[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-199747-en":3,"doc-seo-199747-105":31,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},199747,2336475104736,"วิน","https://ap-avatar.wpscdn.com/avatar/22000c4c5e0e5b17e70?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786591360781797222",4,"Exam","Stat 100a - Exam Notes - Probability","These lecture notes for Stat 100a focus on probability tools used in exam preparation, including counting principles, axioms and rules of probability, permutations and conditional probability. The material emphasizes deeper applications through independence, random variables, expectation, variance, and confidence intervals, plus inequalities and limiting results such as LLN and CLT. Practice sections connect theory to poker reasoning via luck, skill, equity, random walks, and Bayes’ rule examples, with suggested chapter problems and exam logistics.","Stat 100a: Introduction to Probability.  \nOutline for the day:  \n1. Exam notes .  \n2. Luck, skill and equity, ch. 4.4.  \n3. Review list.  \n4. Random walk example .  \n5. Bayes’ rule example .  \n6. Equity gained example .  \n7. Conditional probability examples .  \nExam 2 is Tue Mar 3.  \nThe project is due Tue Mar 3 8pm by email to me .  \nNO CLASS or OH Tue Mar 10.  \nHw3 is due Mar 12.  \nExam 3 is Thu Mar 12.  \n1. Exam notes .  \nExam 2 is Tue Mar 3.  \nThe project is due Tue Mar 3 8pm by email to me .  \nNO CLASS or OH Tue Mar 10.  \nHw3 is due Mar 12.  \nExam 3 is Thu Mar 12.  \nBoth exams are scantron .  \nThey are open book, open note. Bring a calculator and a \\#2 pencil. There’s no R stuff on the exams .  \nSuggested problems from ch 4-7 to look at are 4.5, 4.6, 4.7, 4.8, 4.9, 4.13, 4.14, 4.16, 5.1, 5.2, 5.5, 5.6, 6.2, 6.4, 6.9, 6.10, 6.11, 6.12, 7.1, 7.2, 7.3, 7.4, 7.5, 7.8, 7.13, 7.14, 7.15.  \n2. Luck, skill, and equity.  \nIs poker luck or skill?  \nWho cares? The courts do, for one .  \nHow would you define luck or skill?  \nLet equity = your expected holdings after the hand,  \ni.e . your chips + pot x p, where p = your probability of winning the hand in a showdown, i.e . assuming no future betting.  \nAccording to the book's definitions,  \nluck = equity gained during the dealing of the cards.  \nskill = equity gained during betting.  \nFor example,  \nLederer vs . Minieri, hand number 7. Consider the turn .  \nHand 7. Minieri 10♦ 7♠ , Lederer Q♣ 2♥ . Minieri 43 .57%, Lederer 56.43% . Minieri raises to 3200, Lederer calls 1600.  \nFlop 8♠ 2♠ Q♥ . Minieri 7.27%, Lederer 92.73% . Lederer checks, Minieri bets 3200, Lederer calls .  \nTurn 4♦ . Minieri 0%, Lederer 100%. Lederer checks, Minieri bets 10,000, Lederer calls .  \nRiver A♥ . Lederer checks, Minieri checks.  \nLuck-205.76-2323.20-930.56 = -3459.52.  \nSkill-205.76-2734.72-10000 = -12940.48.  \nLet x = Minieri's number of chips after the hand. The pot size was 6400+6400 = 12800 before the turn . Minieri's gain due to luck on the turn = Minieri's equity right after the turn was dealt – equity before turn dealt = [x + 10,000 + (12,800)(0%)] – [x + 10,000 + (12,800)(7 .27%)] = -930.56.  \nMinieri's gain due to skill on the turn = his equity after turn betting-equity right before turn betting = [x + (12,800)(0%)] – [x + 10,000 + (12,800)(0%)] = -$10,000 .  \nWere there some key hands where one of the two players gained a lot due to either luck or skill? Which ones?  \nDo the definitions seem to fit?  \nWhen would they not fit?  \n3. Review list.  \n1) Basic principles of counting.  \n2) Axioms of probability, and addition rule .  \n3) Permutations & combinations .  \n4) Conditional probability.  \n5) Independence .  \n6) Multiplication rules . P(AB) = P(A) P(B|A) [= P(A)P(B) if ind.]  \n7) Odds ratios .  \n8) Random variables (RVs) .  \n9) Discrete RVs, and probability mass function (pmf) .  \n10) Expected value .  \n11) Pot odds calculations .  \n12) Luck, skill, and deal-making.  \n13) Variance and SD.  \n14) Bernoulli RV. [0-1 . µ = p, σ = √(pq) . ]  \n15) Binomial RV. [\\# of successes, out of n tries . µ = np, σ = √(npq) .]  \n16) Geometric RV. [\\# of tries til 1st success . µ = 1/p, σ = (√q) / p. ]  \n17) Negative binomial RV. [\\# of tries til rth success . µ = r/p, σ = (√rq) / p. ]  \n18) Poisson RV [\\# of successes in some time interval. [µ = λ, σ = √λ .]  \n19) E(X+Y), V(X+Y) (ch. 7. 1) .  \n20) Bayes’s rule (ch. 3.4) .  \n20) Continuous RVs  \n21) Probability density function (pdf)  \n22) Uniform RV  \n23) Normal RV  \n24) Exponential and Pareto RVs  \n25) Moment generating functions  \n26) Markov and Chebyshev inequalities  \n25) Law of Large Numbers (LLN)  \n26) Central Limit Theorem (CLT)  \n27) Conditional expectation .  \n28) Confidence intervals for the sample mean .  \n29) Fundamental theorem of poker  \n30) Random Walks, Reflection Principle, Ballot Theorem, avoiding zero  \n31) Chip proportions and induction .  \nBasically, we’ve done all of ch. 1-7 except 6.7.  \n4. Another random walk example .  \nSuppose that a $10 winne","cbCaipbmj7O3u9hW","https://ap.wps.com/l/cbCaipbmj7O3u9hW","pdf",96162,5,1,10,"English","en",105,"# Exam notes\n## Luck, skill and equity\n## Review list\n## Random walk example\n## Bayes' rule example\n## Equity gained example","[{\"question\":\"What exam topics are included in the review list?\",\"answer\":\"The review list covers counting principles, axioms and addition rule, permutations and combinations, conditional probability and independence, multiplication rules, odds ratios, random variables, expectation, pot odds, variance and standard deviation, several common discrete and continuous distributions, and moment generating functions.\"},{\"question\":\"How is equity defined and used to separate luck vs. skill in poker?\",\"answer\":\"Equity is defined as expected holdings after the hand: chips plus pot times the probability of winning a showdown. Luck is framed as equity gained during dealing, while skill is framed as equity gained during betting.\"},{\"question\":\"How does Bayes’ rule get applied in the Bayes’ rule example with all-in ranges?\",\"answer\":\"The example uses given frequencies for how often the opponent goes all-in with specific hand groups, then computes posterior probability for hands like AK conditioned on all-in by combining likelihoods with prior probabilities and normalizing across all considered hand categories.\"}]","Stat 100a - Exam Notes - Probability | PDF",1788506906,25,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":29},"stat-100a-exam-notes-probability","",{"@graph":37,"@context":86},[38,54,69],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/exam/",3,{"item":53,"name":13,"@type":44,"position":11},"https://docshare.wps.com/document/stat-100a-exam-notes-probability/199747/",{"url":53,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":42,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-09-08","2026-09-04",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What exam topics are included in the review list?","Question",{"text":76,"@type":77},"The review list covers counting principles, axioms and addition rule, permutations and combinations, conditional probability and independence, multiplication rules, odds ratios, random variables, expectation, pot odds, variance and standard deviation, several common discrete and continuous distributions, and moment generating functions.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How is equity defined and used to separate luck vs. skill in poker?",{"text":81,"@type":77},"Equity is defined as expected holdings after the hand: chips plus pot times the probability of winning a showdown. Luck is framed as equity gained during dealing, while skill is framed as equity gained during betting.",{"name":83,"@type":74,"acceptedAnswer":84},"How does Bayes’ rule get applied in the Bayes’ rule example with all-in ranges?",{"text":85,"@type":77},"The example uses given frequencies for how often the opponent goes all-in with specific hand groups, then computes posterior probability for hands like AK conditioned on all-in by combining likelihoods with prior probabilities and normalizing across all considered hand categories.","https://schema.org",{"og:url":53,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,105,109,114,119,124,129,132,135],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":103,"slug":104},70,"exam",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":106,"show_sort_weight":107,"slug":108},"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":47,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":47,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":120,"doc_module":4,"doc_module_name":47,"category_name":121,"show_sort_weight":122,"slug":123},8,"Research & Report",30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":22,"doc_module":4,"doc_module_name":47,"category_name":133,"show_sort_weight":22,"slug":134},"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":47,"category_name":137,"show_sort_weight":20,"slug":138},19,"General","general"]