Final exam PS 30 December 2009
|
|
- Edward Arnold Casey
- 5 years ago
- Views:
Transcription
1 Fial exam PS 30 December 2009 Name: UID: TA ad sectio umber: This is a closed book exam. The oly thig you ca take ito this exam is yourself ad writig istrumets. Everythig you write should be your ow work. Cases of academic dishoesty will be referred to the Dea of Studets office, which has the power to susped ad expel studets. Partial credit will be give: math mistakes will ot jeopardize your grade. There are eight parts i this exam. Each part is weighted equally (12 poits for each part). Please show all steps of your work ad explai what you are doig at each step. Correct aswers aloe are worth othig without a clear ad correct explaatio of where the aswers come from. Clarity ad legibility are factors i the grade. If you have a questio, raise your had ad hold up the umber of figers which correspods to the part you have questios about (if you have a questio o Part 2, hold up two figers). If the TA resposible for a give questio is ot i the room at the time, work o other parts of the exam ad hold the questio util that TA rotates to your exam locatio. Whe the ed of the exam is aouced, please stop workig immediately. People who cotiue workig after the ed of the exam is aouced will have their grades pealized by 25 percet. If you eed to leave the room to use the bathroom durig the exam, please write your ame dow o the bathroom log before you leave. A perso caot leave the room more tha oce durig the exam (a perso who leaves for a secod time will be cosidered to have completed his or her exam). Please tur i your exam to oe of the TAs. Whe you had i your exam, please write your ame dow o the log. Please write all aswers o this exam if you write o the reverse side of pages, please idicate this clearly. Please tur off ad put away all cell phoes ad other electroic gadgets. Please put away all otes ad close all bags. Before you had i your exam, make sure you flip through the exam ad at least look at all questios sometimes two pages get stuck to each other ad you ca miss a etire sectio of the exam. Good luck! total
2 Part 1. Cosider the followig two-perso game. 2a 2b 2c 2d 1a 8, 5 1, 0 2, -1 0, 4 1b 4, 8 3, 6-1, 9 1, 5 1c 7, -1 5, 2 3, 0 2, 3 1d 0, 7 6, 1 7, -2 1, 2 a. Iteratively elimiate strogly ad weakly domiated strategies. Elimiate as may as possible. Show the order of deletio. (4 poits). b. Fid all pure strategy Nash equilibria. (4 poits) c. Fid all mixed strategy Nash equilibria. Please write your aswer out i words (writig p=1/7, q=5/8 is ot sufficiet). (4 poits)
3 Part 2. Cosider a world with two states who are players 1 ad 2. Player 2 has a oil field i its territory that player 1 wats. Player 1 starts off with three choices: it ca do othig (N) ad allow player 2 to keep the oilfield, it ca threate (T) player 2 with force, or it ca lauch a surprise attack (S) at player 2. If player 1 chooses to do othig, the game eds. If player 1 threates, player 2 ca either capitulate (C) ad give up the oil field (i which case the game eds), or reject (R) player 1 s threat ad prepare for war. If player 2 rejects player 1 s threat, the player 1 ca follow up by either attackig (A) player 2, which starts a war, or backig dow (B) from its threat ad allowig player 2 to keep the oil field. Both of these actios ed the game. If player 1 istead lauches a surprise attack, player 2 ca either give up (G) or try to defed (D) itself. Player 1 ideally wats player 2 to capitulate ad give up the oil field without a fight, but gettig player 2 to give up the oil field after a surprise attack is a secod best optio. If player 2 will ot capitulate or give up the oil field, player 1 prefers to attack player 2. However, if he does attack player 2, player 1 would prefer to lauch a surprise attack rather tha attack after player 2 has had a chace to prepare for war. Player 1 s worst optio is to back dow from the use of force after makig a threat, ad oly slightly better tha this is to lose the oil field by doig othig. Player 2 ideally wats to keep the oil field without a fight. It ca do this if player 1 either does othig or backs dow, but player 2 would prefer that player 1 back dow after makig a threat because it makes player 2 look strog i the eyes of the iteratioal commuity. If player 2 caot keep the oil field without a fight, it prefers to fight for the oil field. However, if player 2 must fight, it prefers to fight player 1 after it has had a chace to reject player 1 s threat ad prepare for war, istead of defedig itself from a surprise attack. If player 2 does ot fight for the oil field, the it either capitulates to player 1 s threats or gives up after a surprise attack from player 1. However, give a choice betwee the two, it would prefer to capitulate to threats aloe, sice this avoids ay coflict takig place o player 2 s territory. a. Write out this game i extesive form. For actios, use the choices i the game idicated i brackets (i.e. N, T, S, A, ad B for player 1, ad C, R, G, D for player 2). For each player payoffs, you ca use the payoff values {1, 2, 3, 4, 5, 6}. You will ed up usig all 6 values for both players. (4 poits)
4 b. Write this game i strategic form. (4 poits) c. Fid all pure strategy Nash equilibria of this game. (2 poits) d. Fid all Subgame Perfect Nash Equilibria of this game. (2 poits)
5 Part 3. Say that there are five people V, W, X, Y, ad Z, who choose amog four cadidates a, b, c, ad d. Their prefereces are show i the table below, where the most preferred is listed first ad the least-preferred is listed last. For istace, perso V likes a best ad d worst. V W X Y Z a c b c a b d c d b c b a b c d a d a d a. Is there a Codorcet wier? If so, who? (1 poit) b. Is there a ageda i which they decide o c? If there is, show oe. If ot, explai why ot. (1 poit) c. Who is the Borda cout wier? (1 poit)
6 (Part 3 cotiued) Now suppose there are three groups i society with preferece orders as show below (the best o top ad worst at the bottom). Group X has 5 people, group Y has y people, where y is assumed to be odd, ad group Z has 3 people. X Y Z a c a b b c c a b d. What is the smallest value of y for which c is a Codorcet wier? (3 poits) e. Suppose this society uses the Borda cout system. List all the possible values of y for which a is the Borda cout wier. Remember that y is assumed to be odd. (3 poits)
