{"id":3258,"date":"2023-09-25T06:01:47","date_gmt":"2023-09-25T06:01:47","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=3258"},"modified":"2025-05-17T03:02:40","modified_gmt":"2025-05-17T03:02:40","slug":"probability-trees-learn-it-1-update","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probability-trees-learn-it-1-update\/","title":{"raw":"Probability with Tree Diagrams: Learn It 1","rendered":"Probability with Tree Diagrams: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Learn how to construct and interpret a tree diagram to represent sequential events or decisions<\/li>\r\n\t<li>Calculate conditional probabilities using tree diagrams, considering both dependent and independent events<\/li>\r\n\t<li>Apply tree diagrams to solve problems involving probability of multiple events, such as probability of compound events and conditional probability<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Tree Diagrams<\/h2>\r\n<p>A tree diagram is a visual tool used to show the possible outcomes of a series of events or decisions. It resembles an upside-down tree (or a sideways tree), where each branch represents a different choice or scenario, and the leaves of the tree represent the final outcomes or results. Tree diagrams are commonly used in probability theory to calculate the likelihood of different combinations of events happening. They provide a clear and organized way to understand and analyze various possibilities in a decision-making process or in random events.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>tree diagrams<\/h3>\r\n<p>A <strong>tree diagram<\/strong> is a special type of graph used to determine the outcomes of an experiment.<\/p>\r\n<p>It consists of \u201cbranches\u201d that are labeled with either frequencies or probabilities.\u00a0<\/p>\r\n<\/section>\r\n<p>When creating a tree diagram, it is important to first consider the sample space of a chance experiment. Once you have correctly listed all the possible outcomes, the next step is to consider the probabilities for each of these outcomes.<\/p>\r\n<p>Once you have this information, you can begin constructing the tree diagram by labeling the outcomes at the ends of the \"branches\" with their corresponding probabilities along the \"branch.\" It is important to note that the sum of the probabilities should equal 1 (or 100%) since these represent all the possible outcomes of the chance experiment.<\/p>\r\n<section>\r\n<section class=\"textbox example\">Suppose that a box contains 20 different color LEGOs. Inside, there are 12 blue LEGOs and 8 red LEGOs. Consider the chance experiment where a random LEGO is drawn from the box. Let [latex]A =[\/latex] \"draw a red LEGO\" and [latex]B =[\/latex] \"draw a blue LEGO\"\r\n\r\n<p><strong>(a) <\/strong>What is the sample space [latex]S[\/latex] of the chance experiment?<\/p>\r\n<p><strong>(b) <\/strong>What is the probability of event [latex]A[\/latex]?<\/p>\r\n<p><strong style=\"font-size: 1rem; text-align: initial; background-color: initial;\">(c) <\/strong>What is the probability of event [latex]B[\/latex]?<\/p>\r\n<p><strong>(d) <\/strong>What does the tree diagram look like for the given chance experiment?<\/p>\r\n<p>[reveal-answer q=\"983416\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"983416\"]<\/p>\r\n<p><strong>(a) <\/strong>[latex]S = [\/latex] {red, blue}<\/p>\r\n<p><strong>(b) <\/strong>[latex]P(A)=\\frac{12}{20} = \\frac{3}{5}[\/latex]<\/p>\r\n<p><strong>(c) <\/strong>[latex]P(B)=\\frac{8}{20}=\\frac{2}{5}[\/latex]<\/p>\r\n<p><strong>(d) <\/strong><\/p>\r\n<p><img class=\"aligncenter wp-image-3261 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/25064634\/Tree_Diagram_See_Example_One_Branch-300x228.png\" alt=\"A tree diagram with branch A = 3\/5 and branch B = 2\/5.\" width=\"300\" height=\"228\" \/><\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>In some cases, there might be a situation where the experiment might have more than two outcomes in the sample space or there are multiple stages to the experiment. The result would lead to a greater number of branches for the probability tree.\u00a0<\/p>\r\n<section class=\"textbox example\">\r\n<p>In an urn, there are [latex]11[\/latex] balls. Three balls are red ([latex]R[\/latex]) and eight balls are blue ([latex]B[\/latex]).<\/p>\r\n<p>Experiment: Draw two balls, one at a time, <strong>with replacement<\/strong>.