[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572279-105":53,"doc-detail-572279-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-cheatsheet-v20-reference-sheet","Probability Cheatsheet - v2.0 - reference sheet","","Probability Cheatsheet v2.0 compiles essential probability rules and concepts into a compact reference. It covers counting basics for compound experiments, sampling with and without replacement, naive probability under equal likelihood, and conditional reasoning including independence and conditional independence. The notes explain unions, intersections, complements using De Morgan’s laws, and provide key formulas for joint, marginal, and conditional probabilities, inclusion–exclusion, the law of total probability, and Bayes’ rule (including odds form). It also introduces random variables via PMF and CDF, independence criteria, and expected value with linearity.",{"@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-cheatsheet-v20-reference-sheet/572279/",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-reference-sheet/572279.png","ImageObject",442,249,{"name":88,"@type":89},"Indoniesian Boy","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},"How do you compute the probability of an event when all outcomes are equally likely?","Question",{"text":108,"@type":109},"Use the naive formula: P_naive(A) equals the number of outcomes favorable to A divided by the total number of possible outcomes.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"When are two events considered independent (and how does this differ from conditional independence)?",{"text":113,"@type":109},"Events A and B are independent if knowing whether A occurred gives no information about B, equivalently when P(A ∩ B) = P(A)P(B). They are conditionally independent given C when the conditional version factorizes, and neither conditional independence implies independence nor vice versa.",{"name":115,"@type":106,"acceptedAnswer":116},"What are the key formulas for the law of total probability and Bayes' rule?",{"text":117,"@type":109},"LOTP sums conditional probabilities over a partition: P(A)=Σ P(A|B_i)P(B_i). Bayes' rule updates probabilities using P(A|B)=P(B|A)P(A)/P(B), with an odds-form expression relating posterior odds to likelihood ratio times prior odds.","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},572279,1792403751,{"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":135,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":79},962090760608,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Probability Cheatsheet v2.0  \nCompiled by William Chen ([http://wzchen.com](http://wzchen.com)) and Joe Blitzstein, with contributions from Sebastian Chiu, Yuan Jiang, Yuqi Hou, and Jessy Hwang. Material based on Joe Blitzstein's (@stat110) lectures ([http://stat110.net](http://stat110.net) and Blitzstein/Hwang's Introduction to)[) and Blitzstein/Hwang's Introduction to](http://stat110.net) and Blitzstein/Hwang's Introduction to)[ ](http://stat110.net) and Blitzstein/Hwang's Introduction to)Probability textbook ([http://bit.ly/introprobability](http://bit.ly/introprobability)). Licensed under CC BY-NC-SA 4 .0. Please share comments, suggestions, and errors at [http://github.com/wzchen/probability_cheatsheet](http://github.com/wzchen/probability_cheatsheet).  \nLast Updated September 4, 2015  \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\nNaive De􀀌nition of Probability  \nIf all outcomes are equally likely, the probability of an event A happening is:  \nPnaive (A) = numberonufoutcommber~~ ~~of~~ ~~eosufavoratcomebsle~~ ~~to~~ ~~A  \nThinking Conditionally  \nIndependence  \nIndependent 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)  \nP (Bj A) = P (B)  \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.  \nUnions, Intersections, and Complements  \nDe Morgan's Laws A useful identity that can make calculating probabilities of unions easier by relating them to intersections, and vice versa. Analogous results hold with more than two sets.  \n(A [ B)c = Ac \\ Bc  \n(A \\ B)c = Ac [ Bc  \nJoint, Marginal, and Conditional  \nJoint Probability P (A \\ B) or P (A; B) { Probability of A and B . Marginal (Unconditional) Probability P (A) { Probability of A. Conditional Probability P (Aj B) = P (A; B)=P (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):  \nSimpson's Paradox  \nheart  \nband-aid  \nDr. Hibbert  \nNick  \nIt is possible to have  \nP (A j B; C) \u003C P (A j Bc ; C) and P (A j B; Cc ) \u003C P (A j Bc ; Cc )  \nyet also P (A j B) > P (A j Bc ):  \nLaw of Total Probability (LOTP)  \nLet B1 ; B2 ; B3 ; :::Bn be a partition of the sample space (i.e., they are disjoint and their union is the entire sample space) .  \nP (A) = P (Aj B1 )P (B1 ) + P (Aj B2 )P (B2 ) + 􀀁 􀀁 􀀁 + P(Aj Bn)P (Bn)  \nP (A) = P (A \\ B1 ) + P (A \\ B2 ) + 􀀁 􀀁 􀀁 + P(A \\ Bn)  \nFor LOTP with extra conditioning, just add in another event C!  \nP (Aj C) = P (Aj B1 ; C)P (B1 jC) + 􀀁 􀀁 􀀁 + P(Aj Bn ; C)P (Bn jC)  \nP (Aj C) = P (A \\ B1 jC) + P (A \\ B2 jC) + 􀀁 􀀁 􀀁 + P(A \\ Bn jC)  \nSpecial case of LOTP with B and Bc as partition:","cbCaioM4NCaKyryp","https://ap.wps.com/l/cbCaioM4NCaKyryp","pdf",808202,10,"English","# Counting\n# Independence and Conditional Independence\n# Unions, Intersections, and Complements\n# Joint, Marginal, and Conditional Probability\n# Intersections via Conditioning\n# Unions via Inclusion-Exclusion\n# Simpson's Paradox\n# Law of Total Probability (LOTP)\n# Bayes' Rule\n# Odds Form of Bayes' Rule\n# Random Variables and their Distributions\n## PMF, CDF, and Independence\n## Expected Value and Indicators","[{\"question\":\"How do you compute the probability of an event when all outcomes are equally likely?\",\"answer\":\"Use the naive formula: P_naive(A) equals the number of outcomes favorable to A divided by the total number of possible outcomes.\"},{\"question\":\"When are two events considered independent (and how does this differ from conditional independence)?\",\"answer\":\"Events A and B are independent if knowing whether A occurred gives no information about B, equivalently when P(A ∩ B) = P(A)P(B). They are conditionally independent given C when the conditional version factorizes, and neither conditional independence implies independence nor vice versa.\"},{\"question\":\"What are the key formulas for the law of total probability and Bayes' rule?\",\"answer\":\"LOTP sums conditional probabilities over a partition: P(A)=Σ P(A|B_i)P(B_i). Bayes' rule updates probabilities using P(A|B)=P(B|A)P(A)/P(B), with an odds-form expression relating posterior odds to likelihood ratio times prior odds.\"}]","Probability Cheatsheet - v2.0 - reference sheet | PDF",1791926547]