7 f. Depedig o the value of y, ca b be a Borda cout wier? If so, fid the smallest value of y for which b is a Borda cout wier. Remember that y is assumed to be odd. If ot, explai why ot. (3 poits)
8 Part 4. Say that there are three city coucilpeople, Trudy, Ursula, ad Victor, who make decisios by majority rule. They choose amog four alteratives, c, d, e, ad f. Their prefereces are give by the followig table. Trudy Ursula Victor Best c d e d e f e f c Worst f c d For example, Trudy likes c best, d secod-best, e third-best, ad f worst. a. What is the top cycle? (3 poits) b. Is it possible to write a ageda i which f wis? If so, write oe dow. If ot, explai why ot. (3 poits)
9 (Part 4 cotiued) Now say that you are a iterest group. By cotributig to a coucilperso s campaig, you make that coucilperso s prefereces reverse completely. For example, if you cotribute to Victor, he ow likes d best, c secod-best, f third-best, ad e worst. You ca cotribute to just oe coucilperso, two coucilpersos, or all three, but to save cash you would rather cotribute to as few of them as possible. c. By makig cotributios, is it possible to chage coucilpersos prefereces so that c is chose o matter what the ageda is? If so, which coucilperso(s) should you cotribute to, give that you wat to cotribute to as few as possible? If ot, explai why ot. (3 poits) d. By makig cotributios, is it possible to chage coucilpersos prefereces so that f is chose o matter what the ageda is? If so, which coucilperso(s) should you cotribute to, give that you wat to cotribute to as few as possible? If ot, explai why ot. (3 poits)
10 Part 5. Sister (S) ad Brother (B) are sittig aroud the fireplace ejoyig their holiday whe their mother serves them a plate of m cookies. Sice they just fiished takig a game theory class, they decide to make a game out of it. They take turs eatig the cookies. Each perso ca eat 1, 2, or 3 cookies. Whoever eats the last cookie wis. However, if you choose the same umber that your oppoet chose last time, the the game is immediately over ad you lose. This is true eve if you eat the last cookie. For example, say they start with four cookies (m=4). If Sister eats 1 cookie, the if Brother eats 1 cookie he immediately loses because he chose the same umber (1) as Sister. If Sister eats 1 cookie, the Brother ca wi by eatig 3 cookies. If Sister eats 2 cookies, the if Brother eats 2 cookies, he loses because he chose the same umber as Sister, eve though he ate the last cookie. Here each perso gets a payoff of 1 from wiig, ad a payoff of 0 from losig. For coveiece, write payoffs as (Sister, Brother). Sister always gets to go first. a. Say m=5, i other words they start with 5 cookies. Write dow this game as a extesive form game. (4 poits) b. Write dow a subgame perfect Nash equilibrium (SPNE) of this game by writig arrows i the tree you wrote above (i.e. you do t have to write dow the SPNE i words). Please make your arrows ice ad clear. If there is more tha oe SPNE, just write dow oe of them; you do t have to write dow all of them. (4 poits)
11 c. Now let m, the startig umber of cookies, go from 1 to 20, as show i the table below. For each value of m, fid out which perso wis the game i a SPNE ad write it i the table below. Remember that Sister (S) always goes first. For example, whe m=1, there is oly oe cookie at the start, ad Sister obviously wis by takig oe cookie right at the start. So the table etry whe m=1 is already filled i for you as a example. It is crucial to explai your reasoig here; simply fillig out the table is ot sufficiet without a explaatio of where your aswers come from. (4 poits) m Who wis? S
12 Part 6. Mike ad Sarah are competig i a electio to be their party's omiee for presidet. The oly issue i the electio is how may troops to sed to a peacekeepig missio i aother coutry. Their optios (i army divisios) are to sed 1, 2, 3, 4 or 5. Assume that 32% of the populatio wats to sed 1 divisio, 18% wats to sed 2 divisios, 4% wats to sed 3 divisios, 8% wats to sed 4 divisios, ad 38% wats to sed 5 divisios. Mike ad Sarah each choose to take a positio o this issue: either 1, 2, 3, 4, or 5 divisios. Oce they choose their positios, each voter votes for the cadidate whose positio is closest to their ow. If the two cadidates are equally far away from a voter's positio, the half of the voters at that positio vote for oe cadidate, ad half vote for the other cadidate. For example, if Mike takes positio 1 ad Sarah takes positio 3, the their positios are equally far away from the 18% of voters at positio 2. Thus half of these voters (9%) vote for Mike ad half (9%) vote for Sarah. Each cadidate's payoff is the percetage of votes he or she receives. a. Model this as a strategic form game ad fid all pure strategy Nash equilibria. (6 poits)
13 b. Now there is a third player, Rush. Rush has three possible actios: he ca either edorse Mike, edorse Sarah, or ot make a edorsemet. If he chooses to edorse a cadidate, the the voters act the same as before except for oe chage. Now if two cadidates are equally far away from a voter's positio, the all voters vote for the cadidate whom Rush edorses. For example, if Mike takes positio 1 ad Sarah takes positio 3, the their positios are equally far away from the 18% of voters at positio 2. If Rush edorses Mike, all of these voters (18%) vote for Mike ad oe (0%) vote for Sarah. Of course, voters at positio 1 still vote for Mike ad voters at positios 3, 4, ad 5 still vote for Sarah. Rush's edorsemet affects oly those voters who would otherwise be idifferet betwee Mike ad Sarah. As before, Mike ad Sarah's payoffs are the percetage of votes they get. Rush's payoffs are as follows. If he edorses a cadidate, he gets a payoff of 1 if he edorses a cadidate who gets 50% of the vote or more; otherwise he gets a payoff of 0. If he does ot edorse a cadidate, the he gets a payoff of 1 if Mike ad Sarah tie ad get the same percetage of votes (because he looks like a statesma) but he gets a payoff of 0 if there is a clear wier. Fid all pure strategy Nash equilibria of this game. (Hit: oe ca aswer this questio without writig dow the whole game, although of course writig dow the whole game is oe way to do it.) (3 poits)