<\/p>\r\n<p>Note: \u201cWith replacement\u201d means that you put the first ball back in the urn before you select the second ball. The tree diagram using frequencies that show all the possible outcomes follows.<\/p>\r\n<div class=\"wp-nocaption aligncenter wp-image-3514 size-full\"><img class=\"aligncenter wp-image-3514 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/06\/27161925\/14979036b70be20a0a5ea43a3e48c57e30d8620e2.jpeg\" sizes=\"(max-width: 488px) 100vw, 488px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/06\/27161925\/14979036b70be20a0a5ea43a3e48c57e30d8620e2.jpeg 488w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/06\/27161925\/14979036b70be20a0a5ea43a3e48c57e30d8620e2-300x165.jpeg 300w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/06\/27161925\/14979036b70be20a0a5ea43a3e48c57e30d8620e2-65x36.jpeg 65w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/06\/27161925\/14979036b70be20a0a5ea43a3e48c57e30d8620e2-225x124.jpeg 225w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/06\/27161925\/14979036b70be20a0a5ea43a3e48c57e30d8620e2-350x192.jpeg 350w\" alt=\"This is a tree diagram with branches showing frequencies of each draw. The first branch shows two lines: 8B and 3R. The second branch has a set of two lines (8B and 3R) for each line of the first branch. Multiply along each line to find 64BB, 24BR, 24RB, and 9RR.\" width=\"488\" height=\"268\" \/><\/div>\r\n<p>The first set of branches represents the first draw. The second set of branches represents the second draw. Each of the outcomes is distinct.<\/p>\r\n<p>In fact, we can list each red ball as [latex]R1, R2, \\text{ and } R3[\/latex] and each blue ball as [latex]B1, B2, B3, B4, B5, B6, B7, \\text{ and }B8[\/latex].<\/p>\r\n<ul>\r\n\t<li>Show that there are 9 different outcomes when we draw a red and then a red. Note: this is the most right branch.<\/li>\r\n<\/ul>\r\n<p>[reveal-answer q=\"741894\"]Show Answer[\/reveal-answer][hidden-answer a=\"741894\"]The nine [latex]RR[\/latex] outcomes can be written as:[latex]R1R1, R1R2, R1R3, R2R1, R2R2, R2R3, R3R1, R3R2, R3R3[\/latex][\/hidden-answer]<\/p>\r\n<ul>\r\n\t<li>List out the 24 [latex]BR[\/latex] outcomes.<\/li>\r\n<\/ul>\r\n<p>[reveal-answer q=\"656995\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"656995\"][latex]B1R1, B1R2, B1R3, B2R1, B2R2, B2R3, B3R1, B3R2, B3R3, B4R1, B4R2, B4R3, B5R1, B5R2, B5R3, B6R1, B6R2, B6R3, B7R1, B7R2, B7R3, B8R1, B8R2, B8R3[\/latex][\/hidden-answer]<\/p>\r\n<ul>\r\n\t<li>How many outcomes are there in the sample space?\u00a0<\/li>\r\n<\/ul>\r\n<p>[reveal-answer q=\"122489\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"122489\"]There are a total of [latex]11[\/latex] balls in the urn. Draw two balls, one at a time, with replacement. There are [latex]11*11 = 121[\/latex]\u00a0outcomes, the size of the\u00a0sample space.<\/p>\r\n<p><strong>Question: <\/strong>Why do we multiply the frequencies to find the total possible outcomes?<\/p>\r\n<p><strong>Answer:<\/strong> To understand why we multiply, consider this: for each outcome at the first stage, there are a certain number of outcomes at the second stage. When you multiply these individual outcomes, you are essentially considering all possible combinations of events from each stage. Multiplying the frequencies helps find all the possible combinations of outcomes, providing a systematic way to calculate the total number of outcomes for a sequence of events.<\/p>\r\n<p>You may also obtain the size of the sample space by adding all of the possible outcomes from each branch: Total [latex]= 64+24+24+9 = 121[\/latex] outcomes.[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]13791[\/ohm2_question]<\/p>\r\n<\/section>\r\n<section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]25065[\/ohm2_question]<\/p>\r\n<\/section>\r\n<\/section>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Learn how to construct and interpret a tree diagram to represent sequential events or decisions<\/li>\n<li>Calculate conditional probabilities using tree diagrams, considering both dependent and independent events<\/li>\n<li>Apply tree diagrams to solve problems involving probability of multiple events, such as probability of compound events and conditional probability<\/li>\n<\/ul>\n<\/section>\n<h2>Tree Diagrams<\/h2>\n<p>A tree diagram is a visual tool used to show the possible outcomes of a series of events or decisions. It resembles an upside-down tree (or a sideways tree), where each branch represents a different choice or scenario, and the leaves of the tree represent the final outcomes or results. Tree diagrams are commonly used in probability theory to calculate the likelihood of different combinations of events happening. They provide a clear and organized way to understand and analyze various possibilities in a decision-making process or in random events.