lands on the [latex]1[\/latex] section?\u201d For these types of questions, we will need inequality expressions.<\/p>\r\n<h3>Inequality Review<\/h3>\r\n<p>There are four inequality symbols:<\/p>\r\n<ul>\r\n\t<li>[latex]&lt;[\/latex] (<strong>less than<\/strong>): [latex] a \\lt b [\/latex] means that \u201ca is less than b.\u201d For example, we write [latex] 2 \\lt 5 [\/latex] because [latex]2[\/latex] is less than [latex]5[\/latex].<\/li>\r\n\t<li>[latex]\\leq[\/latex] (<strong>less than or equal to<\/strong>): [latex] a \\leq b [\/latex] means that \u201ca is less than or equal to b.\u201d For example, we can write [latex]2 \\leq 5 [\/latex] and [latex]5 \\leq 5 [\/latex] because the symbol indicates that there are two possibilities: the number on the left is either less than [latex]5[\/latex] or the number on the left equals [latex]5[\/latex].<\/li>\r\n\t<li>[latex]&gt;[\/latex] (<strong>greater than<\/strong>): [latex] a&gt;b [\/latex] means that \u201ca is greater than b.\u201d For example, we write [latex]5 &gt; 2 [\/latex] because [latex]5[\/latex] is greater than [latex]2[\/latex].<\/li>\r\n\t<li>[latex]\\geq[\/latex] (<strong>greater than or equal to<\/strong>): [latex]a \\geq b [\/latex] means that \u201ca is greater than or equal to b.\u201d For example, we can write [latex]5 \\geq 2 [\/latex] and [latex]5 \\geq 5 [\/latex] because the symbol indicates that there are two possibilities: the number on the left is either greater than [latex]5[\/latex] or the number on the left equals [latex]5[\/latex].<\/li>\r\n<\/ul>\r\n<p>One way to help remember how these symbols work is that the pointy end (the smaller side) of the symbol always points to the smaller number, while the open side (the bigger side) of the symbol always opens to the bigger number. (A fun way to remember it is to think of the symbol as a mouth that wants to eat the biggest number!)<\/p>\r\n<p>An important part of answering probability questions is translating the question into a mathematical expression so that you know what probability you\u2019re trying to find.<\/p>\r\n<section class=\"textbox example\"><img class=\"aligncenter wp-image-1505\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/10\/10155303\/Picture14.png\" alt=\"A spinner with 4 quadrants, labeled 1, 2, 3, and 4.\" width=\"101\" height=\"99\" \/>Let\u2019s imagine we are spinning a spinner with [latex]4[\/latex] equally-sized sections [latex]10[\/latex] times.<br \/>\r\nLet the random variable [latex]X[\/latex] be the number of times that we land on the [latex]1[\/latex] in our [latex]10[\/latex] spin attempts.Describe [latex]P(X &gt; 5)[\/latex].[reveal-answer q=\"759455\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"759455\"][latex]P(X &gt; 5)[\/latex] is the probability that we land on [latex]1[\/latex] on more than [latex]5[\/latex] spins OR we want the proportion for the number of spins where the arrow lands on [latex]1[\/latex] to be more than [latex]5[\/latex].[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1019[\/ohm2_question]<\/section>\r\n<h3>Interpretation of Inequality<\/h3>\r\n<p>Sometimes, a description of an inequality can be a little trickier to interpret. Translating intervals in context to notation is an important steps in answering a statistic question. This is necessary for calculating purposes. It is also important to feel comfortable with identifying values that would be located within an interval.<\/p>\r\n<ul>\r\n\t<li>\u201cAt least [latex]x[\/latex]\u201d means [latex]x[\/latex] or more. For example: \u201cat least [latex]3[\/latex]\u201d means [latex]3[\/latex] or more.<\/li>\r\n\t<li>\u201cUp to [latex]x[\/latex]\u201d means less than or equal to [latex]x[\/latex]. For example: \u201cup to [latex]2[\/latex]\u201d means [latex]2[\/latex] or less.<\/li>\r\n\t<li>\u201cAt most [latex]x[\/latex]\u201d means less than or equal to [latex]x[\/latex]. For example: \u201cat most [latex]2[\/latex]\u201d means [latex]2[\/latex] or less.<\/li>\r\n\t<li>\u201cNo more than [latex]x[\/latex]\u201d means less than or equal to [latex]x[\/latex]. For example: \u201cno more than [latex]2[\/latex]\u201d means [latex]2[\/latex] or less.<\/li>\r\n<\/ul>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1536[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1537[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1477[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1020[\/ohm2_question]<\/section>\r\n<section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1538[\/ohm2_question]<\/section>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Interpret statements of inequality<\/li>\n<\/ul>\n<\/section>\n<h2>Inequalities and Probabilities<\/h2>\n<p>Let&#8217;s\u00a0consider probability questions where each trial of an experiment has exactly two outcomes (which we call \u201csuccess\u201d and \u201cfailure\u201d), and we want to count how many of the trials result in successes.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-1505\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/10\/10155303\/Picture14.png\" alt=\"A spinner with 4 quadrants, labeled 1, 2, 3, and 4.\" width=\"101\" height=\"99\" \/>Suppose that we have a spinner with [latex]4[\/latex] sections, each of equal size, and the sections are labeled [latex]1, 2, 3, \\text{and} 4[\/latex]. We might want to count how many times the spinner lands on the [latex]1[\/latex] section if we spin the spinner [latex]10[\/latex] times. We will also consider questions like, \u201cWhat is the probability that in at least [latex]6[\/latex] out of [latex]10[\/latex] spin attempts, the arrow lands on the [latex]1[\/latex] section?\u201d For these types of questions, we will need inequality expressions.