14 c. Agai, Rush has three possible actios: he ca either edorse Mike, edorse Sarah, or ot make a edorsemet. But ow oly the 38% of voters at positio 5 care about Rush's edorsemet. All other voters do ot care. For example, say Rush edorses Sarah. If Mike takes positio 2 ad Sarah takes positio 2, the their positios are equally far away from the 4% of voters at positio 3. Sice the voters at positio 3 could care less about Rush, half of these voters (2%) vote for Mike ad half (2%) vote for Sarah. But the 38% of voters at positio 5 care about Rush, ad all 38% vote for Sarah. Rush's edorsemet affects oly those voters at positio 5, ad oly whe they would otherwise be idifferet betwee Mike ad Sarah. Mike ad Sarah's payoffs, ad Rush's payoffs, are the same as before. Fid all pure strategy Nash equilibria of this game. (Hit: agai, oe ca aswer this questio without writig dow the whole game, although of course writig dow the whole game is oe way to do it.) (3 poits)
15 Part 7. Say there are two me A ad B ad two wome X ad Y. Each perso wats to match up with a member of the opposite sex. Each perso would rather be matched with someoe tha ot have a parter at all. Each perso raks his or her potetial parters from best to worst. a. (2 poits) Say that their prefereces are give by the followig tables. A B Best X Y Worst Y X Best Worst X Y A B I other words, ma A likes woma X best ad woma Y worst ad ma B likes woma Y best ad woma X least. Woma Y likes ma A best ad ma B least. Note that woma X's prefereces are left blak. I other words, X could either like A best ad B worst or he could like B best ad A worst. Is it possible to fill i woma X's prefereces so that there exists exactly oe stable match? If so, fill i her prefereces i the table above ad show that there exists exactly oe stable match. If ot, explai why ot. b. (2 poits) Say that their prefereces are agai give by the followig tables. A B Best X Y Worst Y X Best Worst X Y A B Is it possible to fill i woma X's prefereces so that there exist exactly two stable matches? If so, fill i her prefereces i the table above ad show that there exist exactly two stable matches. If ot, explai why ot.
16 c. (4 poits) Now say there are three me A, B, ad C, ad three wome X, Y, ad Z. Their prefereces are give by the followig tables. A B C Best X Y Z Y Z X Worst Z X Y X Y Z Best C A A B Worst B C Note that woma X's prefereces are left blak. Is it possible to fill i woma X's prefereces so that there exist exactly three stable matches? If so, fill i her prefereces i the table above ad show that there exist exactly three stable matches. If ot, explai why ot.
17 d. (4 poits) Agai, say that their prefereces are give by the followig tables. A B C Best X Y Z Y Z X Worst Z X Y X Y Z Best C A A B Worst B C Note that woma X's prefereces are left blak. Is it possible to fill i woma X's prefereces so that there exists exactly oe stable match? If so, fill i her prefereces i the table above ad show that there exists exactly oe stable match. If ot, explai why ot.
18 Part 8. A revolutioary group is tryig to start a isurgecy to topple the govermet. They eed to gather eough soldiers to costitute a serious threat. However, they face two hurdles: first they eed to covice people to participate ad secod they eed to keep them i the raks oce they ve joied. Sacrificig yourself for the good of others is oble but havig someoe else die for your coutry is easier o the wardrobe. Thus, eve if you support the revolutio, there is a great temptatio to say, Why do they eed me? I ca drop out ad free ride o the isurgets success. These dyamics are depicted below i a threshold model. There are two groups i the populatio, the Sympathizers (S) ad the Moderates (M), ad eve withi these groups there is a good mix of thresholds, as show below. Lower threshold Upper threshold t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 S 0 9 S 1 2 S 1 2 S 2 2 M 3 3 M 3 4 M 4 6 M 6 6 M 7 9 A perso ca either participate (p) i the isurgecy or ot participate (). As explaied i class, a perso participates if the total umber of other people who participate is greater tha or equal to her lower threshold ad less tha or equal to her upper threshold. a. Say that they start from a situatio (t=0) i which o oe participates. Fill i the table above to show how participatio chages over time. How may people will participate i the isurgecy at t=3? How may people will participate i the isurgecy at t=5? (4 poits)
19 (Part 8 cotiued) The isurgets ca wi if they get 7 or more soldiers. However, they will lose ad be crushed by the govermet if they ever drop below 3 soldiers after t=1. The revolutioary leaders have two strategies they ca follow to try to wi: they ca either resort to coercio to keep soldiers or give out beefits to ecourage recruits. If they resort to coercio they will raise the upper thresholds of the sympathizers by 2 but raise the lower thresholds of the moderates by 2. If they decide to give out beefits they lower the lower thresholds of the sympathizers by 2 ad lower the upper thresholds of the moderates by 2. b. Say that they resort to coercio. Agai, assume that they start from a situatio i which o oe participates. Will they wi or lose if they resort to coercio? You ca use the table below to help work out the problem. (4 poits) Lower threshold S S S S M M M M M Upper threshold t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8
20 c. Now say that they give out beefits. Agai, assume that they start from a situatio i which o oe participates. Will they wi or lose if they give out beefits? You ca use the table below to help work out the problem. (4 poits) Lower threshold S S S S M M M M M Upper threshold t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8
lecture notes September 2, Sequential Choice
18.310 lecture otes September 2, 2013 Sequetial Choice Lecturer: Michel Goemas 1 A game Cosider the followig game. I have 100 blak cards. I write dow 100 differet umbers o the cards; I ca choose ay umbers
More informationLogarithms APPENDIX IV. 265 Appendix
APPENDIX IV Logarithms Sometimes, a umerical expressio may ivolve multiplicatio, divisio or ratioal powers of large umbers. For such calculatios, logarithms are very useful. They help us i makig difficult
More informationPart 1. Midterm exam PS 30 November 2012
Last name First name Student ID number Part 1 Midterm exam PS 30 November 2012 This is a closed book exam. The only thing you can take into this exam is yourself and writing instruments. No calculators,
