[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-657344-105":3,"detail-sidebar-cat-0-en-105":80,"doc-detail-657344-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","tree-diagrams-probability-rules-for-combined-experiments","Tree diagrams - Probability rules for combined experiments","","Tree diagrams explain how to identify all possible outcomes when multiple events are combined, then assign probabilities using clear AND and OR rules. The AND rule shows that joint outcomes from independent events are found by multiplying their probabilities, while the OR rule adds probabilities for mutually exclusive outcomes. Examples include coin tosses and spinner or sample-space situations, emphasizing that probabilities for all complete outcomes must sum to 1 and stay within the maximum of 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/tree-diagrams-probability-rules-for-combined-experiments/657344/",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/tree-diagrams-probability-rules-for-combined-experiments/657344.png","ImageObject",300,407,{"name":42,"@type":43},"Riley","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":52,"interactionType":53,"userInteractionCount":30},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What are tree diagrams used for in probability questions?","Question",{"text":62,"@type":63},"Tree diagrams help list and organize all possible outcomes from combined experiments, then support finding the probability of each outcome.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How do you calculate P(A and B) using the AND rule?",{"text":67,"@type":63},"For independent events, multiply the probabilities: P(A and B) = P(A) × P(B).",{"name":69,"@type":60,"acceptedAnswer":70},"When should you use the OR rule, and how do you apply it?",{"text":71,"@type":63},"Use the OR rule when either outcome can happen but the events are mutually exclusive; add their probabilities: P(A or B) = P(A) + P(B).","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},657344,1792977583,{"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":30,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":26,"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":79,"read_time":95},1374391975076,"https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051","Tree diagrams  \nIdentify all the outcomes of a combination of experiments. Use the AND and OR rules. Solve problems using tree diagrams.  \n| Outcomes are a list of all the different possibilities that could happen in an event.\u003Cbr>All the possible outcomes for rolling a dice and throwing a\u003Cbr>coin are shown in purple in the sample space diagram below.\u003Cbr>H = heads, T = tails.\u003Cbr>Dice Coin 1 2 3 4 5 6 H 1H 2H 3H 4H 5H 6H T 1T 2T 3T 4T 5T 6T\u003Cbr>Sometimes we want to know the probability of many things happening, not just one.\u003Cbr>The AND rule\u003Cbr>P(A and B) = P(A) × P(B)\u003Cbr>When you are combining two or more events, if you want both/all to happen, you must multiply their probabilities. These events must be independent, i.e. the outcome of each event does not impact the outcome of the other.\u003Cbr>The OR rule\u003Cbr>P(A or B) = P(A) + P(B)\u003Cbr>When you want either outcome to happen, you need to add the probabilities. The events must be mutually exclusive. This means that they must not be able to happen at the same time as each other. | \u003Cbr>\u003Cbr> Example\u003Cbr>The following spinner is spun twice.\u003Cbr>\u003Cbr>The probability of landing on red once:\u003Cbr>P(R) = 􀫚􀫡~~ ~~ .\u003Cbr>The probability of landing on red twice = probability of landing on red AND red again:\u003Cbr>P(RR) = 􀫚8 × 􀫚8 = 􀫚􀫟􀫝\u003Cbr>The probability of landing on either blue OR yellow = P(B) + P(Y) = 􀫚􀫡 + 􀫚􀫡 = 􀫛􀫡 = 􀫚􀫝 |\n| --- | --- |\n\nCheck that you can:  \n◆ list all the different probabilities that could happen in an event  \n◆ write the probability of an event happening as a fraction or decimal.  \nTree diagrams  \nIdentify all the outcomes of a combination of experiments. Use the AND and OR rules. Solve problems using tree diagrams.  \nTree diagrams  \nWhen considering all the possible outcomes ofa combination of events, it is useful to use a diagram. Sometimes a sample space diagram is an option, as used in the earlier examples. Another option is a probability tree diagram.  \nExample 1  \nThe diagram on the right shows the outcomes of throwing two coins. The first set of branches shows the outcomes of the first coin throw, heads (H) or tails (T) . The second pair of branches shows the outcomes for the second coin throw.  \nThese events are independent of each other, which means that the outcome of the first coin throw has no impact on the outcome of the second.  \nAll possible outcomes are considered with a tree diagram, in this case:  \nHH – landing on heads first and second  \nHT – landing on heads first and tails second  \nTH – landing on tails first and heads second  \nTT – landing on tails first and second.  \nWe have considered every possible outcome after throwing two coins.  \nTo complete the diagram, we must find the probability for each of the outcomes. For this we multiply.  \n1   1   1  \nP(heads AND heads) = × =  \n2 2 4  \n1   1   1  \nP(heads AND tails) = × =  \n2 2 4  \n1   1   1 P(tailsAND heads) = × =  \n2 2 4 P(tails AND tails ) = 12 × 12 = 14  \nNote that the total probabilities of all 4 outcomes adds to 1.  \n1   1   1   1 P(HH 􀛽􀜀 HT 􀛽􀜀 TH 􀛽􀜀 TT) = + + + = 1  \n4 4 4 4 Probability can be written as a fraction, decimal or percentage. It’s important to remember that any probability must never exceed 1. Use this idea when checking your answers.  \nWe can now include the probability of these events on the tree diagram.  \nThe probability of landing on heads after a single throw of the coin is 􀫚 .  \n􀫛  \nThe probability of landing on tails after a single throw of the coin is 􀫚 .  \n􀫛  \nThe probabilities are the same for the second coin.  \nREMEMBER!  \nWhen you add up all the probabilities at the end of the branches on the right-hand side of the tree diagram, they must add up to 1.  \nExample 2  \nThe probability of Sam having school dinners is 0∙6.  \nCalculate:  \na) The probability that he doesn’t have school dinners on either day.  \nb) The probability that he has school dinners on one of the two days.  \nSolution  \na) The probability that he doesn’t have school dinners on either day:  \nP(NN) = 0⋅16  \nb) The probab","cbCaiexMVkNCISDX","https://ap.wps.com/l/cbCaiexMVkNCISDX","pdf",406671,"English","# Outcomes and sample spaces\n# AND rule for independent events\n# OR rule for mutually exclusive events\n# Worked examples with tree diagrams\n## Coin tosses\n## Checking that totals equal 1","[{\"question\":\"What are tree diagrams used for in probability questions?\",\"answer\":\"Tree diagrams help list and organize all possible outcomes from combined experiments, then support finding the probability of each outcome.\"},{\"question\":\"How do you calculate P(A and B) using the AND rule?\",\"answer\":\"For independent events, multiply the probabilities: P(A and B) = P(A) × P(B).\"},{\"question\":\"When should you use the OR rule, and how do you apply it?\",\"answer\":\"Use the OR rule when either outcome can happen but the events are mutually exclusive; add their probabilities: P(A or B) = P(A) + P(B).\"}]","Tree diagrams - Probability rules for combined experiments | PDF"]