[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-657339-105":3,"detail-sidebar-cat-0-en-105":80,"doc-detail-657339-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","probability-tree-diagrams-lec-5-calculating-probabilities-with-tree-diagrams","Probability Tree Diagrams - LEC 5 - Calculating Probabilities with Tree Diagrams","","Probability Tree Diagrams lecture notes explaining how to compute overall probabilities in both independent and dependent scenarios. The content demonstrates coin-toss trees, using multiplication along branches and addition down columns to find outcomes such as the probability of at least one head. It then extends the method to dependent events via a soccer goalkeeper example with different coaches, finishing with overall probability calculations and a validation check that all probabilities sum to 1.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/probability-tree-diagrams-lec-5-calculating-probabilities-with-tree-diagrams/657339/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/probability-tree-diagrams-lec-5-calculating-probabilities-with-tree-diagrams/657339.png","ImageObject",300,407,{"name":42,"@type":43},"Oliver","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":52,"interactionType":53,"userInteractionCount":33},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"How do you calculate overall probabilities from a probability tree diagram?","Question",{"text":62,"@type":63},"Multiply probabilities along the branches to get each path probability, then add probabilities down the relevant columns to combine outcomes.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"What method is used for independent events in the examples?",{"text":67,"@type":63},"For independent events like consecutive coin tosses, the probability of each branch is multiplied with the probabilities on subsequent branches.",{"name":69,"@type":60,"acceptedAnswer":70},"How does the goalkeeper example model dependent events?",{"text":71,"@type":63},"The chance of being goalkeeper depends on who the coach is. The tree branches first split by coach probabilities, then split again by the probability of being goalkeeper given that coach.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},657339,1793065117,{"code":4,"msg":81,"data":82},"success",[83,87,91,94,99,104,109,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":88,"show_sort_weight":89,"slug":90},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":92,"slug":93},70,"exam",{"id":95,"doc_module":4,"doc_module_name":25,"category_name":96,"show_sort_weight":97,"slug":98},5,"Comic",60,"comic",{"id":100,"doc_module":4,"doc_module_name":25,"category_name":101,"show_sort_weight":102,"slug":103},6,"Technology",50,"technology",{"id":105,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},7,"Healthcare",40,"healthcare",{"id":110,"doc_module":4,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},8,"Research & Report",30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":95,"slug":129},19,"General","general",{"code":4,"msg":81,"data":131},{"doc_id":78,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":33,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":105,"language":139,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":12,"update_tm":143,"read_time":144},8796095461610,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Probability Tree Diagrams  \nLEC :5  \nOLA ALI  \nCalculating probabilities can be hard, sometimes we add them, sometimes we multiply them, and often it is hard to figure out what todo ... tree diagrams to the rescue!  \n\n| \u003Cbr>Here is a tree diagram for the toss of a coin:\u003Cbr>There are two \"branches\" (Heads and Tails)\u003Cbr>􀁸 The probability of each branch is written on the branch\u003Cbr>􀁸 The outcome is written at the end of the branch\u003Cbr>We can extend the tree diagram to two tosses of a coin: |\n| --- |\n| |\n| How do we calculate the overall probabilities? |\n\n􀁸 We multiply probabilities along the branches  \n􀁸 We add probabilities down columns  \nNow we can see such things as:  \n􀁸 The probability of \"Head, Head\" is 0.5×0.5 = 0.25  \n􀁸 All probabilities add to 1.0 (which is always a good check)  \n􀁸 The probability of getting at least one Head from two tosses is  \n0. 25+0 . 25+0 .25 = 0.75  \n􀁸   and more  \nThat was a simple example using independent events (each toss of a coin is independent of the previous toss), but tree diagrams are really wonderful for figuring out dependent events (where an event depends on what happens in the previous event) like this example:  \nExample: Soccer Game  \nYou are off to soccer, and love being the Goalkeeper, but that depends who is the Coach today:  \n􀁸 with Coach Sam the probability of being Goalkeeper is 0.5  \n􀁸 with Coach Alex the probability of being Goalkeeper is 0.3  \nSam is Coach more often ... about 6 out of every 10 games (a probability of 0.6) .  \nSo, what is the probability you will be a Goalkeeper today?  \nLet's build the tree diagram. First we show the two possible coaches: Sam or Alex:  \nThe probability of getting Sam is 0.6, so the probability of Alex must be 0.4 (together the probability is 1)  \nNow, if you get Sam, there is 0.5 probability of being Goalie (and 0.5 of not being Goalie):  \nIf you get Alex, there is 0.3 probability of being Goalie (and 0.7 not):  \nThe tree diagram is complete, now let's calculate the overall probabilities. This is done by multiplying each probability along the\"branches\" of the tree.  \nHere is how to do it for the \"Sam, Yes\" branch:  \n(When we take the 0.6 chance of Sam being coach and include the 0.5 chance that Sam will let you be Goalkeeper we end up with an 0.3 chance.)  \nBut we are not done yet! We haven't included Alex as Coach:  \n\n|  |\n| --- |\n| |\n| An 0.4 chance of Alex as Coach, followed by an 0.3 chance gives 0.12. Now we add the column:\u003Cbr>0.3 + 0.12 = 0.42 probability of being a Goalkeeper today\u003Cbr>(That is a 42% chance)\u003Cbr>Check\u003Cbr>One final step: complete the calculations and make sure they add to 1:\u003Cbr>0.3 + 0.3 + 0.12 + 0.28 = 1\u003Cbr>Yes, it all adds up.\u003Cbr>You can see more uses of tree diagrams on Conditional Probability.\u003Cbr>Conclusion |\n\nSo there you go, when in doubt draw a tree diagram, multiply along the branches and add the columns. Make sure all probabilities add to 1 and you are good to go.","cbCaigpUaGjYcor1","https://ap.wps.com/l/cbCaigpUaGjYcor1","pdf",228539,"English","# Probability Tree Diagrams\n## Coin Toss Example\n## Overall Probabilities\n## Dependent Events Example (Soccer Game)\n## Checking Results and Conclusion","[{\"question\":\"How do you calculate overall probabilities from a probability tree diagram?\",\"answer\":\"Multiply probabilities along the branches to get each path probability, then add probabilities down the relevant columns to combine outcomes.\"},{\"question\":\"What method is used for independent events in the examples?\",\"answer\":\"For independent events like consecutive coin tosses, the probability of each branch is multiplied with the probabilities on subsequent branches.\"},{\"question\":\"How does the goalkeeper example model dependent events?\",\"answer\":\"The chance of being goalkeeper depends on who the coach is. The tree branches first split by coach probabilities, then split again by the probability of being goalkeeper given that coach.\"}]","Probability Tree Diagrams - LEC 5 - Calculating Probabilities with Tree Diagrams | PDF",1792977571,18]