[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572283-105":53,"doc-detail-572283-en":127},{"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":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","probability-cheatsheet-v20-quick-reference","Probability Cheatsheet - v2.0 - Quick Reference","","Probability Cheatsheet v2.0 is a compact reference covering core tools in introductory probability. It includes counting principles via the multiplication rule and sampling tables, Cardano’s definition for equally likely finite outcomes, and set-algebra identities with unions, intersections, complements, and De Morgan’s laws. The guide summarizes probability axioms, conditional probability, law of total probability, Bayes’ rule, intersection/union calculations using conditioning and inclusion-exclusion, as well as independence, conditional independence, and foundations for random variables, including discrete PMF/CDF concepts and continuous distributions with PDF and CDF relationships.",{"@graph":63,"@context":119},[64,80,102],{"@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-cheatsheet-v20-quick-reference/572283/",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-cheatsheet-v20-quick-reference/572283.png","ImageObject",442,249,{"name":88,"@type":89},"jayni","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-22","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",6,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"How does the multiplication rule determine the number of outcomes in a compound experiment?","Question",{"text":109,"@type":110},"If component i has n_i possible outcomes for i = 1..r, the whole experiment has n_1 n_2 ... n_r possibilities.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"What is Cardano’s definition of probability for equally likely finite outcomes?",{"text":114,"@type":110},"For event A, the probability is the number of outcomes favorable to A divided by the total number of equally likely outcomes.",{"name":116,"@type":107,"acceptedAnswer":117},"How are conditional probability and independence related in this cheat sheet?",{"text":118,"@type":110},"Conditional probability is given by P(A|B) and is treated as a probability function for fixed B. Independence is characterized by equivalent conditions such as P(A∩B)=P(A)P(B), and conditional independence is defined by a factorization given C.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},572283,1792022905,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":136,"language":137,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":138,"faqs":139,"seo_title":140,"seo_description":61,"update_tm":141,"read_time":79},3985747859343,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Probability Cheatsheet v2.0  \nOriginal work at [http://github.com/wzchen/probability_cheatsheet](http://github.com/wzchen/probability_cheatsheet) , modi􀀌ed by Pablo Angulo for ESTIN@UPM.ES  \nLast Updated August 3, 2020  \nCounting  \nMultiplication Rule  \nLet's say we have a compound experiment (an experiment with multiple components) . If the 1st component has n1 possible outcomes, the 2nd component has n2 possible outcomes, . . . , and the rth component has nr possible outcomes, then overall there are  \nn1 n2 : : : nr possibilities for the whole experiment.  \nSampling Table  \nThe sampling table gives the number of possible samples of size k out of a population of size n, under various assumptions about how the sample is collected.  \n\n|  | Order Matters | Not Matter |\n| --- | --- | --- |\n| With Replacement | nk | 􀀐n + kk 􀀀 1􀀑 |\n| Without Replacement | n!\u003Cbr>(n 􀀀 k)! | 􀀐nk􀀑 |\n\nCardano's De􀀌nition of Probability  \nIf the number of outcomes is 􀀌nite and all outcomes are equally likely, the probability of an event A happening is:  \nPCardano (A) = numberonufoutcommber~~ ~~of~~ ~~eosufavoratcomebsle~~ ~~to~~ ~~A  \nSet algebra  \nUnions, Intersections, and Complements  \nComplements -The following are true.  \nA [ Ac = 􀀊  \nA \\ Ac = ;  \nDe Morgan's Laws  \n(A [ B)c = Ac \\ Bc  \n(A \\ B)c = Ac [ Bc  \nProbability  \nAxioms of probability  \nAny assignment from subsets of E to real numbers is a probability measure if the following holds:  \nProbabilities are positive P (A) 􀀕 0.  \nThe probability of the whole space is 1 P (E) = 1 .  \nProbabilities of a union of disjoint sets  \nP (A [ B) = P (A) + P (B), provided A \\ B = ; .  \nConsequences  \nFor any probability measure, the following are true:  \nProbability of the empty set P (;) = 0 .  \nProbability of the complement P (AC ) = 1 􀀀 P (A) .  \nConditional probability  \nConditional Probability  \nP (Aj B) = P~~ ~~(AP(\\B)B) Probability of A, given that B occurred.  \nConditional Probability is Probability P (Aj B) is a probability function for any 􀀌xed B . Any theorem that holds for probability also holds for conditional probability.  \nProbability of an Intersection or Union  \nIntersections via Conditioning  \nP (A; B) = P (A)P (Bj A)  \nP (A; B; C) = P (A)P (Bj A)P (C jA; B)  \nUnions via Inclusion-Exclusion  \nP (A [ B) = P (A) + P (B) 􀀀 P (A \\ B)  \nP (A [ B [ C) = P (A) + P (B) + P (C)  \n􀀀 P (A \\ B) 􀀀 P (A \\ C) 􀀀 P (B \\ C)  \n+ P(A \\ B \\ C):  \nLaw of Total Probability  \nAssume the n events Ai are pairwise disjoint (Ai \\ Aj = ; for any i  j) and their union is the whole sample space, and let B be any event. Then:  \nP (B) = P (Bj A1 )P (A1 ) + : : : + P(B jAn)P (An)  \n= Pni=1 P (Bj Ai)P (Ai)  \nBayes' Rule  \nP (Bj A)P (A)  \nP (Aj B) = ~~  ~~P (B)  \nIndependence  \n2 Independent Events A and B are independent if knowing whether A occurred gives no information about whether B occurred. More formally, A and B (which have nonzero probability) are independent if and only if one of the following equivalent statements holds:  \nP (A \\ B) = P (A)P (B); P (Aj B) = P (A); P (Bj A) = P (B)  \n3 Independent Events A, B and C are independent if information about two of them gives no information about whether the third one occurred. In other words, P (Aj EB \\ EC ) = P (A), where EB is either B , BC , or E, and EC is either C , C C or E . The relations obtained by permuting A, B and C must also hold.  \nConditional Independence A and B are conditionally independent given C if P(A \\ Bj C) = P (Aj C)P (Bj C) . Conditional independence does not imply independence, and independence does not imply conditional independence.  \nRandom Variables  \nA Random Variable (RV) is a function form a probability space into the real numbers:  \nX : 􀀊 ! R  \nThe support of X is the smallest closed set S such that P (X 2 S) = 1 (morally, the set of values that X can take) .  \nThe distribution of the RV is not the probability distribution of 􀀊, but the induced distribution in the real numbers A ! P (X 2 A) . E.g.  \n􀀊 has a continuous uniform distribution","cbCailcfUFyJgCiX","https://ap.wps.com/l/cbCailcfUFyJgCiX","pdf",1686889,10,"English","# Counting\n# Set Algebra and Probability Axioms\n# Conditional Probability and Bayes' Rule\n# Total Probability and Inclusion-Exclusion\n# Independence and Conditional Independence\n# Random Variables\n## Discrete Random Variables\n## Continuous Random Variables","[{\"question\":\"How does the multiplication rule determine the number of outcomes in a compound experiment?\",\"answer\":\"If component i has n_i possible outcomes for i = 1..r, the whole experiment has n_1 n_2 ... n_r possibilities.\"},{\"question\":\"What is Cardano’s definition of probability for equally likely finite outcomes?\",\"answer\":\"For event A, the probability is the number of outcomes favorable to A divided by the total number of equally likely outcomes.\"},{\"question\":\"How are conditional probability and independence related in this cheat sheet?\",\"answer\":\"Conditional probability is given by P(A|B) and is treated as a probability function for fixed B. Independence is characterized by equivalent conditions such as P(A∩B)=P(A)P(B), and conditional independence is defined by a factorization given C.\"}]","Probability Cheatsheet - v2.0 - Quick Reference | PDF",1791926559]