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>tree diagrams<\/h3>\n<p>A <strong>tree diagram<\/strong> is a special type of graph used to determine the outcomes of an experiment.<\/p>\n<p>It consists of \u201cbranches\u201d that are labeled with either frequencies or probabilities.\u00a0<\/p>\n<\/section>\n<p>When creating a tree diagram, it is important to first consider the sample space of a chance experiment. Once you have correctly listed all the possible outcomes, the next step is to consider the probabilities for each of these outcomes.<\/p>\n<p>Once you have this information, you can begin constructing the tree diagram by labeling the outcomes at the ends of the &#8220;branches&#8221; with their corresponding probabilities along the &#8220;branch.&#8221; It is important to note that the sum of the probabilities should equal 1 (or 100%) since these represent all the possible outcomes of the chance experiment.<\/p>\n<section>\n<section class=\"textbox example\">Suppose that a box contains 20 different color LEGOs. Inside, there are 12 blue LEGOs and 8 red LEGOs. Consider the chance experiment where a random LEGO is drawn from the box. Let [latex]A =[\/latex] &#8220;draw a red LEGO&#8221; and [latex]B =[\/latex] &#8220;draw a blue LEGO&#8221;<\/p>\n<p><strong>(a) <\/strong>What is the sample space [latex]S[\/latex] of the chance experiment?<\/p>\n<p><strong>(b) <\/strong>What is the probability of event [latex]A[\/latex]?<\/p>\n<p><strong style=\"font-size: 1rem; text-align: initial; background-color: initial;\">(c) <\/strong>What is the probability of event [latex]B[\/latex]?<\/p>\n<p><strong>(d) <\/strong>What does the tree diagram look like for the given chance experiment?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q983416\">Show Answer<\/button><\/p>\n<div id=\"q983416\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>(a) <\/strong>[latex]S =[\/latex] {red, blue}<\/p>\n<p><strong>(b) <\/strong>[latex]P(A)=\\frac{12}{20} = \\frac{3}{5}[\/latex]<\/p>\n<p><strong>(c) <\/strong>[latex]P(B)=\\frac{8}{20}=\\frac{2}{5}[\/latex]<\/p>\n<p><strong>(d) <\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3261 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/25064634\/Tree_Diagram_See_Example_One_Branch-300x228.png\" alt=\"A tree diagram with branch A = 3\/5 and branch B = 2\/5.\" width=\"300\" height=\"228\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/25064634\/Tree_Diagram_See_Example_One_Branch-300x228.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/25064634\/Tree_Diagram_See_Example_One_Branch-65x49.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/25064634\/Tree_Diagram_See_Example_One_Branch-225x171.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/25064634\/Tree_Diagram_See_Example_One_Branch-350x266.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/25064634\/Tree_Diagram_See_Example_One_Branch.png 378w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>In some cases, there might be a situation where the experiment might have more than two outcomes in the sample space or there are multiple stages to the experiment. The result would lead to a greater number of branches for the probability tree.\u00a0<\/p>\n<section class=\"textbox example\">\n<p>In an urn, there are [latex]11[\/latex] balls. Three balls are red ([latex]R[\/latex]) and eight balls are blue ([latex]B[\/latex]).<\/p>\n<p>Experiment: Draw two balls, one at a time, <strong>with replacement<\/strong>.<\/p>\n<p>Note: \u201cWith replacement\u201d means that you put the first ball back in the urn before you select the second ball. The tree diagram using frequencies that show all the possible outcomes follows.