[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-608206-105":59,"doc-detail-608206-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","math-464-fall-13-homework-1","Math 464 - Fall 13 - Homework 1","","Homework 1 for Math 464 (Fall 2013) focuses on probability theory using event algebra, set expressions with unions/intersections/complements, and computation of event probabilities from given marginal and joint probabilities. It includes geometric probability by selecting a random point in a rectangle and asking inequalities involving coordinates, as well as geometric/negative binomial style questions for stopping after a roll of 1 on a four-sided die. Additional tasks require simulating dice sums in R, interpreting the simulated event and probability, and solving conditional and inclusion-exclusion proofs.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/math-464-fall-13-homework-1/608206/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/math-464-fall-13-homework-1/608206.png","ImageObject",300,407,{"name":92,"@type":93},"Danger Angel","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-17",true,{"@type":102,"interactionType":103,"userInteractionCount":29},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How do you express events (i)–(v) in Problem 1 using A, B, and C?","Question",{"text":112,"@type":113},"Use unions for “at least” statements, complements for “none,” and intersections with complements to capture “exactly” cases based on which of A, B, C occur.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What do Problem 2 ask for in geometric probability?",{"text":117,"@type":113},"Compute the probability that X > Y, and then compute the same probability under the condition Y ≤ 1 when (X, Y) is uniformly chosen in the given rectangle.",{"name":119,"@type":110,"acceptedAnswer":120},"What is being simulated and asked in Problem 4’s R program?",{"text":121,"@type":113},"The program simulates rolling two dice and estimates the probability that the sum equals 4; you then run it in R, identify the simulated event, and compare with the exact analytical probability.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},608206,1792919724,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":29,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":14,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":24},962090769052,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Math 464-Fall 13-Homework 1  \nIf a problem is marked (Exposition), that means I will grade it more carefully than the others. In particular I will be looking at the quality of your exposition. These will typically be problems where you are asked to prove something.  \n1. Let A, B, C be events.  \n(a) Write each of these events in terms of A, B, C using intersections, unions, and complements:  \n(i) None of A, B, C occurs.  \n(ii) At least two of A, B, C occur.  \n(iii) Exactly one of A, B, C occurs.  \n(iv) Exactly two of A, B, C occurs.  \n(v) Exactly three of A, B, C occur.  \nFor example, the answer to (i) is (A ∪ B ∪ C)c or Ac ∩ Bc ∩ Cc .  \n(b) Now suppose that  \nP (A) = 0 .34, P(B) = 0 .68, P(C) = 0 .32 ,  \nP (A ∩ B) = 0 . 17, P(B ∩ C) = 0 .20, P(A ∩ C) = 0 .07 , P (A ∩ B ∩ C) = 0 .05.  \nFind the probability of each of the events (i) to (v) in part (a) .  \n2. A point (X, Y ) is picked at random (uniformly) from the rectangle whose corners are (0 , 0) ,(0 , 2) ,(1 , 0) ,(1 , 2) . This means that the probability of an event (a subset of the rectangle) is the area of the event divided by the area of the entire sample space (the rectangle) .  \n(a) What is the probability that X > Y ?  \n(b) What is the probability that X > Y given that Y 􀀔 1 ?  \n3. We roll a four-sided die until we get a 1 and then we stop. In the following when we count the number of rolls we include the 􀀌nal roll that came up 1 .  \n(a) What is the probability it takes exactly 3 rolls?  \n(b) What is the probability it takes at least 3 rolls?  \n(c) What is the probability it takes exactly 8 rolls given that it takes at least  \n6 rolls?  \n4. Here is a simple R program that simulates rolling two dice and looking at their sum.  \n1  \nn=100000  \ndie1 \u003C– sample(1:6,n,replace=TRUE)  \ndie2 \u003C– sample(1:6,n,replace=TRUE) diesum \u003C– die1+die2  \nprob \u003C– 0  \nfor (i in 1:n) if (diesum[i]==4) prob \u003C– prob+1 prob \u003C– prob/n  \nprob  \n(a) Run this program in R. It will output a number at the end. What number did you get? There is link on the homework page to a text 􀀌le that contains the above so that you can avoid having to type it in.  \n(b) This program simulates the probability of a certain event. What is the event?  \n(c) What is the exact probability of the event? (Your answer should of course be close to the simulation result you get.)  \n5. I have a six-sided die (with faces numbered 1 to 6) and a four-sided die (with faces numbered 1 to 4) . I 􀀍ip a coin. If heads, I roll the six-sided die and if tails I roll the four-sided die.  \n(a) Find the probability the roll is even.  \n(b) Find the probability the roll is 4 or greater.  \n6. (Exposition) If A, B and C are events and P is a probability measure, prove that  \nP (A ∪ B ∪ C) = P (A) + P(B) + P(C)  \n􀀀 P (A ∩ B) 􀀀 P(A ∩ C) 􀀀 P(B ∩ C) + P(A ∩ B ∩ C)  \nThis is a special case of a more general result called the “inclusion-exclusion formula.”  \n7. (Exposition) Prove that for all events A and B with P(B) > 0, we have  \nP (Ac |B) = 1 􀀀 P(A|B) .  \nYou should not use the fact (which we proved in class) that conditional probability is a probability measure. You should give a direct proof using the de􀀌nition of conditional probability. Caution: P (A|Bc ) is not equal to 1 􀀀 P(A|B) in general.  \n2","cbCaif3RhMn4blDh","https://ap.wps.com/l/cbCaif3RhMn4blDh","pdf",25336,"English","# Problem Set\n## Problem 1\n## Problem 2\n## Problem 3\n## Problem 4\n## Problem 5\n## Problem 6\n## Problem 7","[{\"question\":\"How do you express events (i)–(v) in Problem 1 using A, B, and C?\",\"answer\":\"Use unions for “at least” statements, complements for “none,” and intersections with complements to capture “exactly” cases based on which of A, B, C occur.\"},{\"question\":\"What do Problem 2 ask for in geometric probability?\",\"answer\":\"Compute the probability that X \\u003e Y, and then compute the same probability under the condition Y ≤ 1 when (X, Y) is uniformly chosen in the given rectangle.\"},{\"question\":\"What is being simulated and asked in Problem 4’s R program?\",\"answer\":\"The program simulates rolling two dice and estimates the probability that the sum equals 4; you then run it in R, identify the simulated event, and compare with the exact analytical probability.\"}]","Math 464 - Fall 13 - Homework 1 | PDF",1792223257]