{"id":990,"date":"2023-06-22T01:38:52","date_gmt":"2023-06-22T01:38:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probabilities-of-two-events-learn-it-2\/"},"modified":"2025-05-10T22:32:22","modified_gmt":"2025-05-10T22:32:22","slug":"probabilities-of-two-events-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probabilities-of-two-events-learn-it-2\/","title":{"raw":"Probability of Compound Events: Learn It 2","rendered":"Probability of Compound Events: Learn It 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Calculate and interpret probabilities of simple and compound events.<\/li>\r\n\t<li>Understand the concept of mutually exclusive and independent events.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>AND ([latex]\\cap[\/latex]) vs. OR ([latex]\\cup[\/latex])<\/h2>\r\n<p>There are times when you want to combine two events by using the word AND or the word OR. In statistics, specifically in probability, there is an important distinction between the words AND and OR.<em>\u00a0<\/em><\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>\"AND\" events<\/h3>\r\n<p>AND ([latex]\\cap[\/latex]) means that both events must happen.<\/p>\r\n<p>[latex]P(A \\text{ and }B)= P(A \\cap B)=[\/latex] the relative frequency of events [latex]A[\/latex] and [latex]B[\/latex] must happen in the same outcome.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1414[\/ohm2_question]<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>\"OR\" events<\/h3>\r\n<p>An outcome is in the event [latex]A[\/latex] <span id=\"MathJax-Element-54-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>OR [latex]B[\/latex] if the outcome is in [latex]A[\/latex] or is in [latex]B[\/latex] or is in both [latex]A[\/latex] and [latex]B[\/latex].<\/p>\r\n<p>[latex]P(A \\text{ or }B)= P(A \\cup B)=[\/latex] the relative frequency of either event [latex]A[\/latex] or [latex]B[\/latex] (or both) must happen in the outcome.<\/p>\r\n<\/section>\r\n<p>To find the probability of event [latex]A[\/latex] OR event [latex]B[\/latex] in a two-way table, add all frequencies of event [latex]A[\/latex] and event [latex]B[\/latex] and divide it by the total.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]867[\/ohm2_question]<\/section>\r\n<section>\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p>In the example, the number of students who like Oatmeal cookies is [latex]39[\/latex] and the number of students who like Chocolate Chip cookies is [latex]45[\/latex] but the number of students who prefer Oatmeal OR Chocolate Chip is not [latex]39+45=84[\/latex] because this would include the [latex]36[\/latex] students who like <strong>both<\/strong> twice.\u00a0<br \/>\r\n<br \/>\r\nDo not double count the frequency when both events [latex]A[\/latex] and [latex]B[\/latex] occur.<\/p>\r\n<\/section>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1088[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate and interpret probabilities of simple and compound events.<\/li>\n<li>Understand the concept of mutually exclusive and independent events.<\/li>\n<\/ul>\n<\/section>\n<h2>AND ([latex]\\cap[\/latex]) vs. OR ([latex]\\cup[\/latex])<\/h2>\n<p>There are times when you want to combine two events by using the word AND or the word OR. In statistics, specifically in probability, there is an important distinction between the words AND and OR.<em>\u00a0<\/em><\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>&#8220;AND&#8221; events<\/h3>\n<p>AND ([latex]\\cap[\/latex]) means that both events must happen.<\/p>\n<p>[latex]P(A \\text{ and }B)= P(A \\cap B)=[\/latex] the relative frequency of events [latex]A[\/latex] and [latex]B[\/latex] must happen in the same outcome.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1414\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1414&theme=lumen&iframe_resize_id=ohm1414&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>&#8220;OR&#8221; events<\/h3>\n<p>An outcome is in the event [latex]A[\/latex] <span id=\"MathJax-Element-54-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>OR [latex]B[\/latex] if the outcome is in [latex]A[\/latex] or is in [latex]B[\/latex] or is in both [latex]A[\/latex] and [latex]B[\/latex].<\/p>\n<p>[latex]P(A \\text{ or }B)= P(A \\cup B)=[\/latex] the relative frequency of either event [latex]A[\/latex] or [latex]B[\/latex] (or both) must happen in the outcome.<\/p>\n<\/section>\n<p>To find the probability of event [latex]A[\/latex] OR event [latex]B[\/latex] in a two-way table, add all frequencies of event [latex]A[\/latex] and event [latex]B[\/latex] and divide it by the total.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm867\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=867&theme=lumen&iframe_resize_id=ohm867&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p>In the example, the number of students who like Oatmeal cookies is [latex]39[\/latex] and the number of students who like Chocolate Chip cookies is [latex]45[\/latex] but the number of students who prefer Oatmeal OR Chocolate Chip is not [latex]39+45=84[\/latex] because this would include the [latex]36[\/latex] students who like <strong>both<\/strong> twice.\u00a0<\/p>\n<p>Do not double count the frequency when both events [latex]A[\/latex] and [latex]B[\/latex] occur.<\/p>\n<\/section>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1088\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1088&theme=lumen&iframe_resize_id=ohm1088&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":11,"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\/990"}],"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":14,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/990\/revisions"}],"predecessor-version":[{"id":6537,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/990\/revisions\/6537"}],"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\/990\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=990"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=990"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=990"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=990"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}