More informationGrade 6 Math Review Unit 3(Chapter 1) Answer Key
Grade 6 Math Review Uit (Chapter 1) Aswer Key 1. A) A pottery makig class charges a registratio fee of $25.00. For each item of pottery you make you pay a additioal $5.00. Write a expressio to represet
More informationJoin a Professional Association
Joi a Professioal Associatio 1. The secret job resource: professioal orgaizatios. You may ot kow this, but the career field you re i, or plaig to work i i the future, probably has at least oe professioal
More informationEMCdownload. Acknowledgements. Fair use
EMC_Sulight.idd 1 28/03/2013 09:06 Ackowledgemets Writte by Aa Sarchet, with Kate Oliver Edited by Kate Oliver Frot cover: Rebecca Scambler, 2013 Published by The Eglish ad Media Cetre, 2013 for EMCdowload.co.uk
More information13 Legislative Bargaining
1 Legislative Bargaiig Oe of the most popular legislative models is a model due to Baro & Ferejoh (1989). The model has bee used i applicatios where the role of committees have bee studies, how the legislative
More informationPERMUTATIONS AND COMBINATIONS
www.sakshieducatio.com PERMUTATIONS AND COMBINATIONS OBJECTIVE PROBLEMS. There are parcels ad 5 post-offices. I how may differet ways the registratio of parcel ca be made 5 (a) 0 (b) 5 (c) 5 (d) 5. I how
More informationExtra Practice 1. Name Date. Lesson 1.1: Patterns in Division
Master 1.22 Extra Practice 1 Lesso 1.1: Patters i Divisio 1. Which umbers are divisible by 4? By 5? How do you kow? a) 90 b) 134 c) 395 d) 1724 e) 30 f) 560 g) 3015 h) 74 i) 748 2. Write a 5-digit umber
More informationCh 9 Sequences, Series, and Probability
Ch 9 Sequeces, Series, ad Probability Have you ever bee to a casio ad played blackjack? It is the oly game i the casio that you ca wi based o the Law of large umbers. I the early 1990s a group of math
More informationPERMUTATION AND COMBINATION
MPC 1 PERMUTATION AND COMBINATION Syllabus : Fudametal priciples of coutig; Permutatio as a arragemet ad combiatio as selectio, Meaig of P(, r) ad C(, r). Simple applicatios. Permutatios are arragemets
More informationAS Exercise A: The multiplication principle. Probability using permutations and combinations. Multiplication principle. Example.
Probability usig permutatios ad combiatios Multiplicatio priciple If A ca be doe i ways, ad B ca be doe i m ways, the A followed by B ca be doe i m ways. 1. A die ad a coi are throw together. How may results
More informationCOMBINATORICS 2. Recall, in the previous lesson, we looked at Taxicabs machines, which always took the shortest path home
COMBINATORICS BEGINNER CIRCLE 1/0/013 1. ADVANCE TAXICABS Recall, i the previous lesso, we looked at Taxicabs machies, which always took the shortest path home taxipath We couted the umber of ways that
More information2. There are n letter and n addressed envelopes. The probability that all the letters are not kept in the right envelope, is. (c)
PAGE # CHAPTER EXERCISE I. A sigle letter is selected at radom from the word PROBABILITY. The probability that the selected letter is a vowel is / / / 0. There are letter ad addressed evelopes. The probability
More information5 Quick Steps to Social Media Marketing
5 Quick Steps to Social Media Marketig Here's a simple guide to creatig goals, choosig what to post, ad trackig progress with cofidece. May of us dive ito social media marketig with high hopes to watch
More informationx y z HD(x, y) + HD(y, z) HD(x, z)
Massachusetts Istitute of Techology Departmet of Electrical Egieerig ad Computer Sciece 6.02 Solutios to Chapter 5 Updated: February 16, 2012 Please sed iformatio about errors or omissios to hari; questios
More informationPermutation Enumeration
RMT 2012 Power Roud Rubric February 18, 2012 Permutatio Eumeratio 1 (a List all permutatios of {1, 2, 3} (b Give a expressio for the umber of permutatios of {1, 2, 3,, } i terms of Compute the umber for
More information15 min/ Fall in New England
5 mi/ 0+ -4 Fall i New Eglad Before witer makes its appearace, a particularly warm fall bathes the forest i a golde shimmer. Durig the Idia Summer, New Eglad blossoms oe last time. Treetops are ablaze
More informationMath 7 Flipped Mastery Self Tester Worksheet Name: Class:. Chapter 1 (Unit 1) Patterns and Relationships - Accommodated 1.1 Patterns In Division /36
Chapter 1 (Uit 1) Patters ad Relatioships - Accommodated 1.1 Patters I Divisio /36 Divisibility Rule Cheats; A whole umber is divisible by 2 if it is a eve umber A whole umber is divisible by 4 if the
More informationUnit 5: Estimating with Confidence
Uit 5: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Uit 5 Estimatig with Cofidece 8.1 8.2 8.3 Cofidece Itervals: The Basics Estimatig a Populatio
More informationRoberto s Notes on Infinite Series Chapter 1: Series Section 2. Infinite series
Roberto s Notes o Ifiite Series Chapter : Series Sectio Ifiite series What you eed to ow already: What sequeces are. Basic termiology ad otatio for sequeces. What you ca lear here: What a ifiite series
More informationCopywriting. for your. legacy. giving. website
Copywritig Basics for your legacy givig website www.imarketsmart.com www.imarketsmart.com 301.289.3670 1 You ve decided to tap ito the greatest trasfer of wealth that the world has ever see by buildig
More informationSpeak up Ask questions Find the facts Evaluate your choices Read the label and follow directions
Whe it comes to usig medicie, it is importat to kow that o medicie is completely safe. The U.S. Food ad Drug Admiistratio (FDA) judges a drug to be safe eough to approve whe the beefits of the medicie
More informationPERMUTATIONS AND COMBINATIONS
Chapter 7 PERMUTATIONS AND COMBINATIONS Every body of discovery is mathematical i form because there is o other guidace we ca have DARWIN 7.1 Itroductio Suppose you have a suitcase with a umber lock. The
More informationCombinatorics. Chapter Permutations. Reading questions. Counting Problems. Counting Technique: The Product Rule
Chapter 3 Combiatorics 3.1 Permutatios Readig questios 1. Defie what a permutatio is i your ow words. 2. What is a fixed poit i a permutatio? 3. What do we assume about mutual disjoitedess whe creatig
More information202 Chapter 9 n Go Bot. Hint
Chapter 9 Go Bot Now it s time to put everythig you have leared so far i this book to good use. I this chapter you will lear how to create your first robotic project, the Go Bot, a four-wheeled robot.
