{"id":998,"date":"2023-06-22T01:38:59","date_gmt":"2023-06-22T01:38:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/conditional-probabilities-apply-it-2\/"},"modified":"2025-05-10T22:39:56","modified_gmt":"2025-05-10T22:39:56","slug":"conditional-probabilities-apply-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/conditional-probabilities-apply-it-2\/","title":{"raw":"Conditional Probabilities: Learn It 2","rendered":"Conditional Probabilities: Learn It 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Describe and find conditional probabilities.<\/li>\r\n\t<li>Understand the concept of independent events.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Conditional Probability<\/h2>\r\n<p>A\u00a0<strong>conditional probability<\/strong> is calculated based on the assumption that one event has already occurred. Conditional probabilities restrict the total. The new total is indicated after the word \"given\" in the question.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>conditional probability<\/h3>\r\n<p>The\u00a0<strong>conditional probability\u00a0<\/strong>of [latex]A[\/latex]<span style=\"background-color: initial; font-size: 0.9em;\">\u00a0<\/span>given [latex]B[\/latex]<span style=\"background-color: initial; font-size: 0.9em;\">\u00a0<\/span>is written [latex]P(A\\text{ given }B)[\/latex] or [latex]P(A|B)[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>[latex]P(A\\text{ given }B)[\/latex] is the probability that event [latex]A[\/latex] <span id=\"MathJax-Element-109-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"font-family: proxima-nova, sans-serif; padding: 1px 0px; margin: 0px; font-size: 17.44px; vertical-align: baseline; background: #ffffff; border: 0px; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: 400; letter-spacing: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; color: #373d3f;\" role=\"presentation\" data-mathml=\"&lt;\/p&gt; &lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;A&lt;\/mi&gt;&lt;\/math&gt; &lt;p&gt;\"><\/span><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\">will occur given that the event [latex]B[\/latex] <\/span><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\">has already occurred.<\/span><\/p>\r\n<p>&nbsp;<\/p>\r\n<p>[latex]P(A|B)=\\frac{P(A \\text{ and }B)}{P(B)}[\/latex]<\/p>\r\n<\/section>\r\n<section>To calculate any probabilities, a contingency table is typically used to provide a way of portraying data that can facilitate in calculating probabilities. The table helps in determining conditional probabilities quite easily as well.<\/section>\r\n<section>\r\n<section class=\"textbox example\">\r\n<p>A researcher conducts a survey of 120 randomly selected college students to try to answer the questions: If someone has a laptop, are they likely to own a desktop computer? If someone has a desktop computer, are they likely to own a laptop? The results of the survey are displayed in the following contingency table.<\/p>\r\n<div align=\"center\">\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>Owns laptop<\/td>\r\n<td>Does not own laptop<\/td>\r\n<td><strong>Total<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Owns desktop<\/td>\r\n<td>20<\/td>\r\n<td>20<\/td>\r\n<td><strong>40<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Does not own a desktop<\/td>\r\n<td>60<\/td>\r\n<td>20<\/td>\r\n<td><strong>80<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Total<\/strong><\/td>\r\n<td><strong>80<\/strong><\/td>\r\n<td><strong>40<\/strong><\/td>\r\n<td><strong>120<\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<p>If someone does not own a desktop, what is the probability that they don't own a laptop computer either?<\/p>\r\n\r\n[caption id=\"attachment_5431\" align=\"aligncenter\" width=\"1030\"]<img class=\"wp-image-5431 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/27234437\/Screenshot-2022-12-27-at-4.44.32-PM.png\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"1030\" height=\"346\" \/> Figure 1. This two-way table displays computer ownership among 120 people, showing how many own a laptop, a desktop, both, or neither.[\/caption]\r\n\r\n<p>[reveal-answer q=\"958745\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"958745\"]<br \/>\r\nWe are given that someone does not own a desktop. This means that our sample space has been reduced to 80 college students, as seen highlighted below.<br \/>\r\n<br \/>\r\n[latex]P(\\text{Does not own laptop} \\text{ GIVEN }\\text{Does not own desktop}) = \\dfrac{20}{80}[\/latex][\/hidden-answer]<\/p>\r\n<\/section>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1092[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Describe and find conditional probabilities.<\/li>\n<li>Understand the concept of independent events.