...colour? Who will be going to Macao? When can we get the train tickets? Whose book is this? How many pupils are in the class? Where did you go yesterday? | E.g. Where did you go yesterday? | I went to the cinema. | | 1. | Question 3 is the easiest. | 2. | This is my book. | 3. | My mother and I. | 4. | Thirty-two. | 5. | They will be ready on Friday. | 6. | Green. | Exercise 2: Answer the questions in complete sentences. E.g. Have they brushed the teeth? | Yes, they have brushed the teeth. | | 1. Has he finished his homework? | No, | 2. Has he watered the plants? | Yes, | 3. Have you drawn the picture? | No, | 4. Have you cleaned the floor? | Yes, | 5. Has she closed the door? | Yes, | 6. Has she cooked the dinner? | Yes, | Exercise 3: Answer the questions using ‘ever’ or ‘never’. E.g. Has Josephine ever drunk wine? | No, she has never drunk wine. | | 1. Have you ever eaten beef? | No, | 2. Has Mimi ever killed an insect? | No, | 3. Has he ever drunk coffee? | Yes, | 4. Have you ever been to a theatre? | Yes, | 5. Has Yoyo ever been to a zoo? | No, | 6. Have you ever played ping-pong? | Yes, | Exercise 4: Martin has received a letter from his mother. Fill in the blanks using the correct tense of the given verbs. Dear Martin, How are you? I have got some good news for you. Your sister is getting married this summer. Uncle Paul and Aunt Mary will __________ (come) to attend the wedding ceremony...
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...------------------------------------------------- Monopoly From Wikipedia, the free encyclopedia This article is about the economic term. For the board game, see Monopoly (game). For other uses, seeMonopoly (disambiguation). "I Like a Little Competition"—J. P. Morgan by Art Young. Cartoon relating to the answer J. P. Morgan gave when asked whether he disliked competition at the Pujo Committee.[1] A monopoly (from Greek monos μόνος (alone or single) + polein πωλεῖν (to sell)) exists when a specific person or enterprise is the only supplier of a particular commodity (this contrasts with a monopsony which relates to a single entity's control of a market to purchase a good or service, and with oligopoly which consists of a few entities dominating an industry).[2]Monopolies are thus characterized by a lack of economic competition to produce the good or service and a lack of viable substitute goods.[3] The verb "monopolise" refers to the process by which a company gains the ability to raise prices or exclude competitors. In economics, a monopoly is a single seller. In law, a monopoly is a business entity that has significant market power, that is, the power to charge high prices.[4] Although monopolies may be big businesses, size is not a characteristic of a monopoly. A small business may still have the power to raise prices in a small industry (or market).[4] A monopoly is distinguished from a monopsony, in which there is only one buyer of a product or service; a monopoly may...
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...a) i) P(Beach next year | Beach this year)=0.8 P(Mountains this year | Beach this year)=0.2 b) i) P(Beach in two years | Beach this year)=0.72 P(Mountains in two years | Beach this year) = 0.28 a) ii) P(Beach next year Mountains this year)=0.4 P(Mountains next year Mountains this year) = 0.6 | | b) ii) P(Beach in two years time | Mountains this year)=0.56 P(Mountains in two years | Mountains this year) = 0.44 c) Tried three different initial state vectors and all converged to the same long run probabilities (see next page). The long run probability of you going to the beach is 0.66667 and the long run probability of you going to the mountains is 0.33333 which are 2/3 and 1/3 rounded to 5 decimal places. 1 Practical Problem 2 a) P(Purchasing Brand B four purchases from now|equally likely to purchase each brand initially)=0.36275 P(Purchasing Brand C four purchases from now|equally likely to purchase each brand initially)=0.33586 P(Purchasing Brand A four purchases from now|equally likely to purchase each brand initially)=0.30141 2 Output for b), c) and d) respectively. b) Look at printout in first column of table. P(Purchase Brand A after 4 purchases | purchased brand A this time) =0.6808, P(Purchase Brand B after 4 purchases | purchased brand A this time) =0.29396, P(Purchase Brand C after 4 purchases | purchased brand A this time) =0.02525. c) Look at second column printout. P(Purchase Brand A after 4 purchases | purchased brand B this time)...