More informationVIII. Shell-Voicings
VIII. Shell-Voicigs A. The Cocept The 5th (ad ofte the root as well) ca be omitted from most 7th-chords. Ratioale: Most chords have perfect 5ths. The P5th is also preset as the rd partial i the overtoe
More informationMDM 4U MATH OF DATA MANAGEMENT FINAL EXAMINATION
Caadia Iteratioal Matriculatio rogramme Suway Uiversity College MDM 4U MTH OF DT MNGEMENT FINL EXMINTION Date: November 28 th, 2006 Time: 11.30a.m 1.30p.m Legth: 2 HOURS Lecturers: lease circle your teacher
More informationarxiv: v2 [math.co] 15 Oct 2018
THE 21 CARD TRICK AND IT GENERALIZATION DIBYAJYOTI DEB arxiv:1809.04072v2 [math.co] 15 Oct 2018 Abstract. The 21 card trick is well kow. It was recetly show i a episode of the popular YouTube chael Numberphile.
More informationCrafting Well-Built Sentences. Varying Sentence Patterns. Breaking the Rules to Create Fluency. Capturing a Smooth and Rhythmic Flow
SENTENCE FLUENCY k e y q u a l i t i e s Craftig Well-Built Seteces Varyig Setece Patters Breakig the Rules to Create Fluecy Capturig a Smooth ad Rhythmic Flow crafting WELL-BUILT SENTENCES Do my seteces
More informationTable Of Contents Blues Turnarounds
Table Of Cotets Blues Turarouds Turaroud #1 Turaroud # Turaroud # Turaroud # Turaroud # Turaroud # Turaroud # Turaroud # Turaroud # Blues Turarouds Blues Soloig Masterclass Week 1 Steve Stie A Blues Turaroud
More informationRoom Design [ HOW TO SET UP YOUR EVENT SPACE ]
Room Desig [ HOW TO SET UP YOUR EVENT SPACE ] There are so may compoets of plaig a evet ad so may decisios! I this article you will lear about some factors that will help you choose the best space for
More informationCOLLEGE ALGEBRA LECTURES Copyrights and Author: Kevin Pinegar
OLLEGE ALGEBRA LETURES opyrights ad Author: Kevi iegar 8.3 Advaced outig Techiques: ermutatios Ad ombiatios Factorial Notatio Before we ca discuss permutatio ad combiatio formulas we must itroduce factorial
More informationWe often find the probability of an event by counting the number of elements in a simple sample space.
outig Methods We ofte fid the probability of a evet by coutig the umber of elemets i a simple sample space. Basic methods of coutig are: Permutatios ombiatios Permutatio A arragemet of objects i a defiite
More informationLecture 4: Frequency Reuse Concepts
EE 499: Wireless & Mobile Commuicatios (8) Lecture 4: Frequecy euse Cocepts Distace betwee Co-Chael Cell Ceters Kowig the relatio betwee,, ad, we ca easily fid distace betwee the ceter poits of two co
More informationEnroll now and discover. In one powerful day, you ll learn how to accomplish your goals with
I oe powerful day, you ll lear how to accomplish your goals with Eroll ow ad discover 4 How to set challegig goals that make sese 4 The importace of havig a pla ad the techiques to create oe 4 How to evaluate
More information}, how many different strings of length n 1 exist? }, how many different strings of length n 2 exist that contain at least one a 1
1. [5] Give sets A ad B, each of cardiality 1, how may fuctios map A i a oe-tooe fashio oto B? 2. [5] a. Give the set of r symbols { a 1, a 2,..., a r }, how may differet strigs of legth 1 exist? [5]b.
More informationIntroduction to Wireless Communication Systems ECE 476/ECE 501C/CS 513 Winter 2003
troductio to Wireless Commuicatio ystems ECE 476/ECE 501C/C 513 Witer 2003 eview for Exam #1 March 4, 2003 Exam Details Must follow seatig chart - Posted 30 miutes before exam. Cheatig will be treated
More informationGeneral Model :Algorithms in the Real World. Applications. Block Codes
Geeral Model 5-853:Algorithms i the Real World Error Correctig Codes I Overview Hammig Codes Liear Codes 5-853 Page message (m) coder codeword (c) oisy chael decoder codeword (c ) message or error Errors
More informationDensity Slicing Reference Manual
Desity Slicig Referece Maual Improvisio, Viscout Cetre II, Uiversity of Warwick Sciece Park, Millbur Hill Road, Covetry. CV4 7HS Tel: 0044 (0) 24 7669 2229 Fax: 0044 (0) 24 7669 0091 e-mail: admi@improvisio.com
More informationTeenHealth Quarter. Wise. Making. Choices. free. The way you respond to choices can change your life or someone else s.
TeeHealth Quarter 4 2012 Makig Wise Choices You make choices every day. The choice may be what to wear to school, how to eat healthy, whether to wear a seat belt or drik ad drive. ALL choices have cosequeces.
More informationHomework 5 Answers PS 30 November 2013
Homework 5 Answers PS 30 November 2013 Problems which you should be able to do easily 1. Consider the Battle of the Sexes game below. 1a 2, 1 0, 0 1b 0, 0 1, 2 a. Find all Nash equilibria (pure strategy
More informationCS3203 #5. 6/9/04 Janak J Parekh
CS3203 #5 6/9/04 Jaak J Parekh Admiistrivia Exam o Moday All slides should be up We ll try ad have solutios for HWs #1 ad #2 out by Friday I kow the HW is due o the same day; ot much I ca do, uless you
More informationINSTRUCTIONS: all the calculations on the separate piece of paper which you do not hand in. GOOD LUCK!