<\/li>\n<\/ul>\n<\/section>\n<h2>Conditional Probability<\/h2>\n<p>A\u00a0<strong>conditional probability<\/strong> is calculated based on the assumption that one event has already occurred. Conditional probabilities restrict the total. The new total is indicated after the word &#8220;given&#8221; in the question.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>conditional probability<\/h3>\n<p>The\u00a0<strong>conditional probability\u00a0<\/strong>of [latex]A[\/latex]<span style=\"background-color: initial; font-size: 0.9em;\">\u00a0<\/span>given [latex]B[\/latex]<span style=\"background-color: initial; font-size: 0.9em;\">\u00a0<\/span>is written [latex]P(A\\text{ given }B)[\/latex] or [latex]P(A|B)[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>[latex]P(A\\text{ given }B)[\/latex] is the probability that event [latex]A[\/latex] <span id=\"MathJax-Element-109-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"font-family: proxima-nova, sans-serif; padding: 1px 0px; margin: 0px; font-size: 17.44px; vertical-align: baseline; background: #ffffff; border: 0px; line-height: 0; text-indent: 0px; text-align: left; text-transform: none; font-style: normal; font-weight: 400; letter-spacing: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; color: #373d3f;\" role=\"presentation\" data-mathml=\"&lt;\/p&gt; &lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot;&gt;&lt;mi&gt;A&lt;\/mi&gt;&lt;\/math&gt; &lt;p&gt;\"><\/span><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\">will occur given that the event [latex]B[\/latex] <\/span><span style=\"font-size: 1rem; text-align: initial; background-color: initial;\">has already occurred.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p>[latex]P(A|B)=\\frac{P(A \\text{ and }B)}{P(B)}[\/latex]<br \/>\n<\/section>\n<section>To calculate any probabilities, a contingency table is typically used to provide a way of portraying data that can facilitate in calculating probabilities. The table helps in determining conditional probabilities quite easily as well.<\/section>\n<section>\n<section class=\"textbox example\">\n<p>A researcher conducts a survey of 120 randomly selected college students to try to answer the questions: If someone has a laptop, are they likely to own a desktop computer? If someone has a desktop computer, are they likely to own a laptop? The results of the survey are displayed in the following contingency table.<\/p>\n<div style=\"margin: auto;\">\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>Owns laptop<\/td>\n<td>Does not own laptop<\/td>\n<td><strong>Total<\/strong><\/td>\n<\/tr>\n<tr>\n<td>Owns desktop<\/td>\n<td>20<\/td>\n<td>20<\/td>\n<td><strong>40<\/strong><\/td>\n<\/tr>\n<tr>\n<td>Does not own a desktop<\/td>\n<td>60<\/td>\n<td>20<\/td>\n<td><strong>80<\/strong><\/td>\n<\/tr>\n<tr>\n<td><strong>Total<\/strong><\/td>\n<td><strong>80<\/strong><\/td>\n<td><strong>40<\/strong><\/td>\n<td><strong>120<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>If someone does not own a desktop, what is the probability that they don&#8217;t own a laptop computer either?<\/p>\n<figure id=\"attachment_5431\" aria-describedby=\"caption-attachment-5431\" style=\"width: 1030px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5431 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/27234437\/Screenshot-2022-12-27-at-4.44.32-PM.png\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"1030\" height=\"346\" \/><figcaption id=\"caption-attachment-5431\" class=\"wp-caption-text\">Figure 1. This two-way table displays computer ownership among 120 people, showing how many own a laptop, a desktop, both, or neither.<\/figcaption><\/figure>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q958745\">Show answer<\/button><\/p>\n<div id=\"q958745\" class=\"hidden-answer\" style=\"display: none\">\nWe are given that someone does not own a desktop. This means that our sample space has been reduced to 80 college students, as seen highlighted below.<\/p>\n<p>[latex]P(\\text{Does not own laptop} \\text{ GIVEN }\\text{Does not own desktop}) = \\dfrac{20}{80}[\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1092\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1092&theme=lumen&iframe_resize_id=ohm1092&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":17,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":974,"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\/998"}],"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":11,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/998\/revisions"}],"predecessor-version":[{"id":6546,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/998\/revisions\/6546"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/974"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/998\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=998"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=998"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=998"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=998"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}