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...MAT 1348B Discrete Mathematics for Computer Science Winter 2011 Professor: Alex Hoffnung Dept. of Mathematics & Statistics, 585 King Edward (204B) email: hoffnung@uottawa.ca Important: Please include MAT1348 in the subject line of every email you send me. Otherwise your email may be deleted unread. Please do not use Virtual Campus to send me messages as I may not check them regularly. Course Webpages: This web page will contain detailed and up-to-date information on the course, including a detailed course outline and course policies, homework assignments, handouts to download etc. You are responsible for this information. Consult this page regularly. Timetable: Lectures: Mon. 2:30–4:00 pm, Thurs: 4:00–5:30 pm in STE B0138 Office hours: Mon. 4:00–5:00 pm, Thurs: 3:00 - 4:00 pm DGD: Wed. 10–11:30 am. Textbook: K. H. Rosen, Discrete Mathematics and Its Applications, 6th Edition, McGrawHill. We’ll be covering most of Chapters 1, 2, and 9, and parts of Chapters 4, 5, and 8. The course may contain a small of amount of material not covered by the textbook. This text has been used in Discrete Math courses at Ottawa U. for many years, so secondhand copies can easily be found. Copies of the book are at the bookstore or available from Amazon. Coursework Evaluation: The final grade will be calculated as follows: • 5 homework assignments : 10% • Midterm exam: 30% • Final exam: 60% The midterm test is on February 17 . 1 Note that students must pass the final exam...
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...their ticket, picking a random seat to occupy. All the other passengers are quite normal, and will go to their proper seat unless it is already occupied. If it is occupied, they will then find a free seat to sit in, at random. What is the probability that the last (100th) person to board the plane will sit in their proper seat (#100)? If one tries to solve this problem with conditional probability it becomes very difficult. We begin by considering the following cases if the first passenger sits in seat number 1, then all ∗ wax@alum.mit.edu 1 the remaining passengers will be in their correct seats and certainly the #100’th will also. If he sits in the last seat #100, then certainly the last passenger cannot sit there (in fact he will end up in seat #1). If he sits in any of the 98 seats between seats #1 and #100, say seat k, then all the passengers with seat numbers 2, 3, . . . , k − 1 will have empty seats and be able to sit in their respective seats. When the passenger with seat number k enters he will have as possible seating choices seat #1, one of the seats k + 1, k + 2, . . . , 99, or seat #100. Thus the options available to this passenger are the same options...
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...Notes on Probability Peter J. Cameron ii Preface Here are the course lecture notes for the course MAS108, Probability I, at Queen Mary, University of London, taken by most Mathematics students and some others in the first semester. The description of the course is as follows: This course introduces the basic notions of probability theory and develops them to the stage where one can begin to use probabilistic ideas in statistical inference and modelling, and the study of stochastic processes. Probability axioms. Conditional probability and independence. Discrete random variables and their distributions. Continuous distributions. Joint distributions. Independence. Expectations. Mean, variance, covariance, correlation. Limiting distributions. The syllabus is as follows: 1. Basic notions of probability. Sample spaces, events, relative frequency, probability axioms. 2. Finite sample spaces. Methods of enumeration. Combinatorial probability. 3. Conditional probability. Theorem of total probability. Bayes theorem. 4. Independence of two events. Mutual independence of n events. Sampling with and without replacement. 5. Random variables. Univariate distributions - discrete, continuous, mixed. Standard distributions - hypergeometric, binomial, geometric, Poisson, uniform, normal, exponential. Probability mass function, density function, distribution function. Probabilities of events in terms of random variables. 6. Transformations of a single random variable. Mean, variance, median,...