INSTRUCTIONS: 1) You should hand in ONLY THE ANSWERS ASKED FOR written clearly on this EXAM PAPER. You should do all the calculations on the separate piece of paper which you do not hand in. 2) Problems
More informationSwan Island. Brilliant bikes SAMPLE. Frank s funny shopping trip. A cake for a queen. Katy s favourite song. William s strangest story
2 s t e t Co Swa Islad Brilliat bikes 3 5 2 Frak s fuy shoppig trip 20 A cake for a quee 28 Katy s favourite sog 36 6 William s stragest story The past ad the future 52 The school of sciece 60 Let s have
More informationCS 201: Adversary arguments. This handout presents two lower bounds for selection problems using adversary arguments ëknu73,
CS 01 Schlag Jauary 6, 1999 Witer `99 CS 01: Adversary argumets This hadout presets two lower bouds for selectio problems usig adversary argumets ëku73, HS78, FG76ë. I these proofs a imagiary adversary
More informationChapter (6) Discrete Probability Distributions Examples
hapter () Discrete robability Distributios Eamples Eample () Two balaced dice are rolled. Let X be the sum of the two dice. Obtai the probability distributio of X. Solutio Whe the two balaced dice are
More informationFive things you should know about our hemophilia care program
Five thigs you should kow about our hemophilia care program 1 We re specialists i hemophilia care 1 If you re readig this booklet, you re probably livig with hemophilia or aother bleedig disorder. Or you
More informationALIGNING CHANGE MANAGEMENT WITH AGILE
LIGNING CHNGE MNGEMENT WITH GILE Crowd Sourced Webiar 15 September 2016 Preseted by: Vicky Emery, rector of Chage Capability Services Louise Geoghega, Seior Cosultat PollEv.com/beighuma Itroducig Beig
More informationHow to write reports in plain English. How to write reports in plain English
How to write reports i plai Eglish How to write reports i plai Eglish Plai Eglish Campaig 2001 How to write reports i plai Eglish Itroductio Welcome to the plai Eglish report-writig course. All you eed
More informationTechnical Explanation for Counters
Techical Explaatio for ers CSM_er_TG_E Itroductio What Is a er? A er is a device that couts the umber of objects or the umber of operatios. It is called a er because it couts the umber of ON/OFF sigals
More informationAs an Exceptional Student in Intellectual Disabilities. You Are Cordially Invited to be Seen and Recognized as a Future Leader in the Field
As a Exceptioal Studet i Itellectual Disabilities You Are Cordially Ivited to be See ad Recogized as a Future Leader i the Field You Caot Start Too Early To Begi Your Rise To Leadership i Our Field You
More information20. CONFIDENCE INTERVALS FOR THE MEAN, UNKNOWN VARIANCE
20. CONFIDENCE INTERVALS FOR THE MEAN, UNKNOWN VARIANCE If the populatio tadard deviatio σ i ukow, a it uually will be i practice, we will have to etimate it by the ample tadard deviatio. Sice σ i ukow,
More informationSingle Bit DACs in a Nutshell. Part I DAC Basics
Sigle Bit DACs i a Nutshell Part I DAC Basics By Dave Va Ess, Pricipal Applicatio Egieer, Cypress Semicoductor May embedded applicatios require geeratig aalog outputs uder digital cotrol. It may be a DC
More informationShuffling Cards. D.J.W. Telkamp. Utrecht University Mathematics Bachelor s Thesis. Supervised by Dr. K. Dajani
Shufflig Cards Utrecht Uiversity Mathematics Bachelor s Thesis D.J.W. Telkamp Supervised by Dr. K. Dajai Jue 3, 207 Cotets Itroductio 2 2 Prerequisites 2 2. Problems with the variatio distace................
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 12
EECS 70 Discrete Mathematics ad Probability Theory Sprig 204 Aat Sahai Note 2 Probability Examples Based o Coutig We will ow look at examples of radom experimets ad their correspodig sample spaces, alog
More informationInspire. Your Team. Switched. and. How to Engage. Interview with Chris Atkinson
R Thik Critically i Ifluece Profoudly i Iovate like a Etrepreeur ISSUE 33 Iterview with Chris Atkiso How to Egage Ispire Your Team ad How to Egage Ispire Your Team ad Photo credit: Dmitriy Shiroosov/123RF.com
More informationHOW TO ATTRACT MORE PROSPECTS TO YOUR COMMUNITY
HOW TO ATTRACT MORE PROSPECTS TO YOUR COMMUNITY A Free Guide for Ecoomic Developmet Orgaizatios facebook.com/platiumprfirm @Platium_PR WHAT S THE KEY TO ATTRACTING MORE PROSPECTS TO YOUR COMMUNITY? Perhaps
More informationIdentifying poor performers while the process is running
Idetifyig poor performers while the process is ruig KEYWORDS Michel Ruel P.E. Presidet, TOP Cotrol Ic 4734 Soseeahray Drive 49, Bel-Air St, #103 Hubertus, WI 53033 Levis Qc G6W 6K9 USA Caada mruel@topcotrol.com
More informationMidterm 1 - Solutions
Ec 102 - Aalyi of Ecoomic Data Uiverity of Califoria - Davi Jauary 28, 2010 Itructor: Joh Parma Midterm 1 - Solutio You have util 10:20am to complete thi exam. Pleae remember to put your ame, ectio ad
More informationCP 405/EC 422 MODEL TEST PAPER - 1 PULSE & DIGITAL CIRCUITS. Time: Three Hours Maximum Marks: 100
PULSE & DIGITAL CIRCUITS Time: Three Hours Maximum Marks: 0 Aswer five questios, takig ANY TWO from Group A, ay two from Group B ad all from Group C. All parts of a questio (a, b, etc. ) should be aswered
More information7. Counting Measure. Definitions and Basic Properties
Virtual Laboratories > 0. Foudatios > 1 2 3 4 5 6 7 8 9 7. Coutig Measure Defiitios ad Basic Properties Suppose that S is a fiite set. If A S the the cardiality of A is the umber of elemets i A, ad is
More information1. How many possible ways are there to form five-letter words using only the letters A H? How many such words consist of five distinct letters?
