[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572281-105":53,"doc-detail-572281-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","probability-cheat-sheet-foundations-of-data-analysis","Probability Cheat Sheet - Foundations of Data Analysis","","Concise probability and set-theory reference compiling core foundations used in data analysis and word-problem solving. Covers event notation (intersection, union, complement, difference), key set identities including DeMorgan’s law, counting via permutations and combinations, and major probability rules such as inclusion-exclusion, complement, and difference. Includes conditional probability, multiplication rule, total probability, Bayes’ rule, plus a step-by-step workflow for translating word problems into events, formulas, and computed expressions.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/probability-cheat-sheet-foundations-of-data-analysis/572281/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/probability-cheat-sheet-foundations-of-data-analysis/572281.png","ImageObject",442,249,{"name":88,"@type":89},"A glass of water","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-22","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":47},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What do A ∩ B, A ∪ B, and Ac mean in event notation?","Question",{"text":108,"@type":109},"A ∩ B means both events A and B happen. A ∪ B means either event A or B (or both) happens. Ac means event A does not happen.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How do you compute P(A ∪ B) using the inclusion-exclusion rule?",{"text":113,"@type":109},"Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B). This accounts for overlap between the two events.",{"name":115,"@type":106,"acceptedAnswer":116},"What are the key steps to solve a probability word problem from the cheat sheet?",{"text":117,"@type":109},"First identify the events, then express the asked probability in terms of those events. Map each given probability number to its mathematical expression (e.g., P(A), P(Bc), P(A|B)), and finally use the relevant formula so the right-hand side is known or computable.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},572281,1792015512,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":47,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":73,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":9},962090760832,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Probability Cheat Sheet Foundations of Data Analysis  \nReview of “English translation” for events:  \n• A \\ B = “both events A and B happen”  \n• A [ B = “either event A or B (or both) happens”  \n• Ac = “event A does not happen”  \n• A 􀀀 B = A \\ Bc = “event A happens, but event B does not happen”Set Theory Rules: (try drawing Venn diagrams of these)  \n• Associative Law:  \n• Commutative Law:  \n• Distributive Law:  \n• DeMorgan's Law:  \n(A [ B) [ C = A [(B [ C)  \n(A \\ B) \\ C = A \\ (B \\ C)  \nA [ B = B [ A  \nA \\ B = B \\ A  \n(A [ B) \\ C = (A \\ C) [(B \\ C)  \n(A \\ B) [ C = (A [ C) \\ (B [ C)(A [ B)c = Ac \\ Bc  \n(A \\ B)c = Ac [ Bc  \nCounting:  \n• Number of permutations of n items: n! = n 􀀂 (n 􀀀 1) 􀀂 (n 􀀀 2) 􀀂 􀀁 􀀁 􀀁 􀀂 2 (number of unique orderings)  \n• Number of ways to select k items out of n choices: 􀀀nk􀀁 = k!(~~n~~n􀀀~~! ~~k)!(here order does not matter, just which k items you select)  \nProbability Rules:  \n• Equally likely outcomes: P (A) = jjA􀀊jj  \n• Inclusion-Exclusion Rule: P (A [ B) = P(A) + P(B) 􀀀 P(A \\ B)  \n• Complement Rule: P (Ac ) = 1 􀀀 P(A)  \n• Difference Rule: P (A 􀀀 B) = P(A) 􀀀 P(A \\ B)  \nExercise: Try deriving these rules from the definition of a probability function. Draw a Venn diagram to convince yourself they work.  \nConditional Probability:  \nP (A j B) = “the probability of event A given that we know B happened”  \nFormula:  \nP (A j B) = P~~ ~~(AP(\\BB))  \nMultiplication Rule:  \nP (A \\ B) = P(A j B)P (B)  \nTotal Probability:  \nP (A) = P(A j B)P (B) + P(A j Bc)P (Bc)  \nBayes' Rule:  \nP (B j A) = P~~ ~~(APj~~ ~~B(A)P)(B)  \nNote: The denominator, P(A); in Bayes' Rule is often computed using the Total Probability formula above!  \nSteps to Solve a Word Problem:  \n1. Identify the events in the problem.  \n2. In terms of these events, write down the probability that is being asked for in the problem.  \n3. For each probability (number) in the problem, write down what its mathematical expression is, e.g., P (A); P (Bc ); P (A j B); etc.  \n4. Use the formula for the probability being asked. The right hand side should all be known, or computable, from what is given in the problem.","cbCait5miabWP5q4","https://ap.wps.com/l/cbCait5miabWP5q4","pdf",96997,"English","# Set Theory Rules\n## Event notation\n## Associative, commutative, distributive, and DeMorgan’s law\n# Counting\n## Permutations and combinations\n# Probability Rules\n## Equally likely outcomes\n## Inclusion-exclusion, complement, difference\n# Conditional Probability and Bayes’ Rule\n## Conditional probability\n## Multiplication, total probability, Bayes’ rule\n# Solving Word Problems\n## Step-by-step procedure","[{\"question\":\"What do A ∩ B, A ∪ B, and Ac mean in event notation?\",\"answer\":\"A ∩ B means both events A and B happen. A ∪ B means either event A or B (or both) happens. Ac means event A does not happen.\"},{\"question\":\"How do you compute P(A ∪ B) using the inclusion-exclusion rule?\",\"answer\":\"Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B). This accounts for overlap between the two events.\"},{\"question\":\"What are the key steps to solve a probability word problem from the cheat sheet?\",\"answer\":\"First identify the events, then express the asked probability in terms of those events. Map each given probability number to its mathematical expression (e.g., P(A), P(Bc), P(A|B)), and finally use the relevant formula so the right-hand side is known or computable.\"}]","Probability Cheat Sheet - Foundations of Data Analysis | PDF",1791926552]