<\/p>\n<h3>Inequality Review<\/h3>\n<p>There are four inequality symbols:<\/p>\n<ul>\n<li>[latex]<[\/latex] (<strong>less than<\/strong>): [latex]a \\lt b[\/latex] means that \u201ca is less than b.\u201d For example, we write [latex]2 \\lt 5[\/latex] because [latex]2[\/latex] is less than [latex]5[\/latex].<\/li>\n<li>[latex]\\leq[\/latex] (<strong>less than or equal to<\/strong>): [latex]a \\leq b[\/latex] means that \u201ca is less than or equal to b.\u201d For example, we can write [latex]2 \\leq 5[\/latex] and [latex]5 \\leq 5[\/latex] because the symbol indicates that there are two possibilities: the number on the left is either less than [latex]5[\/latex] or the number on the left equals [latex]5[\/latex].<\/li>\n<li>[latex]>[\/latex] (<strong>greater than<\/strong>): [latex]a>b[\/latex] means that \u201ca is greater than b.\u201d For example, we write [latex]5 > 2[\/latex] because [latex]5[\/latex] is greater than [latex]2[\/latex].<\/li>\n<li>[latex]\\geq[\/latex] (<strong>greater than or equal to<\/strong>): [latex]a \\geq b[\/latex] means that \u201ca is greater than or equal to b.\u201d For example, we can write [latex]5 \\geq 2[\/latex] and [latex]5 \\geq 5[\/latex] because the symbol indicates that there are two possibilities: the number on the left is either greater than [latex]5[\/latex] or the number on the left equals [latex]5[\/latex].<\/li>\n<\/ul>\n<p>One way to help remember how these symbols work is that the pointy end (the smaller side) of the symbol always points to the smaller number, while the open side (the bigger side) of the symbol always opens to the bigger number. (A fun way to remember it is to think of the symbol as a mouth that wants to eat the biggest number!)<\/p>\n<p>An important part of answering probability questions is translating the question into a mathematical expression so that you know what probability you\u2019re trying to find.<\/p>\n<section class=\"textbox example\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1505\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/10\/10155303\/Picture14.png\" alt=\"A spinner with 4 quadrants, labeled 1, 2, 3, and 4.\" width=\"101\" height=\"99\" \/>Let\u2019s imagine we are spinning a spinner with [latex]4[\/latex] equally-sized sections [latex]10[\/latex] times.<br \/>\nLet the random variable [latex]X[\/latex] be the number of times that we land on the [latex]1[\/latex] in our [latex]10[\/latex] spin attempts.Describe [latex]P(X > 5)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q759455\">Show Solution<\/button><\/p>\n<div id=\"q759455\" class=\"hidden-answer\" style=\"display: none\">[latex]P(X > 5)[\/latex] is the probability that we land on [latex]1[\/latex] on more than [latex]5[\/latex] spins OR we want the proportion for the number of spins where the arrow lands on [latex]1[\/latex] to be more than [latex]5[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1019\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1019&theme=lumen&iframe_resize_id=ohm1019&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Interpretation of Inequality<\/h3>\n<p>Sometimes, a description of an inequality can be a little trickier to interpret. Translating intervals in context to notation is an important steps in answering a statistic question. This is necessary for calculating purposes. It is also important to feel comfortable with identifying values that would be located within an interval.<\/p>\n<ul>\n<li>\u201cAt least [latex]x[\/latex]\u201d means [latex]x[\/latex] or more. For example: \u201cat least [latex]3[\/latex]\u201d means [latex]3[\/latex] or more.<\/li>\n<li>\u201cUp to [latex]x[\/latex]\u201d means less than or equal to [latex]x[\/latex]. For example: \u201cup to [latex]2[\/latex]\u201d means [latex]2[\/latex] or less.<\/li>\n<li>\u201cAt most [latex]x[\/latex]\u201d means less than or equal to [latex]x[\/latex]. For example: \u201cat most [latex]2[\/latex]\u201d means [latex]2[\/latex] or less.<\/li>\n<li>\u201cNo more than [latex]x[\/latex]\u201d means less than or equal to [latex]x[\/latex]. For example: \u201cno more than [latex]2[\/latex]\u201d means [latex]2[\/latex] or less.<\/li>\n<\/ul>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1536\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1536&theme=lumen&iframe_resize_id=ohm1536&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1537\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1537&theme=lumen&iframe_resize_id=ohm1537&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1477\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1477&theme=lumen&iframe_resize_id=ohm1477&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1020\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1020&theme=lumen&iframe_resize_id=ohm1020&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1538\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1538&theme=lumen&iframe_resize_id=ohm1538&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n","protected":false},"author":8,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":2912,"module-header":"learn_it","content_attributions":[],"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\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1031"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1031\/revisions"}],"predecessor-version":[{"id":5530,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1031\/revisions\/5530"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/2912"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1031\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1031"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1031"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1031"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1031"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}