COMBINATORICS EXERCISES Stepha Wager 1. How may possible ways are there to form five-letter words usig oly the letters A H? How may such words cosist of five distict letters? 2. How may differet umber
More informationFingerprint Classification Based on Directional Image Constructed Using Wavelet Transform Domains
7 Figerprit Classificatio Based o Directioal Image Costructed Usig Wavelet Trasform Domais Musa Mohd Mokji, Syed Abd. Rahma Syed Abu Bakar, Zuwairie Ibrahim 3 Departmet of Microelectroic ad Computer Egieerig
More informationPRACTICAL FILTER DESIGN & IMPLEMENTATION LAB
1 of 7 PRACTICAL FILTER DESIGN & IMPLEMENTATION LAB BEFORE YOU BEGIN PREREQUISITE LABS Itroductio to Oscilloscope Itroductio to Arbitrary/Fuctio Geerator EXPECTED KNOWLEDGE Uderstadig of LTI systems. Laplace
More informationRefer to the guidance notes when you see this icon. The notes will give you extra information to help you complete the form correctly.
PR1 Applicatio for studet fiace - cotiuig studets 2017/18 Foreame(s) Surame Refer to the guidace otes whe you see this ico. The otes will give you extra iformatio to help you complete the form correctly.
More informationLX 422/722 Intermediate Syntax KEY (with notes) SPRING 2018 FINAL 38 points total; 22 for #1, 2 for #2, 7 for #3, 1 for #4, 6 for #5
LX 422/722 Itermediate Sytax KEY (with otes) SPRIG 2018 FIAL 38 poits total; 22 for #1, 2 for #2, 7 for #3, 1 for #4, 6 for #5 SEEES FOR PROBLEM #1 (i) (ii) (iii) Pat seems to kow who will wi. Pat s fried
More informationTHE LUCAS TRIANGLE RECOUNTED. Arthur T. Benjamin Dept. of Mathematics, Harvey Mudd College, Claremont, CA Introduction
THE LUCAS TRIANLE RECOUNTED Arthur T Bejami Dept of Mathematics, Harvey Mudd College, Claremot, CA 91711 bejami@hmcedu 1 Itroductio I 2], Neville Robbis explores may properties of the Lucas triagle, a
More informationONDURA-9. 9-Corrugation Asphalt Roofing Sheets I N S T A L L A T I O N I N S T R U C T I O N S
ONDURA-9 9-Corrugatio Asphalt Roofig Sheets I N S T A L L A T I O N I N S T R U C T I O N S Thak you for choosig ONDURA-9 for your roofig project. ONDURA-9 should be carefully istalled. Mistakes i istallatio
More informationGames in Extensive Form
Games in Extensive Form the extensive form of a game is a tree diagram except that my trees grow sideways any game can be represented either using the extensive form or the strategic form but the extensive
More informationIn the history of woodworking joinery,
joiery made simple Box-Joit Basics Create Simple, Strog, ad Hadsome Corers By Craig Betzley I the history of woodworkig joiery, box joits (sometimes called figer joits) are a fairly recet arrival. That
More informationBench slave. Tips & Tricks
Tips & Tricks Bech slave I have a couple of these slaves mouted uder my workbech top to allow for easy clamp-free support of log stock for edge plaig. Each uit is basically a bracket that houses a stick
More informationCounting III. Today we ll briefly review some facts you dervied in recitation on Friday and then turn to some applications of counting.
6.04/18.06J Mathematics for Computer Sciece April 5, 005 Srii Devadas ad Eric Lehma Lecture Notes Coutig III Today we ll briefly review some facts you dervied i recitatio o Friday ad the tur to some applicatios
More information8. Combinatorial Structures
Virtual Laboratories > 0. Foudatios > 1 2 3 4 5 6 7 8 9 8. Combiatorial Structures The purpose of this sectio is to study several combiatorial structures that are of basic importace i probability. Permutatios
More informationAMC AMS AMR ACS ACR ASR MSR MCR MCS CRS
Sectio 6.5: Combiatios Example Recall our five frieds, Ala, Cassie, Maggie, Seth ad Roger from the example at the begiig of the previous sectio. The have wo tickets for a cocert i Chicago ad everybody
More informationMath 140 Introductory Statistics
6. Probability Distributio from Data Math Itroductory Statistics Professor Silvia Ferádez Chapter 6 Based o the book Statistics i Actio by A. Watkis, R. Scheaffer, ad G. Cobb. We have three ways of specifyig
More informationString and Fan inlay Made Simple
Strig ad Fa ilay Made Simple Pre-made parts let you focus o the fu. By Gle Jewell with Joe Hurst-Wajszczuk www.woodcraftmagazie.com straightforward ad easy to master. Like a clever magic trick, ilay looks
More informationHELIARC. THE FIRST NAME IN TIG.
HELIARC. THE FIRST NAME IN TIG. YOU AND HELIARC. NOT EVERYONE APPRECIATES THE BEAUTY OF A TRULY GREAT WELD. BUT YOU DO. YOU VE PUT IN THE YEARS AND MASTERED THE ART AND CRAFT OF GTAW (TIG). AND EVER SINCE
More informationCOS 126 Atomic Theory of Matter
COS 126 Atomic Theory of Matter 1 Goal of the Assigmet Video Calculate Avogadro s umber Usig Eistei s equatios Usig fluorescet imagig Iput data Output Frames Blobs/Beads Estimate of Avogadro s umber 7.1833
More informationLaboratory Exercise 3: Dynamic System Response Laboratory Handout AME 250: Fundamentals of Measurements and Data Analysis
Laboratory Exercise 3: Dyamic System Respose Laboratory Hadout AME 50: Fudametals of Measuremets ad Data Aalysis Prepared by: Matthew Beigto Date exercises to be performed: Deliverables: Part I 1) Usig
More informationA generalization of Eulerian numbers via rook placements
A geeralizatio of Euleria umbers via rook placemets Esther Baaia Steve Butler Christopher Cox Jeffrey Davis Jacob Ladgraf Scarlitte Poce Abstract We cosider a geeralizatio of Euleria umbers which cout
More informationMEI Core 2. Logarithms and exponentials. Section 2: Modelling curves using logarithms. Modelling curves of the form y kx
MEI Core 2 Logarithms ad eoetials Sectio 2: Modellig curves usig logarithms Notes ad Eamles These otes cotai subsectios o: Modellig curves of the form y = k Modellig curves of the form y = ka Modellig
More informationCombinatorics. ChaPTer a The addition and multiplication principles introduction. The addition principle
ChaPTer Combiatorics ChaPTer CoTeTS a The additio ad multiplicatio priciples b Permutatios C Factorials D Permutatios usig P r e Permutatios ivolvig restrictios F Arragemets i a circle G Combiatios usig
More informationAQA Level 2 Further mathematics Further algebra. Section 3: Inequalities and indices
AQA Level Further mthemtics Further lgebr Sectio : Iequlities d idices Notes d Emples These otes coti subsectios o Iequlities Lier iequlities Qudrtic iequlities Multiplyig epressios The rules of idices
More informationRUBE GOLDBERG MACHINE CONTEST INTRODUCTION TO HOSTING. Apprentice Division (Ages 8-11) New This Year:
1988 RUBE GOLDBERG MACHINE CONTEST Rube Goldberg, Ic. WEB: rubegoldberg.com EMAIL: rube@rubegoldberg.com OCTOBER New This Year: Appretice Divisio (Ages 8-11) HOST MANUAL 1 Eve though he d be 135 years
More informationWindWise Education LESSON 11: HOW CAN I DESIGN BETTER BLADES?