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...= 5 10 là số nguyên tố • Các mệnh đề thường được ký hiệu bởi P, Q, R,…,p, q… và chân trị của mệnh đề ký hiệu T (đúng) F (sai) 3 PHÉP TOÁN MỆNH ĐỀ • Các phép toán mệnh đề còn gọi là phép nối logic (Logical connectives): ¬ (phủ định), ∧ (hội), ∨ (tuyển), → (kéo theo) • Tập các mệnh đề cùng với các phép toán logic tạo thành một đại số mệnh đề 4 PHÉP PHỦ ĐỊNH p T F ¬p F T p =“4 là số nguyên tố” thì ¬ p =“4 không là số nguyên tố” có chân trị là T 5 PHÉP HỘI p F F T T q F T F T p ∧q F F F T p =“12 là một số nguyên”, q =“12 chia hết cho 5” thì p ∧ q =“12 là một số nguyên chia hết cho 5”, có chân trị là F 6 PHÉP TUYỂN p F F T T q F T F T p∨q F T T T p =“12 là một số nguyên”, q =“12 chia hết cho 5” thì p ∨ q =“12 là một số nguyên hoặc 12 chia hết cho 5”, có chân trị là T 7 PHÉP TUYỂN p F F T T q F T F T p→ q T T F T p =“12 là một số nguyên”, q =“12 chia hết cho 5” thì p → q =“nếu 12 là một số nguyên thì chia hết cho 5”, có chân trị là F 8 LƯU Ý • p ∨ q sai khi và chỉ khi p và q đều sai • p ∧ q đúng khi và chỉ khi p và q đều đúng • p → q sai khi và chỉ khi p đúng q sai • Còn có phép p ↔ q (kéo theo hai chiều) phép toán này đúng khi và chỉ khi cả p và q cùng đúng hoặc cùng sai 9 DẠNG MỆNH ĐỀ • Dạng mệnh đề (Propositional form-PF), E(p, q, r,…), là một biểu thức logic chứa các hằng và các biến mệnh đề • Nếu P, Q là hai dạng mệnh đề thì ¬ P, P ∨ Q, P ∧ Q, P → Q, P ↔ Q cũng là các dạng mệnh đề • Thứ tự ưu tiên: ( ), ¬, ∧, ∨, → ...
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...Chapter 7—Introduction to Sampling Distributions When applicable, selected problems in each section will be done following the appropriate stepby-step procedures outlined in the corresponding sections of the chapter. Other problems will provide key points and the answers to the questions, but all answers can be arrived at using the appropriate steps. Section 7-1 Exercises 7.1. Step 1: The population mean of 125 is given Step 2: Compute the sample mean using Equation 7-3 x = (103 + 123 + 99 + 107 + 121 + 100 + 100 + 99)/8 = 852/8 = 106.5 Step 3: Compute the sampling error using Equation 7-1 Sampling Error = x - µ = 106.5 – 125 = -18.50 The sample of eight has a sampling error of -18.50. The sample has a smaller mean than the population as a whole. 7.3. The following steps can be used to determine the sampling error: Step 1: Determine the population mean. The population mean is computed as follows: x 273 µ=∑ = = 11.38 N 24 Step 2: Compute the sample mean. ∑ x = 61 = 10.17 x= n 6 Step 3: Compute the sampling error. Sampling error = x − µ = 10.17 − 11.38 = −1.21 7.5. a. Step 1: Compute the population mean using Equation 7-2. µ= 10+14+32+9+34+19+31+24+33+11+14+30+6+27+33+32+28+30+ 10+31+19+13+6+35)/24 = 531/24 = 22.125 Step 2: Compute the sample mean using Equation 7-3 x = (32+19+6+11+10+19+28+9+13+33)/10 = 180/10 = 18 Step 3: Compute the sampling error using Equation 7-1 Sampling Error = x - µ = 18 – 22.125 = -4.125 The sample of ten has a sampling error of -4.125. The sample...
