{"id":3022,"date":"2023-08-30T19:44:08","date_gmt":"2023-08-30T19:44:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=3022"},"modified":"2025-05-10T22:21:27","modified_gmt":"2025-05-10T22:21:27","slug":"probability-learn-it-2-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probability-learn-it-2-2\/","title":{"raw":"Probability: Learn It 2","rendered":"Probability: Learn It 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Calculate the probability of an event in a chance experiment.<\/li>\r\n\t<li>Recognize the differences between theoretical and empirical probability.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section><\/section>\r\n<h2>Estimating Probabilities<\/h2>\r\n<p>To estimate the probability of an outcome, you can carry out a chance experiment and analyze an outcome's frequency.<\/p>\r\n<p>Recall the experiment of flipping a fair coin. We conducted [latex]1[\/latex], [latex]10[\/latex], [latex]100[\/latex], [latex]1,000[\/latex], and [latex]10,000[\/latex] simulations. Through many simulations, we can compare the observed proportions of heads and tails. This comparison helps us understand how empirical probability converges to theoretical probability, based on observed outcomes and mathematical reasoning, respectively.<\/p>\r\n<section class=\"textbox proTip\">The <strong>theoretical probability<\/strong> of an outcome is the proportion of times the outcome would occur in the long run.<\/section>\r\n<p>In the case of a fair coin, the more you flip the coin, the closer the empirical probability will get to the theoretical probability of [latex]50\\%[\/latex] for heads and [latex]50\\%[\/latex] for tails.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>theoretical probabilities<\/h3>\r\n<p class=\"student12ptnumberlist\"><strong>Theoretical probability<\/strong> is the probability that an event will happen based on pure mathematics, not by carrying out an experiment.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p><strong>Notation:<\/strong> [latex]P(\\text{event})[\/latex] indicates \"probability of an event\".<\/p>\r\n<p>&nbsp;<\/p>\r\n<p class=\"student12ptnumberlist\">When the outcomes of the sample space are <b>equally likely<\/b>, the <b>probability<\/b> of an event is the number of outcomes in the event divided by the number of possible outcomes in the sample space.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]P(\\text{event}) = \\dfrac{\\text{number of outcomes in event}}{\\text{number of all possible outcomes}}[\/latex]<\/p>\r\n<\/section>\r\n<p class=\"student12ptnumberlist\" style=\"margin-left: 0in; text-indent: 0in;\">Notice that a probability can be determined by thinking of it as two counting problems followed by the computation of a related fraction.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]17609[\/ohm2_question]<\/section>\r\n<section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1082[\/ohm2_question]<\/section>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate the probability of an event in a chance experiment.<\/li>\n<li>Recognize the differences between theoretical and empirical probability.<\/li>\n<\/ul>\n<\/section>\n<section><\/section>\n<h2>Estimating Probabilities<\/h2>\n<p>To estimate the probability of an outcome, you can carry out a chance experiment and analyze an outcome&#8217;s frequency.<\/p>\n<p>Recall the experiment of flipping a fair coin. We conducted [latex]1[\/latex], [latex]10[\/latex], [latex]100[\/latex], [latex]1,000[\/latex], and [latex]10,000[\/latex] simulations. Through many simulations, we can compare the observed proportions of heads and tails. This comparison helps us understand how empirical probability converges to theoretical probability, based on observed outcomes and mathematical reasoning, respectively.<\/p>\n<section class=\"textbox proTip\">The <strong>theoretical probability<\/strong> of an outcome is the proportion of times the outcome would occur in the long run.<\/section>\n<p>In the case of a fair coin, the more you flip the coin, the closer the empirical probability will get to the theoretical probability of [latex]50\\%[\/latex] for heads and [latex]50\\%[\/latex] for tails.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>theoretical probabilities<\/h3>\n<p class=\"student12ptnumberlist\"><strong>Theoretical probability<\/strong> is the probability that an event will happen based on pure mathematics, not by carrying out an experiment.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Notation:<\/strong> [latex]P(\\text{event})[\/latex] indicates &#8220;probability of an event&#8221;.<\/p>\n<p>&nbsp;<\/p>\n<p class=\"student12ptnumberlist\">When the outcomes of the sample space are <b>equally likely<\/b>, the <b>probability<\/b> of an event is the number of outcomes in the event divided by the number of possible outcomes in the sample space.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]P(\\text{event}) = \\dfrac{\\text{number of outcomes in event}}{\\text{number of all possible outcomes}}[\/latex]<\/p>\n<\/section>\n<p class=\"student12ptnumberlist\" style=\"margin-left: 0in; text-indent: 0in;\">Notice that a probability can be determined by thinking of it as two counting problems followed by the computation of a related fraction.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm17609\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=17609&theme=lumen&iframe_resize_id=ohm17609&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1082\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1082&theme=lumen&iframe_resize_id=ohm1082&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n","protected":false},"author":12,"menu_order":5,"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\/3022"}],"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\/12"}],"version-history":[{"count":15,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3022\/revisions"}],"predecessor-version":[{"id":6526,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3022\/revisions\/6526"}],"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\/3022\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=3022"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=3022"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=3022"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=3022"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}