WidWise Educatio Wid Eergy Activities for Studets Cotets LESSON 11: HOW CAN I DESIGN BETTER BLADES? Forward...1 Itroductio...7 Uit 1: Eergy...17 Uit 2: Wid... 75 Uit 3: Turbies.... 147 Lesso 8 How Does
More informationArithmetic Sequences and Series Sequences and Series Preliminary Maths
Arithmetic Sequeces ad Series Arithmetic Sequeces ad Series Sequeces ad Series Prelimiary Maths www.primeeducatio.com.au Arithmetic Sequeces ad Series Sequeces ad Series 1 Questio 1 The first 5 terms of
More informationA Novel Three Value Logic for Computing Purposes
Iteratioal Joural o Iormatio ad Electroics Egieerig, Vol. 3, No. 4, July 23 A Novel Three Value Logic or Computig Purposes Ali Soltai ad Saeed Mohammadi Abstract The aim o this article is to suggest a
More informationThe Institute of Chartered Accountants of Sri Lanka
The Istitute of Chartered Accoutats of Sri Laka Postgraduate Diploma i Busiess ad Fiace Quatitative Techiques for Busiess Hadout 02:Presetatio ad Aalysis of data Presetatio of Data The Stem ad Leaf Display
More informationControl Charts MEC-13. Causes of Variation 12/3/2016
Variatio due to Assigable Causes Variatio mostly due to Commo Causes Variatio due to Assigable Causes Outlie Basic Terms MEC-13 Cotrol Charts Types of Cotrol Charts with their purpose Creatig Cotrol Charts
More informationPUBLIC VALUE STUDENT AWARD 2016
PUBLIC VALUE STUDENT AWARD 2016 Youg Talets Create Societies Good ideas move the world. Good projects chage society. This is exactly what the ew if PUBLIC VALUE STUDENT AWARD is about: cocepts ad product
More informationONDURA. Corrugated Asphalt Roofing Sheets I N S T A L L A T I O N I N S T R U C T I O N S
ONDURA Corrugated Asphalt Roofig Sheets I N S T A L L A T I O N I N S T R U C T I O N S Thak you for choosig ONDURA for your roofig project. ONDURA should be carefully istalled. Mistakes i istallatio ca
More information9/26/18. + Where to Start??? + Take Inventory: What Do They Want? Crafting a Successful Personal Statement. Sample Prompts.
Craftig a Successful Persoal Statemet Deae Gute, 2018 Where to Start??? Take Ivetory: What Do They Wat? Sample Prompts Please provide a statemet, o more tha two pages i legth, about your persoal history,
More informationStudents and Income Tax
Studets ad Icome Tax 3(5HY Visually impaired persos ca order publicatios i braille or large prit, or o audio cassette or computer diskette, by callig 1-800-267-1267 weekdays from 8:15 a.m. to 5:00 p.m.
More informationA Math Learning Center publication adapted and arranged by. EUGENE MAIER and LARRY LINNEN
A Math Learig Ceter publicatio adapted ad arraged by EUGENE MAIER ad LARRY LINNEN ALGEBRA THROUGH VISUAL PATTERNS, VOLUME 1 A Math Learig Ceter Resource Copyright 2005, 2004 by The Math Learig Ceter, PO
More informationX-Bar and S-Squared Charts
STATGRAPHICS Rev. 7/4/009 X-Bar ad S-Squared Charts Summary The X-Bar ad S-Squared Charts procedure creates cotrol charts for a sigle umeric variable where the data have bee collected i subgroups. It creates
More informationColor profile: Disabled Composite Default screen BC60XLT. Programmable Hand-Held Scanner OPERATING GUIDE
BC60XLT Programmable Had-Held Scaer OPERATING GUIDE Mo Aug 30 15:33:43 1999 PRECAUTIONS Before you use this scaer, please read ad observe the followig: WARNING! Uide DOES NOT represet this uit to be waterproof.
More informationYou Think You ve Got Problems? Marc Brodie Associate Professor of Mathematics, WJU
You Thik You ve Got Problems? Marc Brodie Associate Professor of Mathematics, WJU Itroductio. My life, like that of ay other s, has its share of problems. I cosider myself fortuate, however, to have more
More informationApplication of Improved Genetic Algorithm to Two-side Assembly Line Balancing
206 3 rd Iteratioal Coferece o Mechaical, Idustrial, ad Maufacturig Egieerig (MIME 206) ISBN: 978--60595-33-7 Applicatio of Improved Geetic Algorithm to Two-side Assembly Lie Balacig Ximi Zhag, Qia Wag,
More informationDepartment of Electrical and Computer Engineering, Cornell University. ECE 3150: Microelectronics. Spring Due on April 26, 2018 at 7:00 PM
Departmet of Electrical ad omputer Egieerig, orell Uiersity EE 350: Microelectroics Sprig 08 Homework 0 Due o April 6, 08 at 7:00 PM Suggested Readigs: a) Lecture otes Importat Notes: ) MAKE SURE THAT
More information