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...three issues: 1. How to represent uncertain data 2. How to combine two or more pieces of uncertain data 3. How to draw inference using uncertain data We will introduce three ways of handling uncertainty: Probabilistic reasoning. Certainty factors Dempster-Shafer Theory 1. Classical Probability The oldest and best defined technique for managing uncertainty is based on classical probability theory. Let us start to review it by introducing some terms. Sample space: Consider an experiment whose outcome is not predictable with certainty in advance. However, although the outcome of the experiment will not be known in advance, let us suppose that the set of all possible outcomes is known. This set of all...
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...and producers. 1.2 Variability compromises quality Mass produced items are not identical. Some variation is inevitable and can cause problems. Too much variation might mean that parts which should fit together don’t. e.g. A screw might be too small/large to fit the corresponding bolt. There is a need to identify items which exhibit too much variation and deal with them, perhaps even scrap them. 1 1.3 Inspection versus Prevention How should we deal with variation? An early strategy was to inspect goods at the end of the production line and release, to the consumer, only those of a sufficiently high standard. It was argued that the marginal cost of each item was small, so the cost of the rejected items was small. However, this argument ignored a number of...
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...can form two different predicates. Let P(x) be “x is a student at UT”. Let Q(x, y) be “x is a student at y”. Definition: A predicate is a property that a variable or a finite collection of variables can have. A predicate becomes a proposition when specific values are assigned to the variables. P(x1, x2, ..., xn) is called a predicate of n variables or n arguments. Example: She lives in the city. P(x,y): x lives in y. P(Mary, Austin) is a proposition: Mary lives in Austin. Example: Predicates are often used in if statements and loop conditions. if(x > 100) then y:= x ∗ x predicate T(x): x > 100 2 Domains and Truth Sets Definition: The domain or universe or universe of discourse for a predicate variable is the set of values that may be assigned to the variable. Definition: If P(x) is a predicate and x has domain U, the truth set of P(x) is the set of all elements t of U such that P(t) is true, ie {t ∈ U |P (t) is true} Example: U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} P(x): “x is even”. The truth set is: {2, 4, 6, 8, 10} 3 The Universal Quantifier: ∀ Turn predicates into propositions by assigning values to all variables: Predicate P(x): “x is even” Proposition P(6): “6 is even” The other way to turn a predicate into a proposition: add a quantifier like “all” or “some” that indicates the number of values for which the predicate is true. Definition: The symbol ∀ is called the universal quantifier. The universal quantification of P(x) is the statement “P(x) for all values x in the universe”...
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...and puts on the same stock: i) Straddles; ii) Strips; iii) Straps; iv) Strangles COVERED CALL Consider a portfolio that consists of a long position (buy) in a stock plus a short position (sell) in a call option The investment strategy represented by this portfolio is known as writing a covered call This is because the long stock position "covers" or protects the investor from the possibility of a sharp rise in the stock price. lp on St and a sp on C z Pro…t ( ): ST S T < X : ST payo¤ }| max(ST St + C {z X; 0) cost }| { St + C When ST < X the pro…t is a positive function of ST (slope +1) When ST = 0: = St + C (maximum loss). From upper bound of a call this is negative: C St S T > X : ST (ST X) St + C = X St + C When ST > X the pro…t is not a¤ected by changes in ST (slope 0) When ST = X : = X St + C (maximum gain). From lower bound of a call this is positive: C St C+X Xe r(T t) =) C + Xe r(T t) St 0 Finally, the pro…t is zero when S T = St C = ST St 0) St + C = 0 ) An investor will invest in this strategy if he/she believes that ST St C See graph covered call Covered call: Lp on St and a sp on C But from the put call parity we have: C +Xe r(T t) = P +St =) St C = Xe r(T t) P That is why the pro…t of this strategy is similar to the one of selling a put The reverse of writing a covered call is a short position (sell) in a stock combined with a long position (buy) in a call option Pro…t ( ): ST < X : ST > X : z payo¤ }| max(ST St + C ) {z cost }| { (St...