<\/p>\n<div class=\"wp-nocaption aligncenter wp-image-3514 size-full\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3514 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/06\/27161925\/14979036b70be20a0a5ea43a3e48c57e30d8620e2.jpeg\" sizes=\"(max-width: 488px) 100vw, 488px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/06\/27161925\/14979036b70be20a0a5ea43a3e48c57e30d8620e2.jpeg 488w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/06\/27161925\/14979036b70be20a0a5ea43a3e48c57e30d8620e2-300x165.jpeg 300w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/06\/27161925\/14979036b70be20a0a5ea43a3e48c57e30d8620e2-65x36.jpeg 65w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/06\/27161925\/14979036b70be20a0a5ea43a3e48c57e30d8620e2-225x124.jpeg 225w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/06\/27161925\/14979036b70be20a0a5ea43a3e48c57e30d8620e2-350x192.jpeg 350w\" alt=\"This is a tree diagram with branches showing frequencies of each draw. The first branch shows two lines: 8B and 3R. The second branch has a set of two lines (8B and 3R) for each line of the first branch. Multiply along each line to find 64BB, 24BR, 24RB, and 9RR.\" width=\"488\" height=\"268\" \/><\/div>\n<p>The first set of branches represents the first draw. The second set of branches represents the second draw. Each of the outcomes is distinct.<\/p>\n<p>In fact, we can list each red ball as [latex]R1, R2, \\text{ and } R3[\/latex] and each blue ball as [latex]B1, B2, B3, B4, B5, B6, B7, \\text{ and }B8[\/latex].<\/p>\n<ul>\n<li>Show that there are 9 different outcomes when we draw a red and then a red. Note: this is the most right branch.<\/li>\n<\/ul>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q741894\">Show Answer<\/button><\/p>\n<div id=\"q741894\" class=\"hidden-answer\" style=\"display: none\">The nine [latex]RR[\/latex] outcomes can be written as:[latex]R1R1, R1R2, R1R3, R2R1, R2R2, R2R3, R3R1, R3R2, R3R3[\/latex]<\/div>\n<\/div>\n<ul>\n<li>List out the 24 [latex]BR[\/latex] outcomes.<\/li>\n<\/ul>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q656995\">Show Answer<\/button><\/p>\n<div id=\"q656995\" class=\"hidden-answer\" style=\"display: none\">[latex]B1R1, B1R2, B1R3, B2R1, B2R2, B2R3, B3R1, B3R2, B3R3, B4R1, B4R2, B4R3, B5R1, B5R2, B5R3, B6R1, B6R2, B6R3, B7R1, B7R2, B7R3, B8R1, B8R2, B8R3[\/latex]<\/div>\n<\/div>\n<ul>\n<li>How many outcomes are there in the sample space?\u00a0<\/li>\n<\/ul>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q122489\">Show Answer<\/button><\/p>\n<div id=\"q122489\" class=\"hidden-answer\" style=\"display: none\">There are a total of [latex]11[\/latex] balls in the urn. Draw two balls, one at a time, with replacement. There are [latex]11*11 = 121[\/latex]\u00a0outcomes, the size of the\u00a0sample space.<\/p>\n<p><strong>Question: <\/strong>Why do we multiply the frequencies to find the total possible outcomes?<\/p>\n<p><strong>Answer:<\/strong> To understand why we multiply, consider this: for each outcome at the first stage, there are a certain number of outcomes at the second stage. When you multiply these individual outcomes, you are essentially considering all possible combinations of events from each stage. Multiplying the frequencies helps find all the possible combinations of outcomes, providing a systematic way to calculate the total number of outcomes for a sequence of events.<\/p>\n<p>You may also obtain the size of the sample space by adding all of the possible outcomes from each branch: Total [latex]= 64+24+24+9 = 121[\/latex] outcomes.<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm13791\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13791&theme=lumen&iframe_resize_id=ohm13791&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm25065\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=25065&theme=lumen&iframe_resize_id=ohm25065&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<\/section>\n<\/section>\n","protected":false},"author":51,"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":2910,"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\/3258"}],"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\/51"}],"version-history":[{"count":20,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3258\/revisions"}],"predecessor-version":[{"id":6941,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3258\/revisions\/6941"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/2910"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3258\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=3258"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=3258"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=3258"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=3258"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}