[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-657359-105":59,"doc-detail-657359-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","independent-events-and-tree-diagrams-starter-probability-lesson","Independent Events and Tree Diagrams - Starter - Probability lesson","","Independent Events and Tree Diagrams explains how to handle probability when successive events may or may not have equal likelihoods. It defines independent events and contrasts them with dependent ones, then applies the multiplication rule for P(A,B) when events are independent. The lesson introduces tree diagrams as a structured method: probabilities are placed on branches, branch totals sum to 1, and moving across versus down the tree indicates multiplying or adding probabilities. Worked examples cover odd totals, independence scenarios, and selecting discs with replacement.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/independent-events-and-tree-diagrams-starter-probability-lesson/657359/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/independent-events-and-tree-diagrams-starter-probability-lesson/657359.png","ImageObject",300,407,{"name":92,"@type":93},"Margaret","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":24},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What does it mean for two events to be independent?","Question",{"text":112,"@type":113},"Two events are independent when the outcome of one does not depend on the outcome of the other, so one event does not affect the other. In general, they are independent when they are physically independent.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How do you calculate P(A and B) for independent events?",{"text":117,"@type":113},"If events A and B are independent, then P(A followed by B) equals P(A) × P(B). The lesson also states P(A, B) = P(A) × P(B).",{"name":119,"@type":110,"acceptedAnswer":120},"What rules apply when using a tree diagram?",{"text":121,"@type":113},"Put probabilities on the branches and the outcomes at the end of each branch. The probabilities on branches must sum to 1. Going across the tree means multiply probabilities, and going down the tree means add probabilities.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},657359,1793065899,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":24,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":52},137451207643,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Independent Events and Tree Diagrams  \nStarter  \n1. (Review of last lesson)  \nStella and Paul each have three cards numbered 1 , 2 and 3. They each take one of the other’s cards and add the total. What is the probability the total will be an odd number? Hint: list the possible outcomes in a probability space diagram.  \n2. (Review of previous material)  \nWithout using a calculator, find: (a) ~~ ~~611 × ~~ ~~511~~ ~~ (b) ~~ ~~49 × ~~ ~~35  \n3. (Review of previous material)  \nWithout using a calculator, find: (a) 0.6 × 0.4 (b) 0.7 × 0.1  \nNotes  \nSuccessive events are when we have one event followed by another (e.g. rolling a dice and flipping a coin) . In the previous lesson we looked at two events whose outcomes all had the same probability (e.g. flipping two coins) . What happens when the outcomes do not have the same probability?  \nIndependent events  \nEvents are independent when the outcome of one event does not depend on the outcome of the other i.e. one event does not affect the outcomes of another event.  \nE.g. 1 Decide whether the following events are independent or not.  \n(a) Flipping two coins.  \n(b) Penalties in a penalty shoot-out.  \n(c) Rolling two dice.  \n(d) The weather on two consecutive days.  \n(e) The weather on days separated by 6 months.  \nIn general, events are independent when they are physically independent from each other.  \nIf events A and B are independent, the probability of event A happening followed by event B is P (A) × P(B)  \nP (A , B) = P (A) × P(B)  \nE.g. 2 The probability that Irina is late for school is 0.3; the probability that Helena is late is 0.2.  \nWhat is the probability that both girls are late for school?  \nWorking: P (late, late) = P (late) × P(late) = 0.3 × 0.2 = 0.06  \nE.g. 3 A coin is flipped 3 times. What is the probability of getting 3 heads?  \nE.g. 4 A dice is rolled twice. Find the probability of getting a two on just one of these rolls.  \n[www.mathspanda.com](www.mathspanda.com)  \nTree diagrams  \nA tree diagram is a useful tool to solve probability problems and would make it easier to do E.g. 4.  \nImagine a bag has 9 discs —4 red and 5 blue. A disc is chosen at random, its colour noted and then replaced in the bag. Another disc is then chosen.  \n2nd pick Outcomes  \n1st pick  \n4  \n9  \n5  \n9  \nR  \nB  \n4  \n9  \nMULTIPLY  \nR  \nB  \nR  \nB  \nR, R  \nR, B  \nB, R  \nB, B  \nADD  \nKey ideas  \nProbabilities go on the branches and outcomes at the end of each branch.  \n The sum of the probabilities on the branches must be 1.  \n When going across the tree diagram, we multiply probabilities.  \nE.g. P (2 red discs) = P (R , R)  \n= P (R) × P(R)  \n4 4 = ~~ ~~ × ~~ ~~  \n9 9 16  \n= ~~ ~~  \n81  \n When going down the tree diagram, we add the probabilities.  \nE.g. P (same colour) = P (R , R) + P(B, B)  \n= P (R) × P(R) + P(B) × P(B)  \n4 4 5 5 = ~~ ~~ × ~~ ~~ + ~~ ~~ × ~~ ~~  \n9 9 9 9 16 25  \n= ~~ ~~ + ~~ ~~  \n81 81  \n41  \n= ~~ ~~  \n81  \n[www.mathspanda.com](www.mathspanda.com)  \nTwo common questions  \n1. What is the probability of getting only one red disc?  \nWorking: To get one red disc, we could choose a red disc first and then a blue disc or could a blue one and then a red one.  \nP (only 1 red disc) = P (R , B) + P(R , B)  \n= P (R) × P(B) + P(B) × P(R)  \n=  \n=  \n=  \n4 5 5 ~~ ~~ × ~~ ~~ + ~~  ~~9 9 9 20 20 ~~ ~~ + ~~ ~~  \n81 81 40  \n81  \n×  \n4 9  \nN.B. This is the same as finding the probability of getting different colours.  \n2. What is the probability of getting at least one blue disc?  \nWorking: “At least one blue disc” is the complementary event of “no blue discs”.  \n“No blue discs” means that two red discs have been chosen  \nP (at least 1 blue disc) = 1 − P(no blue discs)  \n= 1 − P(2 red discs)  \n= 1 − P(R, R)  \n= 1 − ~~ ~~49 × ~~ ~~49  \n16  \n= 1 − ~~ ~~  \n81  \n65  \n= ~~ ~~  \n81  \nN.B. You could also do P(at least 1 blue disc) = P (1 blue disc) + P(2 blue discs)  \nNotation  \nR′ is used to denote “not R”and is called complement notation. For example, when rolling a dice P(6) = ~~ ~~16 and P(6′) = ~~ ~~56  ","cbCaitbk0HW24ydb","https://ap.wps.com/l/cbCaitbk0HW24ydb","pdf",210138,"English","# Starter\n## Review of last lesson\n## Review of previous material\n# Notes\n## Successive events\n## Independent events\n# Tree diagrams\n## Key ideas\n## Two common questions\n# Notation\n# Video and Solutions\n# Exercise\n# Summary","[{\"question\":\"What does it mean for two events to be independent?\",\"answer\":\"Two events are independent when the outcome of one does not depend on the outcome of the other, so one event does not affect the other. In general, they are independent when they are physically independent.\"},{\"question\":\"How do you calculate P(A and B) for independent events?\",\"answer\":\"If events A and B are independent, then P(A followed by B) equals P(A) × P(B). The lesson also states P(A, B) = P(A) × P(B).\"},{\"question\":\"What rules apply when using a tree diagram?\",\"answer\":\"Put probabilities on the branches and the outcomes at the end of each branch. The probabilities on branches must sum to 1. Going across the tree means multiply probabilities, and going down the tree means add probabilities.\"}]","Independent Events and Tree Diagrams - Starter - Probability lesson | PDF",1792977990]