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...là m =1. Vậy: P(A)= = b) Gọi B là biến cố “mặt có số chẵn chấm xuất hiện”. Số kết cục thuận lợi cho B là n = 3. Vậy: P(B) = = = 0.5 1.2 a) Gọi A là biến cố “lấy ra tấm bìa có xuất hiện chữ số 5”. khi đó là biến cố không xuất hiện chữ số 5. Vì số kết cục đồng khả năng là 100, trong khi số kết cục thuận lợi cho A là 19, nên số kết cục thuận lợi cho là 81. Vậy P ( ) = 0.81. b) từ 1 đến 100 có 50 số chẵn nên có 50 số chia hết cho 2. Có 20 số chia hết cho 5, trong đó 10 số vừa chia hết cho 5 vừa chia hết cho 2. Do vậy số kết cục thuận lợi cho biến cố lấy lên bìa có số hoặc chia hết cho 2, hoặc chia hết cho 5, hoặc chia hết cho cả 2 và 5 là 50 +20-10 = 60. Vậy P(A)= =0.6. 1.3 a) A = “quả cầu thứ nhất là trắng” Số kết cục duy nhất đồng khả năng là tất cả các phương pháp để lấy được 1 quả cầu ra khỏi (a+b) quả cầu. Vậy n = a+b. Số kết cục thuận lợi lấy ra quả cầu thứ nhất màu trắng là a. Vậy xác suất P(A) = b) Nếu quả thứ nhất trắng thì chọn quả thứ 2 sẽ còn a+b-1 kết cục đồng khả năng. Số kết cục thuận lợi để quả thứ 2 màu trắng là a-1 Vậy xác suất P(B) = c) tương tự câu b), vì quả thứ hai là trắng nên số kết cục đồng khả năng khi chọn quả thứ nhất là a+b-1 trong khi số kết quả thuận lợi là a-1. Vậy P(C) = 1.4. a) Số kết quả đồng khả năng thực ra hoán vị của a + b quả cầu nên m = (a+b)! nếu quả cầu thứ 2 là trắng thì số kết quả thuận lợi cho biến cố này là chỉnh hợp chập a+b-1 phần tử của a+b phần tử. Vậy xác suất P(A) = ,A)) . ...
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...Chapter 1 INTRODUCTION Objective of the Study The study is conducted, which aimed to identify the feasibility of the proposed business “Knights’ Fruit Malabon” at Poblacion, Mawab, Compostela Valley Province. Determining and identifying the procedures of the whole operation of the chosen business from its production, marketing and financing the said business. The objective of the study is to promote the product and create a harmonious relationship towards the customers and in the society. Moreover, to offer good quality products and services not only for the purpose of earning profit but also to meet the customer’s needs and wants. Methodology Having teamwork, hard work and patience are some of the effective keys in order to have a successful feasibility study. Those techniques are being applied by the researchers in order to have a good output of the said research. Different techniques and strategies are being used prior to the day of interview. These techniques such as questionnaires are being distributed to the prospect buyers in order to gather data like their preferences in terms of our offered products. And then all date gathered are analyze by the researchers in order to come up with a comprehensive feasibility study. Scope and Limitation of the Study One of the limitations of the study is that the business is limited on its target market capacity of purchasing. In terms of its production, the proposed business hence, the limitations of the tools equipment...
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...by the Rams, a loss or tie by the Saints and Bears, a win by the Seahawks and a tie between the Redskins and Cowboys. (Copied verbatim from the sports page 12/26/2004.) Propositions may be true or false and no preference is given one way or the other. This is sometimes difficult to grasp as we have a “natural” preference for true statements. But “snow is chartreuse” and “snow is white” are both propositions of equal standing though one is true and the other false. Non-propositions: What is today? Is today Monday? Questions are not propositions. You can’t judge whether the question itself is true or false, even though the answer to the question may be true or false. Show me some ID! Similarly, imperative statements lack a truth value. 2x=4 x=y Statements with undetermined variables do not have truth value xy=yx looks like a true proposition but is it multiplication or concatenation? If you don’t know, it’s not a proposition. This sentence is false (paradoxes are not propositions) Propositions may be abbreviated by letters: p,q,r, etc. Thus logic may sometimes be called symbolic logic because we use symbols like p or q to stand for propositions. E.g., let p be the...
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